P
r
e
f
The pur-pose of this handbook is to supply a collection of mathematical formulas and
tables which will prove to be valuable to students and research workers in the fields of
mathematics, physics, engineering and other sciences. TO accomplish this, tare has been
taken to include those formulas and tables which are most likely to be needed in practice
rather than highly specialized results which are rarely used. Every effort has been made
to present results concisely as well as precisely SOthat they may be referred to with a maximum of ease as well as confidence.
Topics covered range from elementary to advanced. Elementary topics include those
from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics
include those from differential equations, vector analysis, Fourier series, gamma and beta
functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions
and various other special functions of importance. This wide coverage of topics has been
adopted SOas to provide within a single volume most of the important mathematical results
needed by the student or research worker regardless of his particular field of interest or
level of attainment.
The book is divided into two main parts. Part 1 presents mathematical formulas
together with other material, such as definitions, theorems, graphs, diagrams, etc., essential
for proper understanding and application of the formulas. Included in this first part are
extensive tables of integrals and Laplace transforms which should be extremely useful to
the student and research worker. Part II presents numerical tables such as the values of
elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as
advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion,
especially to the beginner in mathematics, the numerical tables for each function are separated, Thus, for example, the sine and cosine functions for angles in degrees and minutes
are given in separate tables rather than in one table SOthat there is no need to be concerned
about the possibility of errer due to looking in the wrong column or row.
1 wish to thank the various authors and publishers who gave me permission to adapt
data from their books for use in several tables of this handbook. Appropriate references
to such sources are given next to the corresponding tables. In particular 1 am indebted to
the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates, F.R.S.,
and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their
book S
T
tf
B
a
Aao
i
b a gtMy
o R l n ir e
l
e e d si d
1 also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin
for their excellent editorial cooperation.
M. R. SPIEGEL
Rensselaer Polytechnic Institute
September, 1968
o
s s
tc i
CONTENTS
Page
1.
Special
Constants..
.............................................................
1
2. Special Products and Factors ....................................................
2
3. The Binomial Formula and Binomial Coefficients .................................
3
4. Geometric Formulas ............................................................
5
5. Trigonometric Functions ........................................................
11
6. Complex Numbers ...............................................................
21
7. Exponential and Logarithmic Functions .........................................
23
8. Hyperbolic Functions ...........................................................
26
9. Solutions of Algebraic Equations ................................................
32
10. Formulas from Plane Analytic Geometry ........................................
...................................................
34
40
11.
Special Plane Curves........~
12.
Formulas from Solid Analytic Geometry ........................................
46
13.
Derivatives .....................................................................
53
14.
Indefinite Integrals ..............................................................
57
15.
Definite Integrals ................................................................
94
16.
The Gamma
Function .........................................................
..10 1
17.
The Beta Function ............................................................
18.
Basic Differential Equations and Solutions .....................................
19.
Series of Constants..............................................................lO
20.
Taylor Series...................................................................ll
21.
Bernoulliand
22.
Formulas from Vector Analysis..
23.
Fourier Series ................................................................
..~3 1
24.
Bessel Functions..
..13 6
2s.
Legendre Functions.............................................................l4
26.
Associated Legendre Functions .................................................
.149
27.
28.
Hermite Polynomials............................................................l5
Laguerre Polynomials ..........................................................
1
.153
29.
Associated Laguerre Polynomials ................................................
30.
Chebyshev Polynomials..........................................................l5
Euler Numbers .................................................
.............................................
............................................................
..lO 3
.104
7
0
..114
..116
6
KG
7
Part
I
FORMULAS
THE
GREEK
Greek
name
G&W
ALPHABET
Greek
name
Greek
Lower case
tter
Capital
Alpha
A
Nu
N
Beta
B
Xi
sz
Gamma
l?
Omicron
0
Delta
A
Pi
IT
Epsilon
E
Rho
P
Zeta
Z
Sigma
2
Eta
H
Tau
T
Theta
(3
Upsilon
k
Iota
1
Phi
@
Kappa
K
Chi
X
Lambda
A
Psi
*
MU
M
Omega
n
1.1
1.2
= natural
base of logarithms
1.3
fi
=
1.41421
35623 73095 04889..
1.4
fi
=
1.73205
08075 68877 2935.
1.5
fi
=
2.23606
79774
1.6
h
=
1.25992
1050..
.
1.7
&
=
1.44224
9570..
.
1.8
fi
=
1.14869
8355..
.
1.9
b
=
1.24573
0940..
.
1.10
eT = 23.14069
26327 79269 006..
.
1.11
re = 22.45915
77183 61045 47342
715..
1.12
ee =
22414
.
1.13
logI,, 2
=
0.30102
99956 63981 19521
37389.
..
1.14
logI,, 3
=
0.47712
12547
19662 43729
50279..
.
1.15
logIO e =
0.43429
44819
03251 82765..
1.16
logul ?r =
0.49714
98726
94133 85435 12683.
1.17
loge 10
In 10
1.18
loge 2 =
ln 2
=
0.69314
71805
59945 30941
1.19
loge 3 =
ln 3 =
1.09861
22886
68109
1.20
y =
1.21
ey =
1.22
fi
=
1.23
6
=
15.15426
=
0.57721
56649
1.78107
r(&)
=
79264
2.30258
190..
12707
6512.
9852..
00128 1468..
1.77245
2.67893
85347 07748..
.
1.25
r(i)
3.62560
99082 21908..
.
1-26
1 radian
1.27
1”
=
~/180
radians
.
=
=
..
.
57.29577
0.01745
..
7232.
.
69139 5245..
.. = Eukr's co%stu~t
[see 1.201
.
38509 05516
II’(&) =
180°/7r
.
02729
~ZLYLC~~OTZ
[sec pages
1.24
=
.
50929 94045 68401 7991..
01532 86060
F is the gummu
=
.
99789 6964..
24179 90197
1.64872
where
=
..
8167..
.O
95130 8232..
32925
.
101-102).
19943 29576 92.
1
..
radians
THE
4
BINOMIAL
FORMULA
PROPERTIES
OF
AND
BINOMIAL
BINOMIAL
COElFI?ICIFJNTS
COEFFiClEblTS
3.6
This
leads
to Paseal’s
[sec page 2361.
triangk
3.7
(1)
+
(y)
+
(;)
+
...
3.8
(1)
-
(y)
+
(;)
-
..+-w(;)
3.10
(;)
+
(;)
+
(7)
+
.*.
=
2n-1
3.11
(y)
+
(;)
+
(i)
+
..*
=
2n-1
+
(1)
=
27l
=
0
3.9
3.12
3.13
-d
3.14
MUlTlNOMlAk
3.16
(zI+%~+...+zp)~
where
q+n2+
the
mm,
...
denoted
+np =
72..
by
2,
=
FORfvlUlA
~~~!~~~~~..~~!~~1~~2...~~~
is taken over
a11 nonnegative
integers
% %,
. . , np fox- whkh
1
4
GEUMElRlC
FORMULAS
&
RECTANGLE
4.1
Area
4.2
Perimeter
OF LENGTH
b AND
WIDTH
a
= ab
= 2a + 2b
b
Fig. 4-1
PARAllELOGRAM
4.3
Area
=
4.4
Perimeter
bh =
OF ALTITUDE
h AND
BASE b
ab sin e
= 2a + 2b
1
Fig. 4-2
‘fRlAMf3i.E
Area
4.5
=
+bh
OF ALTITUDE
h AND
BASE b
= +ab sine
*
ZZZI/S(S - a)(s - b)(s - c)
where s = &(a + b + c) = semiperimeter
b
Perimeter
4.6
n_
L,“Z
.,
.,,
= u+ b+ c
Fig. 4-3
:
‘fRAPB%XD
4.7
Area
4.8
Perimeter
C?F At.TlTUDE
fz AND
PARAl.lEL
SlDES u AND
b
= 3h(a + b)
=
=
/c-
a + b + h
Y&+2
sin 4
C
a + b + h(csc e + csc $)
1
Fig. 4-4
5
/
-
GEOMETRIC
6
REGUkAR
4.9
Area
= $nb?- cet c
4.10
Perimeter
=
POLYGON
inbz-
FORMULAS
OF n SIDES EACH CJf 1ENGTH
b
COS(AL)
sin (~4%)
= nb
7,’
0.’
0
Fig. 4-5
CIRÇLE OF RADIUS
4.11
Area
4.12
Perimeter
r
= &
=
277r
Fig. 4-6
SEClOR
4.13
4.14
Area
=
&r%
OF CIRCLE OF RAD+US Y
[e in radians]
T
Arc length s = ~6
A
8
0
T
Fig. 4-7
RADIUS
4.15
OF C1RCJ.E INSCRWED
r=
where
&$.s-
tN A TRtANGlE
*
OF SIDES a,b,c
U)(S Y b)(s -.q)
s
s = +(u + b + c) = semiperimeter
Fig. 4-6
RADIUS- OF CtRClE
4.16
R=
where
CIRCUMSCRIBING
A TRIANGLE
OF SIDES a,b,c
abc
4ds(s - a)@ -
b)(s - c)
e = -&(a.+ b + c) = semiperimeter
Fig. 4-9
G
4
A
=.
4
P
.
&
sr s
=
2e
s
1=
n
+
1
=
FE
3
ise n
7
r
n
OO
6
ni a
2 nr s i y 8
2r
RM
0
n
n ri i n
M7E
UT
°
2
r mn z
e
t
e
!
?
Fig. 4-10
4
A
=.
4
P
.
= 1 n r t a eL T
n
t rZ
n
n
=
2e
2
t
9 r 2 a n a!
0
2 nr t a
=
2
n
n ri a n
T
!
I
:
e?
r m nk
T
t
e
0
F
SRdMMHW W
4
o .s
A
f=2 h +
pr
( -ae s
C%Ct&
e) 1 a r
e
OF RADWS
ra i
d2
4
i
-
g
1
T
tn
e
T
e
d
r
tz!?
Fig. 4-12
4
A
=.
4
P
.
r
r
2
a
e
2
2 4 1 - kz rs
e c3
b
a
7r/2
=
e 5
4a
ii
m
+
l
e
@
t
e
0
=
w
k = ~/=/a.h
4
A
4
A
l
[
27r@sTq
See
p
e254 f
=.
$ab
r
2
.
ABC
r = e -&2dw
a
n a
e
r
to
4
c +n E5
p
u g
e
ar
p
m e
b F
r
4e
l
i
-r
o
e g
a
4
gl
1
a
)
tn
+
h
AOC
@
T
b
Fig. 4-14
- f
1i
GEOMETRIC
8
RECTANGULAR
4.26
Volume
=
4.27
Surface
area
PARALLELEPIPED
FORMULAS
OF
LENGTH
u, HEIGHT
r?, WIDTH
c
ubc
Z(ab + CLC
+ bc)
=
a
Fig. 4-15
PARALLELEPIPED
4.28
Volume
=
Ah
=
OF CROSS-SECTIONAL
AREA
A AND
HEIGHT
h
abcsine
Fig. 4-16
SPHERE
4.29
Volume
=
OF RADIUS
,r
+
1
---x
,-------
4.30
Surface
area
=
4wz
@
Fig. 4-17
RIGHT
4.31
Volume
4.32
Lateral
=
CIRCULAR
CYLINDER
OF RADIUS
T AND
HEIGHT
h
77&2
surface
area
=
h
25dz
Fig. 4-18
CIRCULAR
4.33
Volume
4.34
Lateral
=
m2h
surface
area
CYLINDER
=
OF RADIUS
r AND
SLANT
HEIGHT
2
~41 sine
=
2777-1 =
2wh
z
=
2wh csc e
Fig. 4-19
.
GEOMETRIC
CYLINDER
=
OF CROSS-SECTIONAL
4.35
Volume
4.36
Lateral surface area
Ah
FORMULAS
9
A AND
AREA
SLANT
HEIGHT
I
Alsine
=
=
pZ =
GPh
--
ph csc t
Note that formulas 4.31 to 4.34 are special cases.
Fig. 4-20
RIGHT
=
CIRCULAR
4.37
Volume
4.38
Lateral surface area
CONE
OF RADIUS
,r AND
HEIGHT
h
jîw2/z
=
77rd77-D
=
~-7-1
Fig. 4-21
PYRAMID
4.39
Volume
=
OF
BASE
AREA
A AND
HEIGHT
h
+Ah
Fig. 4-22
SPHERICAL
4.40
Volume (shaded in figure)
4.41
Surface area
=
CAP
=
OF RADIUS
,r AND
HEIGHT
h
&rIt2(3v - h)
2wh
Fig. 4-23
FRUSTRUM
=
OF RIGHT
4.42
Volume
4.43
Lateral surface area
+h(d
CIRCULAR
CONE
OF RADII
u,h
AND
HEIGHT
h
+ ab + b2)
=
T(U + b) dF
=
n(a+b)l
+ (b - CL)~
Fig. 4-24
10
SPHEMCAt hiiWW
4.44
Area of triangle ABC
=
GEOMETRIC
FORMULAS
OF ANG%ES
A,&C
Ubl SPHERE OF RADIUS
(A + B + C - z-)+
Fig. 4-25
TOW$
&F
lNN8R
4.45
Volume
4.46
w
Surface area = 7r2(b2- u2)
4.47
Volume
=
RADlU5 a
AND
OUTER RADIUS
b
&z-~(u+ b)(b - u)~
= $abc
Fig. 4-27
T.
4.4a
Volume
=
PARAWlO~D
aF REVOllJTlON
&bza
Fig. 4-28
Y
5
TRtGOhiOAMTRiC
D
OE T
FF R
WNCTIONS
F
l I FU
A R N G T
ON
Triangle ABC bas a right angle (9Oo) at C and sides of length u, b, c.
angle A are defined as follows.
sintz
. of A
5
5
5
5
5
sin A
1=
:
=
opposite
hypotenuse
i
=
adjacent
hypotenuse
cosine
. of
A
=
~OSA
2=
. of
A
=
tanA
3= f = -~
. of
A
=
of A
tangent
c
5.5
=
secant
cosecant
. of
A
4=
k
=
adjacent
t
opposite
=
sec A
=
t
=
-~
=
csc A
6=
z
=
hypotenuse
opposite
E
l O R
RC
functions
G T
N I T
of
B
opposite
adjacent
A
o cet
The trigonometric
I
TX A
c
z
A
n
g
hypotenuse
adjacent
W OT
Fig. 5-1
N M
3 HG
E
G A
TE I R
N9L Y
H C E
S0 E
A H A
I ’
Consider an rg coordinate system [see Fig. 5-2 and 5-3 belowl. A point P in the ry plane has coordinates
(%,y) where x is eonsidered as positive along OX and negative along OX’ while y is positive along OY and
negative along OY’. The distance from origin 0 to point P is positive and denoted by r = dm.
If it is described dockhse
from
The angle A described cozmtwcZockwLse
from OX is considered pos&ve.
OX it is considered negathe.
We cal1 X’OX and Y’OY the x and y axis respectively.
The various quadrants are denoted by 1, II, III and IV called the first, second, third and fourth quadrants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A
is in the third quadrant.
Y
Y
II
1
II
1
III
IV
III
IV
Y’
Y’
Fig. 5-3
Fig. 5-2
11
f
TRIGONOMETRIC
12
FUNCTIONS
For an angle A in any quadrant the trigonometric
functions of A are defined as follows.
5.7
sin A
=
ylr
5.8
COSA
=
xl?.
5.9
tan A
=
ylx
5.10
cet A
=
xly
5.11
sec A
=
v-lx
5.12
csc A
=
riy
RELAT!ONSHiP BETWEEN DEGREES AN0
RAnIANS
N
A radian is that angle e subtended at tenter 0 of a eircle by an arc
MN equal to the radius r.
Since 2~ radians = 360° we have
5.13
1 radian
= 180°/~
5.14
10 = ~/180 radians
=
1
r
e
0
57.29577 95130 8232. . . o
r
B
= 0.01745 32925 19943 29576 92.. .radians
Fig. 5-4
REkATlONSHlPS
5.15
tanA
= 5
5.16
&A
~II ~ 1
5.17
sec A
=
~
5.18
cscA
=
-
tan A
AMONG
COSA
sin A
zz -
1
COS
A
TRtGONOMETRK
5.19
sine A +
~OS~A
5.20
sec2A
-
tane
5.21
csceA
- cots A
II
III
IV
1
A = 1
=
1
1
sin A
SIaNS AND VARIATIONS
1
=
FUNCTItB4S
+
0 to 1
+
1 to 0
+
1 to 0
0 to -1
0 to -1
-1 to 0
OF TRl@ONOMETRK
+
0 to m
-mtoo
+
0 to d
-1 to 0
+
0 to 1
+
CCto 0
oto-m
+
Ccto 0
-
--
too
oto-m
FUNCTIONS
+
1 to uz
+
m to 1
-cc to -1
+
1 to ca
-1to-m
+
uz to 1
--COto-1
-1 to --
M
TRIGONOMETRIC
E
Angle A
in degrees
00
X
F
Angle A
in radians
A
T
A
O
RL
FC
R
1
IU
O
UT
O
S
sec A
csc A
0
1
0
w
1
cc
ii/6
1
+ti
450
zl4
J-fi
$fi
60°
VI3
Jti
750
5~112
900
z.12
105O
7~112
*(fi+&)
-&(&-Y%
-(2+fi)
-(2-&)
120°
2~13
*fi
-*
-fi
-$fi
1350
3714
+fi
-*fi
150°
5~16
4
-+ti
#-fi)
2-fi
&(&+fi)
fi
1
0
fi)
-&(G+
0
-*fi
-fi
-(2-fi)
-(2+fi)
180°
?r
-1
1950
13~112
210°
7716
225O
5z-14
-Jfi
240°
4%J3
-#
255O
17~112
270°
3712
-1
285O
19?rll2
-&(&+fi)
3000
5ïrl3
-*fi
2
315O
7?rl4
-4fi
*fi
-1
330°
117rl6
*fi
-+ti
345O
237112
360°
2r
-$(fi-fi)
-*(&+fi)
2-fi
-
1
4
-*fi
-i(fi-
2+fi
0
-(2+6)
&(&+
-ti
fi)
1
0
see pages
206-211
-(2
- fi)
0
++
-fi
\h
-+fi
2
-(fi-fi)
f
-(&-fi)
-2
-(&+?cz)
-@-fi)
&+fi
-(2+6)
T-J
i
-36
-(fi-fi)
-1
-(fi-fi)
2
-1
f
-fi
Tm
-*fi
-ti
-2
g
-fi
0
*ca
-(&+fi)
i -
&fi
2-6
Vz+V-c?
-1
3
1
km
*(&-fi)
6)
angles
ti
-&(&-fi)
1
l
1
-4
-&&+&Q
6
fi-fi
-2
2 + ti
&
1
-(&+fi)
Tm
0
fi-fi
km
-1
-1
TG
;G
&+fi
0
N
fi
2
2-&
*CU
fi)
fi
.+fi
2+&
R
2
$fi
1
C
N
3
&+fi
fi-fi
fi
1
@-fi)
$(fi-
2+*
*fi
r1
i(fi+m
other
A
cet A
300
involving
FN
A
tan A
rIIl2
tables
GE
COSA
0
llrll2
V
sin A
15O
165O
For
V
FUNCTIONS
fi
$fi
fi-fi
-$fi
-fi
-2
-(&+fi)
1
?m
and 212-215.
f
I
TRIGONOMETRIC
5.89
y = cet-1%
5.90
y
=
FUNCTIONS
19
sec-l%
5.91
_--/
y
=
csc-lx
Y
I
T
---
,
/A--
/’
/
-77
-//
,
Fig. 5-14
Fig. 5-15
RElAilONSHfPS
BETWEEN
The following results hold for
sides a, b, c and angles A, B, C.
5.92
ANGtGS
any plane triangle
ABC
OY A PkAtM
with
TRlAF4GlG
’
A
Law of Sines
a
-=Y=sin A
5.93
SIDES AND
Fig. 5-16
1
b
c
sin B
sin C
C
Law of Cosines
/A
cs = a2 +
bz -
Zab
COS
f
C
with similar relations involving the other sides and angles.
5.94
Law of Tangents
tan $(A + B)
a+b
-a-b
= tan i(A -B)
with similar relations involving the other sides and angles.
5.95
sinA
where
s = &a + b + c)
=
:ds(s
is the semiperimeter
- a)(s - b)(s - c)
of the triangle.
B and C cari be obtained.
See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
Spherieal triangle ABC is on the surface of
in Fig. 5-18. Sides a, b, c [which are arcs of
measured by their angles subtended at tenter 0 of
are the angles opposite sides a, b, c respectively.
results hold.
5.96
Law of Sines
sin a
-z-x_
sin A
5.97
sin b
sin B
a sphere as shown
great circles] are
the sphere. A, B, C
Then the following
sin c
sin C
Law of Cosines
sinbsinccosA
cosa
=
cosbcosc
COSA
=
- COSB COSC +
+
Fig. 5-1’7
sinB sinccosa
with similar results involving other sides and angles.
Similar relations involving angles
2
T
0
L
5
o.
w
T
a
s
5
f
9
ri
s = &
S = +
f
e
E
g
i
c
S(
8 n
&
+ B
a
t(
t
&
a=
t(
4l
op
f
r x
F s ii
1e
o mn s
e
I
g
U
$ ) + n
b
)
sh
a t ai v i
r)
G
e
aA
(
N
)
n
a
A
n
O
n
C
T
t
u
n
u n h nl o d
N
s
i
l d e a
g l e
t
r
O
(
rl v s
s
e i
u
i hr
f e. o
.s
r)
A i rh
ra
0
FGR
RtGHT
o C it c na
i b -va b i
m
+ ose
o sa t a i l i
r u
n h n l dd
ld e g a e
t
r
lr s
0
( Se
RlJlES
a
wn
- B
0
h+ B + C
NAPIER’S
a
t
1
w
a
et
F
9
1+
h
.
S
w
9
w
5
a
i i
.
R
4o
f e. o
m
g
.
meos
4
eu
ANGLED
rf , gh
p e gor s i
c gB e le, , u , . .
o sa t a i l i
,
l
SPHERICAL
ha
e p t lef
9n A l
pv
Atr
r un h n l dd
ld e
ga
e
t
r
lr
a
TRIANGLES
t rwe
, d
he i
Z
aet
ih
ei f t r3s o a i
n
r
h rC r nc
a
C
F
S
[
c
A
a
t
i
o
p
5
uq
ot
i
h
hn
o t n p n
da
t
a
t
-
g
a pu a
fi hce
ri a m s ia
ea d a e c to a om
c r
of th y ac e
rj n
h
w
r p
T
s.
o
a
h m i1
fp
5.102
T
s
o
a
h m i
fp n ee i n
S
T
x
C
c
= 9i
o
ch
a-
n ee i n0 t
O C
s
a
s
(
ba
=
t
n0= m
9A
t
=
o f
.
oe
F
9
ri p aar c
s a en io nr Fi
a
n l
p A a
npi B
e
m oc ayd
5
E
1
5
5e s n
w
s t a ir
ndoc
g
. c
p rt
ti
tsl
a ps r p ea Te
an
cs
laN
u d f oeh t oa a a l
a p y q d eo ht r rc
u d fo eht t ooo
O w c° h 0p,
B
A
-
e e a °l
n s
,
(
a
na
C
o
C
i
C
a C
(
nO
Oo O C ~
uts
rl i rt 5 rhe o es p
a er fb
g
e 2p t
hd
ead
wl
xi hr
pv le
r l aao s
f p eh
B
dsp
et Cg.l h
eoc
tn eu t
a
-ei
e p
sr
.
0r t
ri O
ee e
ni n s
rl
da
sl
i n
ae ug j s
l e
ae
ve
r
uip s
r
frg
di ie
ce a
n
co
e:
i
AS =-r SC OaOs
2
ee
et
f eph dn d l e
a l
n = rOt
b it
-
w - a ehi ngc t
dta
l
m
p
t
a p y q d eo1 ht r rt
--
i
O
-a
B A
oa 1. s e n os a
n
O -
mi 99 e
Bn
S
i
)
ug
a
)b
SB n
n .7
l e
e
A complex number
is generally written as a + bi where a and b are real numbers and i, called the
imaginaru unit, has the property that is = -1. The real numbers a and b are called the real and ima&am
parts of a + bi respectively.
The complex numbers a + bi and a - bi are called complex
6.1
a+bi
=
c+di
if and only if
conjugates
a=c
and b=cZ
6.2
(a + bi) + (c + o!i) =
(a + c) + (b + d)i
6.3
(a + bi) - (c + di) =
(a - c) + (b - d)i
6.4
(a+ bi)(c+
di) =
(ac- bd) + (ad+
of each other.
bc)i
Note that the above operations are obtained by using the ordinary rules of algebra and replacing 9 by
-1 wherever it occurs.
21
22
COMPLEX
GRAPH
NUMBERS
OF A COMPLEX
NtJtWtER
A complex
number a + bi cari be plotted as a point (a, b) on an
xy plane called an Argand
diagram
or Gaussian
plane.
For example
in Fig. 6-1 P represents
the complex
number -3 + 4i.
A
eomplex
number
cari
also
be
interpreted
as
a
wector
p,----.
y
from
0 to P.
-
0
X
*
Fig. 6-1
POLAR
FORM
OF A COMPt.EX
NUMRER
In Fig. 6-2 point P with coordinates
(x, y) represents
the complex
number x + iy. Point P cari also be represented
by polar coordinates
(r, e). Since
x = r COS6, y = r sine
we have
6.6
x + iy = ~(COS
0+
called
the poZar form
the mocklus
of the complex
and t the amplitude
i sin 0)
number.
L
We often
-
X
cal1 r = dm
of x + iy.
Fig. 6-2
tWJLltFltCATt43N
[rl(cos
6.7
AND
DtVlStON
OF CWAPMX
el + i sin ei)] [re(cos ez + i sin es)]
V-~(COSe1 + i sin el)
6.8
ZZZ 2
rs(cos ee + i sin ez)
If p is any real
number,
De Moivre’s
[r(cos
rrrs[cos
1bJ POLAR
ilj
0”
FtMM
tel + e2) + i sin tel + e2)]
[COS(el - e._J + i sin (el - .9&]
DE f#OtVRtt’S
6.9
=
NUMBRRS
THEORRM
theorem
states
e + i sin e)]p
=
that
rp(cos pe + i sin pe)
.
RCWTS
If
p = l/n
where
k=O,l,2
integer,
[r(cos e + i sin e)]l’n
6.10
where
n is any positive
OF CfMMWtX
k is any
,...,
integer.
n-l.
From
this
the
=
n nth
NUtMB#RS
6.9 cari be written
rl’n
roots
L
e + 2k,,
~OSn
of
a complex
+
e + 2kH
i sin ~
number
n
cari
1
be
obtained
by
putting
”
In the following p, q are real numbers, CL,t are positive numbers and WL,~are positive integers.
7.1
cp*aq z aP+q
7.2
aP/aqE @-Q
7.3
(&y E rp4
7.4
u”=l,
7.5
a-p = l/ap
7.6
(ab)p = &‘bp
7.7
&
7.8
G
7.9
Gb
a#0
z aIIn
= pin
=%Iî/%
In ap, p is called the exponent, a is the base and ao is called the pth power of a. The function
is called an exponentd function.
If a~ = N where a # 0 or 1, then p = loga N is called the loga&hm
N = ap is called t,he antdogatithm of p to the base a, written arkilogap.
Example:
Since
The fumAion
3s = 9 we have
y = ax
of N to the base a. The number
log3 9 = 2, antilog3 2 = 9.
v = loga x is called a logarithmic
jwzction.
7.10
logaMN
=
loga M + loga N
7.11
log,z ;
=
logG M -
7.12
loga Mp
=
p lO& M
loga N
Common logarithms and antilogarithms [also called Z?rigg.sian]
are those in which the base a = 10.
The common logarit,hm of N is denoted by logl,, N or briefly log N. For tables of common logarithms and
antilogarithms, see pages 202-205. For illuskations using these tables see pages 194-196.
23
EXPONENTIAL
24
AND LOGARITHMIC
NATURAL LOGARITHMS
FUNCTIONS
AND ANTILOGARITHMS
Natural logarithms and antilogarithms [also called Napierian] are those in which the base a = e =
2.71828 18. . . [sec page 11. The natural logarithm of N is denoted by loge N or In N. For tables of natural
logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving ex for values of z]
see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF lO@ARlTHMS
The relationship between logarithms of a number N to different bases a and b is given by
7.13
loga N
=
hb
iv
hb
a
-
In particular,
= ln N
7.14
loge N
7.15
logIO
N = logN
RElATlONSHlP
= 2.30258 50929 94.. . logio N
=
0.43429
44819 03.. . h& N
BETWEEN EXPONBNTIAL ANO TRl@ONOMETRlC
eie =
7.16
These are called Euler’s
COS
0 + i sin 8,
dent&es.
e-iO
=
COS 13 -
sin 6
Here i is the imaginary unit [see page 211.
7.17
sine
7.18
case =
=
eie- e-ie
2i
eie+ e-ie
2
7.19
7.20
2
7.21
sec 0
=
&O + e-ie
7.22
csc 6
=
eie
7.23
i
2i
eiCO+2k~l
From this it is seen that @ has period 2G.
-
e-if3
=
eie
k =
integer
FUNCT#ONS
;;
E
POiAR
T
p
XA
FORfvl OF COMPLEX
f
7
o h a co
o n
.
2
6
t
o
6
NUMBERS
.o hp r 2
(reiO)l/n E
LOGARITHM
7.29
COMPLEX
a.
l
OD
ym
i a tm e
(
ffUMBERS
e7n ra m 2 t 1r t
(q-eio)Pzz q-P&mJ [
7.2B
OF
GU
EXPRESSE$3 AS AN
oxl + i r c u b w m
a
WITH
7.27
PN
or rpe
N
AN 25
E
RC
N
EXPONENTNAL
n re b
[if lx 6
pi r e 2 st a ep .
a mr 2 et s x o6
g
4 6 + i sin 0) = 9-ei0
x + iy = ~(COS
OPERATIONS
F
fe
L
[~&O+Zk~~]l/n
q f og
M
t
=
n
u
D
FORM
o 0eh uo ue
o
h
l
e
e
i
il
g
a
e
vl
v
h
s
o
NUMBER
k e=e i k
@n z
) t -
ao
r
rl/neiCO+Zkr)/n
OF A COMPLEX
= l r n
+ iT + 2
IN POLAR
e i
DEIWWOPI
OF HYPRRWLK
8.1
Hyperbolic
sine of x
=
sinh x
=
8.2
Hyperbolic
cosine
=
coshx
=
8.3
Hyperbolic
tangent
= tanhx
=
8.4
Hyperbolic
cotangent
8.5
Hyperbolic
secant
8.6
Hyperbolic
cosecant
RELATWNSHIPS
of x
of x
coth x
of x =
of x
AMONG
ez + e-=
2
~~~~~~
2
ez + eëz
HYPERROLIC FUWTIONS
=
sinh x
a
coth z
=
1
tanh x
sech x
=
1
cash x
8.10
cschx
=
1
sinh x
8.11
coshsx - sinhzx
=
1
8.12
sechzx + tanhzx
=
1
8.13
cothzx - cschzx
=
1
FUNCTIONS
2
= csch x = &
tanhx
8.7
# - e-z
ex + eCz
= es _ e_~
= sech x =
of x
.:‘.C,
FUNCTIONS
cash x
sinh x
=
OF NRGA’fWE
ARGUMENTS
8.14
sinh (-x)
=
- sinh x
8.15
cash (-x)
= cash x
8.16
tanh (-x)
= - tanhx
8.17
csch (-x)
=
-cschx
8.18
sech(-x)
=
8.19
coth (-x)
=
26
sechx
-~OUIS
HYPERBOLIC
AWMWM
FUNCTIONS
27
FORMWAS
0.2Q
sinh (x * y)
=
sinh x coshg
8.21
cash (x 2 g)
=
cash z cash y * sinh x sinh y
8.22
tanh(x*v)
=
tanhx f tanhg
12 tanhx tanhg
8.23
coth (x * y)
=
coth z coth y 2 1
coth y * coth x
8.24
sinh 2x
=
2 ainh x cash x
8.25
cash 2x
=
coshz x + sinht x
8.26
tanh2x
=
2 tanh x
1 + tanh2 x
=
* cash x sinh y
2 cosh2 x -
1
=
1 + 2 sinh2 z
HAkF ABJGLR FORMULAS
8.27
sinht
=
8.28
CoshE
2
=
8.29
tanh;
=
k
Z
sinh x
cash x + 1
.4
[+ if x > 0, - if x < O]
cash x + 1
-~
2
cash x - 1
cash x + 1
’ MUlTWlE
[+ if x > 0, - if x < 0]
ZZ cash x - 1
sinh x
A!Wlfi WRMULAS
8.30
sinh 3x
=
3 sinh x + 4 sinh3 x
8.31
cosh3x
=
4 cosh3 x -
8.32
tanh3x
=
3 tanh x + tanh3 x
1 + 3 tanhzx
8.33
sinh 4x
=
8 sinh3 x cash x + 4 sinh x cash x
8.34
cash 4x
=
8 coshd x -
8.35
tanh4x
=
4 tanh x + 4 tanh3 x
1 + 6 tanh2 x + tanh4 x
3 cash x
8 cosh2 x -t- 1
2
H
8
YF
P
O
PU
HO
E N
FY& W
P
J
R C
E
E
B T
f
R
R
8
.
3
s
6=
&i c
2
-
4 na
8
.
3
c
7=
4 oc
2
+
$ sa
8
.
3
s
x
8=
&i s
3
-
8
.
3
c
x
9=
&o c
+
8
.
4
s
0=
8i -
4 c
2
n+
4 ca
4x
h
as
% 4
sh
x
h
8
.
4
c
1=
#o +
+ c
2
s+
& ca
4x
h
as
x 4
sh
x
h
S
D
8
U
.
AI
F
A
hs
zh
x
x
hs
zh
x
2 sn i
xx
ihn
nsh
2 cs o
x
ahs
ssh
K
NFO
x
W
R
&
DFF F O
P
Sl
h
h3
E
x
UR
D
R
s
4+
s
i
=
2 si2
&
n
+ y
cn
i
$ hx - y)
anh
(x
)
s hy
x
h
x
h
kR
U
8
.
4s
-
s
3i
=
2 ci
n&
+ y
s an
$ hx - Y)
i sh
(x
)
n hy
8
.
4c
+
c
4o
=
2 co
is
+ y
c as
#(h
- Y)
a sh
xxx
)
s hy
8
.
4c
-
c
5o
=
2 so
$s
+ y
s is
$ (h - Y)
i nh
( xx
)
n hy
8
.
4s
x s
y 6i=
* i
n
{- n c
h
c ho
o
s
s
h
h
(
8
.
4c
x c
y 7 a=
+ a
s
{+ s c
h
c ho
o
s
s
h
h
(
s
x 4c
y
i=
+ a
n+ y
{- s s
x @ h- ) Y sl h i
) -i
n
} n
h
h
8
.
E
I
t
t
OX H
f
n
hw
.o
8
s
FP FY
x e>e 0 ls I
oa
1
x = u
i c
8(
= u
!R UPT
x < 0 u. l s t f
a
9
.
n o t
t
s
x
i
n
h
c
x
a
s
h
t
x
a
n
h
c
x
o
t
h
s
x
e
c
h
c
x
s
c
h
= uh s a c
s ou h
s p
O
N
‘ E NEE
a e i wme
x = 1h n o s
i p b s fn i e
x =1 xu h t e c
h
x
h
F OSC
RR
g r 8y
o dn
x = xwh c s
T SB
n o .
rig
h c
HYPERBOLIC
GRAPHS
8.49
y = sinh x
OF HYPERBOkfC
8.50
29
FUNCltONS
8.51
y = coshx
Fig. S-l
8.52
FUNCTIONS
Fig. 8-2
y = coth x
8.53
/i
y
y = tanh x
Fig. 8-3
8.54
y = sech x
y = csch x
Y
\
X
1
7
10
X
0
-1
iNVERSE
HYPERROLIC
L
X
Fig. 8-6
Fig. 8-5
Fig. 8-4
0
FUNCTIONS
If x = sinh g, then y = sinh-1 x is called the inverse hyperbolic sine of x. Similarly we define the
The inverse hyperbolic functions are multiple-valued and. as in the
other inverse hyperbolic functions.
case of inverse trigonometric functions [sec page 171 we restrict ourselves to principal values for which
they ean be considered as single-valued.
The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic
functions expressed in terms of logarithmic functions which are taken as real valued.
8.55
sinh-1 x
=
ln (x + m
8.56
cash-lx
=
ln(x+&Z-ï)
8.57
tanh-ix
=
8.58
coth-ix
=
8.59
sech-1 x
8.60
csch-1 x
)
-m<x<m
XZl
-l<x<l
+ln
X+l
( x-l
)
x>l
O<xZl
=
[cash-r x > 0 is principal value]
ln(i+$$G.)
x+O
or xc-1
[sech-1 x > 0 is principal value]
HYPERBOLIC
30
FUNCTIONS
8.61
eseh-]
x
=
sinh-1
(l/x)
8.62
seeh-
x
=
coshkl
(l/x)
8.63
coth-lx
=
tanh-l(l/x)
8.64
sinhk1
(-x)
=
- sinh-l
x
8.65
tanhk1
(-x)
=
- tanh-1
x
8.66
coth-1
(-x)
=
- coth-1
x
8.67
eseh-
(-x)
=
- eseh-
x
GffAPHS
8.68
y =
OF fNVt!iffSft HYPfkfftfUfX
8.69
sinh-lx
FfJNCTfGNS
X
7
-ll
8.72
coth-lx
Y
0
\
\
\
\
‘-.
Fig. 8-9
L
11
x
y =
8.73
sech-lx
y =
Y
Il
0
I
I’
Fig. 8-11
csch-lx
Y
I
Fig. 8-10
\
Fig. 8-8
Fig. 8-7
l
l
l
x
-1
\
y =
tanhkl
l
Y
Y
8.71
y =
8.70
y = cash-lx
,
,
/
X
3
L
0
Fig. 8-12
-x
HYPERBOLIC
tan (ix) == i tanhx
sec (ix) = sechz
8.79
cet (ix)
8.81
cash (ix) = COSz
8.82
tanh (iz)
= i tan x
8.84
sech (ix) = sec%
8.85
coth (ix)
=
sin (ix) = i sinh x
8.75
COS(iz)
8.77
csc(ix)
8.78
8.80
sinh (ix) = i sin x
8.83
csch(ti)
-i
=
cschx
-icscx
In the following
31
8.76
8.74
=
FUNCTIONS
= cash x
== -<cothx
k is any integer.
8.86
sinh (x + 2kd)
=
sinh x
8.87
cash (x + 2kd)
=
cash x
8.88
tanh(x+
8.89
csch (x +2ks-i)
=
cschx
8.90
sech (x + 2kri)
=
sech x
8.91
coth (S + kri)
= i sinh-1 x
8.93
sinh-1 (ix)
2 i cash-1 x
8.95
cash-lx
tan-1 (ix) = i tanh-1 x
8.97
tanh-1 (ix) = i tan-1 x
8.98
cet-1 (ix)
8.99
coth-1 (ix)
8.100
sec-l x
8.101
sech-* x
8.102
C~C-1 (iz)
8.103
eseh-
8.92
sin-1 (ix)
8.94
Cos-ix
8.96
=
=
-icotz
=
*i
=
- i coth-1 x
sech-lx
- i csch-1 z
=
=
(ix)
i sin-1 x
k i COS-~x
= - i cet-1 x
=
*i
=
sec-l x
- i C~C-1 x
kri)
= tanhx
=
coth z
9
S
QUAURATIC
9.1
S
o
I a b, c a
fa , i
(
r
a
u i
(
r
a
e i
(
c
I
9.2
r
a
t
L
9.3
Dr = eb2 - n
4
ei
I
u
t
2a
f i te
discriminant,
a
d
a
s ht
l t
i
cr
aeh
e
l
q
u
d
u)
l
a
l
n i
p
j )
o i
fr r
D <o 0i
mf
t r
h
x + ox , = -bla
h e
Q
t
3a2 -
a;
=
9
Xl
=
S + T -
x
=
-
=
-
7
,
w
x
x
x ah t
rt
s
-
-
( -
S
--
+3
-
+ &
a
T/ S
1
)Z +
( T
S )
-
discriminant,
a
d
s ht
wo
l
o i a D >do 0 ot
eh
e i af
y
mf l
dt e
n
) l
s mfp r
e
= 2
C
(
a
O @
x2 =
2
C
(
+ 1m
O +w
2C
=
2
C
(
+ 2G
O +
4
+ Ca + x r
,h ,
fa
i )
u
s im
o t
2 ar r e h
a
-
.
)f T
D
1n =C
o 0.
n
x
ar
,s
1
1a o u
xI + x2 + xs = -
a
a
Ti +
a i r
9.5
sx
x
a
r
3
g
+ S
a
x
e
l
+
(
if D < 0:
t
$
1a
n o et e e )i awD = rd0t q a f l o
Solutions
en
&
1a o a l r i et
b
o
o er e
32
e
p
n
j
l
e s u s
a t
s q
u
pu i
u
e
l
a
l tg
i ao
S )
= Qr r
r
s
1
a i r
so
e
n
a
d
-
4
r
r i ei
nl
a
x
nce
-
re
u
tr s
- 2a
5
+
2
cs )e
- 2
-
n
a
i
to
=e c
o
f
an
oex x
9
R=
’
ra 2 i s Dr = eQ3 +, nR2, f i te
i ri
are
l
o
o
d
(
c
h
af
Xl
9.4
2 ~/@-=%c-
(
D < 0,
r
-b
=
f a
iL
rf
uz2 + bx -t c = 0
Q G
U
Dn > 0n )
Solutions:
a a
EQUATION:
L
i e D =n 0 q i
e
I a a
Ef
lx
c i
x
o O
A
o
Th
0 O
e =S -RI&@
x s x
e
S e
0
= - ,s ,
r z s
e
t
’
e
S
)
r
’
ax
e
x r
s
)
ss
2
.
SOLUTIONS
QUARTK
Let y1 be a real root
9.7
Solutions:
ALGEBRAIC
EQUATION:
of the cubic
The 4 roots
OF
x* -f- ucx3 + ctg9 +
of ~2 +
xl, x2, x3, x4 are the four
u
3
+
a
3
=
0
4
3
$
equation
+{a1
2
a; -4uz+4yl}z
If a11 roots of 9.6 are real, computation
is simplified
a11 real coefficients
in the quadratic
equation
9.7.
where
EQUATIONS
by using
+
that
$&
* d-1
particular
=
real
root
0
which
produces
roots.
-
FURMULAS
Pt.ANE ANALYTIC
10
fram
GEOMETRY
DISTANCE d BETWEEN TWO POINTS F’&Q,~~) AND &(Q,~~)
10.1
d=
-
Fig. 10-1
10.2
mzz-z
EQUATION
10.3
OF tlNE JOlNlN@
Y -
Y1
x -
ccl
m
Y2 -
Y1
F2 -
Xl
TWO POINTS ~+%,y~)
Y2 - Y1
xz -
10.4
cjr
Xl
y =
where
b = y1 - mxl =
XZYl xz -
EQUATION
XlYZ
51
tan 6
Y -
Y1 =
mb
ANiI
l%(cc2,1#2)
- Sl)
mx+b
is the intercept
on the y axis, i.e. the y intercept.
OF LINE IN ‘TEMAS OF x INTERCEPT a # 0 AN0 3 INTERCEPT b + 0
Y
b
a
Fig. 10-2
34
2
FORMULAS
FROM
ffQRMAL
10.6
ANALYTIC
FORA4 FOR EQUATION
+ Y sin a
x cosa
PLANE
=
where
p
=
perpendicular
and
a
1
angle of inclination
positive
z axis.
GEOMETRY
OF 1lNE
y
p
distance
35
from
origin
0 to line
of perpendicular
P/
,
with
,
L
LX
0
I
Fig. 10-3
GENERAL
10.7
Ax+BY+C
KIlSTANCE
where
FROM
the sign is chosen
ANGLE
10.9
s/i BETWEEN
tan $
Lines are parallel
Lines
POINT
SO that
=
(%~JI)
the distance
TWO
OF LINE
EQUATION
0
TO LINE
AZ -l- 23~ -l- c = Q
is nonnegative.
l.lNES
HAVlNG
SlOPES
wsx AN0
%a2
m2 - ml
1 + mima
=
or coincident
are perpendicular
if and only if mi = ms.
if and only
if ma = -Ilmr.
Fig. 10-4
AREA
10.10
Area
=
z=
where
*T
OF TRIANGLE
1
*;
the sign
If the area
Xl
Y1
1
~2
ya
1
x3
Y3
1
(Xl!~/2
+
?4lX3
is chosen
WiTH
VERTIGES
AT @I,z&
@%,y~), (%%)
(.% Yd
+
Y3X2
SO that
is zero the points
-
!!2X3
-
the area
YlX2
-
%!43)
is nonnegative.
a11 lie on a line.
Fig. 10-5
FORMULAS
36
TRANSFORMATION
1
10.11
FROM
PLANE
ANALYTIC
OF COORDINATES
x
=
x’ + xo
Y
=
Y’ + Y0
1
x’
or
y’
x
x
GEOMETRY
INVGisVlNG
x -
xo
Y -
Y0
PURE
TRANSlAliON
Y
l Y’
l
l
where (x, y) are old coordinates [i.e. coordinates relative to
xy system], (~‘,y’) are new coordinates [relative to x’y’ system] and (xo, yo) are the coordinates of the new origin 0’
relative to the old xy coordinate system.
Fig. 10-6
TRANSFORMATION
10.12
1
=
x’ cas L -
OF COORDIHATES
y’ sin L
or
-i y = x’ sin L + y’ cas L
x’ z
INVOLVING
PURE
x COSL + y sin a
ROTATION
\Y!
\
\
\
\
yf z.z y COSa - x sin a
where the origins of the old [~y] and new [~‘y’] coordinate
systems are the same but the z’ axis makes an angle a with
the positive x axis.
,
,
,
,
Y
,
\o/
,
’
\
,
/
/
/
,x’
L
CL!
\
Fig. 10-7
TRANSFORMATION
OF COORDINATES
1
1
02 =
10.13
lNVGl.VlNG
TRANSLATION
x’ cas a - y’ sin L + x.
y = 3~’sin a + y’ COSL + y0
or
ANR
x’
ZZZ
(X - XO) cas L + (y - yo) sin L
y!
rz
(y - yo) cas a - (x - xo) sin a
1
\
ROTATION
/
,‘%02
\
where the new origin 0’ of x’y’ coordinate system has coordinates (xo,yo) relative to the old xy eoordinate system
and the x’ axis makes an angle CYwith the positive x axis.
Fig. 10-8
POLAR
COORDINATES
(Y, 9)
A point P cari be located by rectangular coordinates (~,y) or
polar eoordinates (y, e). The transformation between these coordinates
is
10.14
x
=
1 COS 0
y = r sin e
or
T=$FTiF
6 = tan-l
(y/x)
Fig. 10-9
FORMULAS
RQUATIQN
10.15
FROM PLANE
OF’CIRCLE
(a-~~)~ + (g-vo)2
ANALYTIC
OF RADIUS
GEOMETRY
37
R, CENTER AT &O,YO)
= Re
Fig. 10-10
RQUATION
10.16
OF ClRClE
OF RADIUS
R PASSING
T = 2R COS(~-a)
THROUGH
ORIGIN
Y
where (r, 8) are polar coordinates of any point on the
circle and (R, a) are polar coordinates of the center of
the circle.
Fig. 10-11
CONICS
[ELLIPSE,
PARABOLA
OR HYPEREOLA]
If a point P moves SO that its distance from a fixed point
[called the foc24 divided by its distance from a fixed line [called
the &rectrkc] is a constant e [called the eccen&&ty], then the
curve described by P is called a con& [so-called because such
curves cari be obtained by intersecting a plane and a cane at
different angles].
If the focus is chosen at origin 0 the equation of a conic
in polar coordinates (r, e) is, if OQ = p and LM = D, [sec
Fig. 10-121
10.17
T =
P
1-ecose
=
CD
1-ecose
The conic is
(i)
an ellipse if e < 1
(ii)
a parabola if e = 1
(iii) a hyperbola if c > 1.
Fig. 10-12
38
FORMULAS
FROM PLANE
10.18
Length of major axis A’A
=
2u
10.19
Length of minor axis B’B
=
2b
10.20
Distance from tenter C to focus F or F’ is
ANALYTIC
GEOMETRY
C=d--
= c =
E__
10.21
Eccentricity
10.22
Equation in rectangular
a
-
~
0
a
coordinates:
(r - %J)Z + E
b2
a2
Fig. 10-13
=
3
re zz
a2b2
10.23
Equation in polar coordinates if C is at 0:
10.24
Equation in polar coordinates if C is on x axis and F’ is at 0:
10.25
If P is any point on the ellipse, PF + PF’
=
a2 sine a + b2 COS~6
r =
a(1 - c2)
l-~cose
2a
If the major axis is parallel to the g axis, interchange x and y in the above or replace e by &r - 8 [or
9o” - e].
PARAR0kA
WlTJ4 AX$S PARALLEL
TU 1 AXIS
If vertex is at A&,, y,,) and the distance from A to focus F is a > 0, the equation of the parabola is
10.26
(Y - Yc?
10.27
(Y - Yo)2 =
=
4u(x - xo)
if parabola opens to right [Fig. 10-141
-4a(x - xo)
if parabola opens to left [Fig. 10-151
If focus is at the origin [Fig. 10-161 the equation in polar coordinates is
10.28
T
=
2a
1 - COSe
Y
Y
-x
0
Fig. 10-14
Fig. 10-15
x
Fig. 10-16
In case the axis is parallel to the y axis, interchange x and y or replace t by 4~ - e [or 90” - e].
FORMULAS
FROM PLANE
ANALYTIC
GEOMETRY
39
Fig. 10-17
10.29
Length of major axis A’A
= 2u
10.30
Length of minor axis B’B
=
10.31
Distance from tenter C to focus F or F’
10.32
Eccentricity
10.33
Equation in rectangular
10.34
Slopes of asymptotes G’H and GH’
10.35
Equation in polar coordinates if C is at 0:
10.36
Equation in polar coordinates if C is on X axis and F’ is at 0:
10.37
If P is any point on the hyperbola,
e = ;
= -
2b
=
c = dm
a
coordinates:
=
(z - 2#
os
(y - VlJ2
-7=
1
* a
”
PF - PF!
=
=
If the major axis is parallel to the y axis, interchange
[or 90° - e].
a2b2
b2 COS~e - a2 sin2 0
22a
r =
Ia~~~~~O
[depending on branch]
5 and y in the above or replace 6 by &r - 8
11.1
E
i
p
qc
n
r
E
1
1 i
1
A
b1
1
A
o 1o
r
l
+ y
An
o
r=
uo
= a c
2
. cn
u
q
(
o
e
l
2
2 a
0
c
a
2 o
= C S - y*
A
e.
a
&f . n
B
xga r
o
e
ao
a
o
’lx
o
B
w
a
E
i
p q
fn
[
C
=
CE
L-
1y = a
1
A
1
A
T
a r
o 1o
l
a
1
2
o r ae
i a c
dh
a
x ao
s
l
.=
o
r
8f
(
F o
o
E
1 i
r
q
%
E
1
1
i
p q
A
11.11
A
u
b
l
T
i a c
a i r
o /t
brc
o
e r
e
u
y
2 Z
a
=
a
s
li
dh
s
bu a p ei
P o
si t o o a
n c4h n o rl f
d
A
o
’
s
\
l ’
eB,
/
xg
n
n
i 1
1
g-
l
m
i
2
,
tY
n-
nn
j
m i
O
:
e o
t n
n
S
a
#
h
)
2
c g
h t
v i
ic f
a
g
is
h
er
nr
n. F
1
d
c
ti
g i
1
i
l
b
g
-
u
ViflTH FOUR CUSf’S
/ Z
c
2
a
f
9o
o
3 Z
r
O
0
f= n6 c
a
1
o ss o r n
n
8 o
Z
l
u
a
r
a
a
t
m
3
ar
ta
n
gr
ya
r c o ss o r n v i
ai e s f al
r.
i
d
m i
:
n
o
i
g
n
e o
t n
9
n
a
40
t
3
r
S
nu
t
3
i
o = & yeu
ec
r
e
i
F
)
a
a
ya
r c
. fn a u
x = a C
y
11.10
. cn
+
c
7
HYPOCYCLOID
1
,
e
o i
C
O
= 6e
nc rn
ei
p
a
i
e
bu a p ei
x
l
r
a
&
Y
- C
r = 3f . n
o
a u
(s + +
r
i d\
,
,
\
)!
5d
Y
r
t
(
C
11.5
t
s)
t’=3 4 n
\
s
G
a 4e
tA
r
z
2
v
i
ic f a
i
sd
d
e
tv
er
c
r
nr
F
d
1
e
d
he
d
i
e
c
l
ti
i
e
i
1
u
l
e
b
g
-
u
s
.
SPECIAL
PLANE
CURVES
41
CARDIOID
11 .12
Equation:
11 .13
Area bounded by curve
11 .14
Arc length of curve
r = a(1 + COS0)
= $XL~
= 8a
This is the curve described by a point P of a circle of radius
a as it rolls on the outside of a fixed circle of radius a. The
curve is also a special case of the limacon of Pascal [sec 11.321.
Fig. 11-4
CATEIVARY
11.15
Equation:
Y z : (&/a + e-x/a)
= a coshs
This is the eurve in which a heavy uniform cham would
hang if suspended vertically from fixed points A anda. B.
Fig. 11-5
THREEdEAVED
11.16
Equation:
ROSE
\
r = a COS39
The equation T = a sin 3e is a similar curve obtained by
rotating the curve of Fig. 11-6 counterclockwise through 30’ or
~-16 radians.
In general
n is odd.
v = a cas ne
or
r = a sinne
‘Y
\
\
\
\
\
,
/
has n leaves if
/
+
,/
,
Fig. 11-6
FOUR-LEAVED
11.17
Equation:
ROSE
r = a COS20
The equation r = a sin 26 is a similar curve obtained by
rotating the curve of Fig. 11-7 counterclockwise through 45O or
7714radians.
In general
n is even.
y = a COSne
or
r = a sin ne has 2n leaves if
Fig. 11-7
a
X
42
SPECIAL
11.18
PLANE
CURVES
Parametric equations:
X
=
(a + b) COSe -
b COS
Y
=
(a + b) sine -
b sin
This is the curve described by a point P on a circle of
radius b as it rolls on the outside of a circle of radius a.
The cardioid
[Fig. 11-41 is a special case of an epicycloid.
Fig. 11-8
GENERA&
11.19
HYPOCYCLOID
Parametric equations:
z
=
(a - b) COS@ + b COS
Il
=
(a-
b) sin + -
b sin
This is the curve described by a point P on a circle of
radius b as it rolls on the inside of a circle of radius a.
If
b = a/4,
the curve is that of Fig. 11-3.
Fig. 11-9
TROCHU#D
11.20
Parametric equations:
x =
a@ - 1 sin 4
v = a-bcos+
This is the curve described by a point P at distance b from the tenter of a circle of radius a as the
circle rolls on the z axis.
If
1 < a, the curve is as shown in Fig. 11-10 and is called a cz&ate c~cZOS.
If b > a, the curve is as shown in Fig. ll-ll
If
and is called a proZate c&oti.
1 = a, the curve is the cycloid of Fig. 11-2.
Fig. 11-10
Fig. ll-ll
SPECIAL
PLANE
CURVES
43
TRACTRIX
11.21
PQ
x
Parametric equations:
u(ln cet +$ - COS#)
=
y = asin+
This is the curve described by endpoint P of a taut string
of length a as the other end Q is moved along the x
axis.
Fig. 11-12
WITCH
11.22
Equation in rectangular
11.23
Parametric equations:
coordinates:
OF AGNES1
u =
8~x3
x2 + 4a2
x = 2a cet e
y = a(1 - cos2e)
Andy
-q-+Jqx
In Fig. 11-13 the variable line OA intersects
and the circle of radius a with center (0,~) at A
respectively. Any point P on the “witch” is located oy constructing lines parallel to the x and y axes through B and
A respectively and determining the point P of intersection.
y = 2a
FOLIUM
11.24
OF DESCARTRS
Y
3axy
\
Parametric equations:
1
x=m
y =
11.26
Area of loop = $a2
11.27
Equation of asymptote:
3at
1
3at2
l+@
\
x+y+u
Z
Fig. 11-14
0
INVOLUTE
il.28
Fig. 11-13
Equation in rectangular coordinates:
x3 + y3 =
11.25
l
OF A CIRCLE
Parametric equations:
x = ~(COS+ + @ sin $J)
I y = a(sin + - + cas +)
This is the curve described by the endpoint P of a string
as it unwinds from a circle of radius a while held taut.
jY!/--+$$x
.
Fig. Il-15
I
44
S
11.29
E
i
r
q
(axy’3
+
P
11.30
e
x
(bvp3
1b =
i t he u
= 1e s
z
d
e
o
u
tu3
-
1
P
of
6a
so h
t i n r
/i h lF
1a
+ qa4 .
a
s
z
i
2
G
2
T
I
i t
2 a
c
c
d
hd s h
i a c a p
i a ih F u 1 s
t
b = u
cf
-
u
b a p
ie
ib s
o
t
u
A
C
a
m
r
R
N
I
V
E
A
a
i
d
t
e
n
o
i
g
2 u 3=
n
o he
2 s wg
lre
yr t sos t
a 2s
n
s[
a r
e1e
e lm
l
n.
1e
F
1
F
o
i
rS
W
i oa
6d
1
p pi
g
L
t
i
m
1
v
i
r.
-k
o
1 7 eo
e
g
-
S
p u v h o i cih d r c e a f nrt e t i o hf
trp t is ws d i
.r t
s
t a
t
] n
a
c
i g va1 - b <c a og .1 e>
s- a1 rc
r
t
s
e
-h
A
aO e a 4
ba
Pe s
i
)
OF CASSINI
V
so nFe i r1 1
i , a Zh
U
t
)
tf the s eov v
a
nb o i
1s
l~i
2
o
_--\
++Y
!---
T
[
e 1
a
u
C
O
1
E
by3
COS3
z 8
- b ys
L
ELLIPSE
c
r
- b
(
C
OF Aff
q
= (
P
EVOWTE
=
a
c
T
c
+ y
cn
P
8
1c
oo b
n
sr
.
1a
P
X
a
F
1
i
1
g
LIMACON
11.32
t
P
L
c
T
c
O b
i t
c
i a c
e
o r = qb
l
-
.
1
i
1
g
-
u+
a
aa
r
tc
y i gai p f
a n
s
s os nFe i 1r 1 a r i g a1v -b >c a og .b s<
-e a1 r c
r F r 1
v id
e ig
4
o
ii
h
r.
.
t
io
os
r in a0
t t
aTn h c nt s.
s
2 I9 e o1 = a
1
i
t
0 f sr ,
. d
-
F
1
i
1
.
OF PASCAL
a l
ej
Q eo i 0to t a
rp n Q ioo an c io eo dnn
h
l
u o a s ph oe P rs f 1t oe Pc = vub 1h i Q u . ec
i a ih F u 1
u [ s a1
17
F
g
-
.
1
F
1
9
i 1
g-
m
hg r
h
SPECIAL
PLANE
C
11.33
Equation
in rectangular
y
11.34
Parametric
OF
CURVES
Ll
IS
x
2a -
2
3
x
equations:
i
=
2a sinz t
?4 =-
2a sin3 e
COSe
This is the curve described
by a point P such that the
distance
OP = distance RS.
It is used in the problem
of
duplicution
of a cube, i.e. finding the side of a cube which has
twice the volume of a given cube.
SPfRAL
Polar
BS
coordinates:
ZZZ
x
11.35
45
equation:
Y =
a6
Fig. 11-21
OF ARCHIMEDES
Y
Fig. 11-22
OO
C
FORMULAS
APJALYTK
12
SCXJD
GEOMETRY
from
Fig. 12-1
RlRECTlON
12.2
COSINES OF LINE ,lOfNlNG
1 =
COS L
=
% - Xl
~
d
’
m
=
where a, ,8, y are the angles which line PlP2
d is given by 12.1 [sec Fig. 12-lj.
FO!NTS &(zI,~z,zI)
COS~
=
Y2 d,
Y1
n
=
AND &(ccz,gz,rzz)
c!o?, y
=
22 -
-
21
d
makes with the positive x, y, z axes respectively
and
RELATIONSHIP EETWEEN DIRECTION COSINES
12.3
or
cosza+ COS2
p + COS2
y = 1
lz + mz +
nz
=
1
DIRECTION NUMBERS
Numbers L,iVl, N which are proportional
The relationship between them is given by
12.4
1 =
L
dL2+Mz+
to the direction cosines 1,m, n are called direction
M
m=
N2’
dL2+M2+Nz’
46
n=
N
j/L2 + Ar2 i N2
numbws.
FORMULAS
OF LINE JOINING
EQUATIONS
12.5
These
FROM
x-
x,
% -
Xl
are also valid
Y-
~~~~
Y2 -
Y1
z -
Y1
752 -
ANGLE
are
also valid
+ BETWEEN
if 1, m, n are replaced
TWO
LINES WITH
12.7
12.8
x -
OF PLANE
AND
y
=
Y-
12.9
xz -
Xl
x3 -
Xl
2 -
Y1
m
FORM
Zl
n
IN PARAMETRIC
1 =
FORM
.zl + nt
by L, M, N respectively.
DIRECTION
mlm2
THROUGH
X
x -
Y =p=p
I’&z,y~,zz)
y1 + mt,
EQUATION
PASSING
Xl
1
COSINES
L,~I,YZI
AND
h
,
+ nln2
OF A PLANE
.4x + By + Cz + D
EQUATION
IN STANDARD
~&z,yz,zz)
or
47
by L, M, N respeetively.
COS
$ = 1112
+
GENERAL
GEOMETRY
21
I’I(xI,~,,zI)
x = xI + lt,
These
AND
.z,
if Z, m, n are replaced
12.6
ANALYTIC
~I(CXI,~I,ZI)
OF LINE JOINING
EQUATIONS
SOLID
=
[A, B, C, D are constants]
0
POINTS
Y1l
2 -
.zl
Y2 -
Y1
22 -
21
Y3 -
Y1
23 -
Zl
(XI, 31, ZI), (a,yz,zz),
=
(zs,ys, 2s)
cl
or
12.10
Y2 -
Y1
c! -
21
Y3 -
Y1
z3 -
21
~x _ glu
+
EQUATION
z+;+;
12.11
where
a, b,c
respectively.
are
the
z2 -
Zl
% -
Xl
23 -
21
x3 -
Xl
OF PLANE
z
intercepts
~Y _
yl~
+
IN INTERCEPT
xz -
Xl
Y2 -
Y1
x3
Xl
Y3 -
Y1
-
(z-q)
FORM
1
on
the
x, y, z
axes
Fig. 12-2
=
0
48
FOkMULAS
FROM
E
A
z
t
N
YB
X”
A
N
t
A B C
OQ
P
x -
-
Yn
P z
-
F I
2
P
T
T
”
R( S y
O
2
w
.
t
s
i hc
x
=
N
I
(
x,, + At,
r oe
T
O
+ B
F
A N
x
R
T E
+ C q+ D
3
d
EO
= yo + Bf, z =
y
R
y
O
I
N
.z(j
,
oees s
N
M
R
T
,
r e ir n
,NC
~
,
,
B
n e t
FUM
L
U
E
+
t
,
ct
+
A
OQ R
P
O
nA b
o +rhB c +l C + eDx =e p
0ey at a z
z
nas
o
E
I AZO + eM
By
L A ,+ Cz
N + L ~
=A N0,
s teh d e Ogh i nhro i
F
H
ti mt et e pr
O
xP T ,
1
k
h S it
GEOMETRY
R Ax O+ By
L + C.z
P + L =A 0
a al u ep
A
1
FU
PD
or
C
i ft
ANALYTIC
L
E
d o
h n h
, , .
D
SOLID
n na
A
A A
e
L
T N
1
1
2 x cas L + y COS,8
.
i- z COSy
w
P a
a
p = p
C/ y a
h
an
x
de
Xb3
=1
ef
On d a
e
r
4
p
0 i tr p r r a
,e,p
e P
xg y nz
s
s op l
to
eo t
,l , d
.
t e a
ws
m
e
a n n
ei
n d e
et
s
Fig. 12-3
T
R
22
1
=
2y =
z
w
(
t
t r
t
t x
o t
n
s
=
x’ +
O
A
F
x’
x()
y’ + yo
d
C
c
x -
y’ ZZZ
1Y -r
. o
+
O IN
x
ON
PS
(
T
RV
UF
R
DO
RO
A
l
J
5
Y0
z
(
a h o c%
r e [l
oc ,
e rird
oy
i
(o y z y a v n a c z’ ’ s r e e[ ? o , ) t e
s i
oa ’ ( y vz a n yt q c s0 ee r d ’ h , o t,
o
f 0h r e
r t t ’ e o e wq i c o h
l l z go
y
s
t
s
‘
J
e.e
ro,
rw , o
) e ze
e da
e
e
o e
io
el
e
m
X
Fig. 12-4
d
r~
r
’
t
m
r m
nr
.a
l
d
i)
d
i
.
]
d ]
d
FORMULAS
FROM
TRANSFORMATION
x
=
1
2
=
n
+ &
+
ANALYTIC
OF COORDINATES
+ 1
n
l+
y
1
y = WQX’+ wtzyf+
12.16
SOLID
3
r
INVOLVING
!
x
n
n
2
x
y
'
z
PURE ROTATION
*
% 1
p
3
49
GEOMETRY
?
'
\
’
%
\
'
\
O
i
=
Z
+' m
+I
y'
l=
1
+
x
=
z
+' m
m ?
T
1
X
z
y
l
+2
n
2
x
p
y
.
+z
?
a
x
%
y
g
\
Z
\
\
z
where the origins of the Xyz and x’y’z’ systems are the
same and li, '
n
1 mm
nl
1 2 m 2 l n 2 ; are
3, 3
the
, , sdirection
;
, ,
cosines of the x’, ,y’, z’ axes relative
to the x, y, .z axes
respectively.
3
1
, ,
,
\
X
’
\
, Y
, ?/‘
’
’
~
’
Y
,,/
X
Fig. 12-5
TRANSFORMATION
z
12.17
OF COORDINATES
Z
=
+ &
+ l&
+
I x.
INVOLVING
y
X
TRANSLATION
’
’
y = miX’ + mzy’ + ma%’ + yo
2
or
i
=
n
+
n
l+
2+
zX
3
y
.'
y!
zz &z(X- Xo) + mz(y - yo) + n&
- 4
x’
=
- zO)
d-
y
x I+
F’
\
\
=
+t m
z
ROTATION
'
X
4
-' X
n
AND
n
-d z t
&(X - X0) + ms(y - Y& + 42
z
'
l
d y
COORDINATES
/
/
‘X’
(r, 0,~)
A point P cari be located by cylindrical
coordinates
(r, 6, z.)
[sec Fig. 12-71 as well as rectangular
coordinates
(x, y, z).
The
transformation
x
12.18
=
between
these
coordinates
is
r COS0
y = r sin t
or
0 =
tan-i
r
(y/X)
z=z
Fig. 12-7
-
Y
1
'
$
l
Fig. 12-6
CYLINDRICAL
/
o
/
where the origin 0’ of the x’y’z’ system has coordinates
(xo, y,,, zo) relative
to the Xyz system
and Zi,mi,rri;
cosines
of the
la, mz, ‘nz; &,ms, ne are the direction
X’, y’, z’ axes relative
to the x, y, 4 axes respectively.
y
,
\
b
,
,
l
'
"
FORMULAS
50
FROM
SPHERICAL
[sec
SOLID
ANALYTIC
COORDINATES
GEOMETRY
(T, @,,#I)
A point P cari be located
by spherical
coordinates
(y, e, #)
Fig. 12-81 as well as rectangular
coordinates
(x,y,z).
The
transformation
12.19
between
those
=
x sin .9 cas .$J
=
r sin 6 sin i$
=
r COSe
coordinates
is
x2 + y2 + 22
or
$I =
tan-l
(y/x)
e =
cosl(ddx2+y~+~~)
Fig. 12-8
EQUATION
12.20
OF
SPHERE
(x - x~)~ + (y - y#
where
the sphere
has tenter
IN
RECTANGULAR
+ (,z - zo)2 =
COORDINATES
R2
(x,,, yO, zO) and radius
R.
Fig. 12-9
EQUATION
12.21
OF
SPHERE
CYLINDRICAL
COORDINATES
rT - 2x0r COS(e - 8”) + x; + (z - zO)e
where
the sphere
If the tenter
has tenter
(yo, tio, z,,) in cylindrical
is at the origin
the equation
12.22
7.2+ 9
EQUATION
12.23
OF
SPHERE
rz + rt where
the sphere
If the tenter
12.24
IN
has tenter
IN
and radius
= Re
SPHERICAL
COORDINATES
2ror sin 6 sin o,, COS(# - #,,)
the equation
r=R
R’2
is
(r,,, 8,,, +0) in spherical
is at the origin
coordinates
=
is
coordinates
=
Rz
and radius
R.
R.
FORMULAS
E
FROM
OQ E
SOLID
ANALYTIC
C
tA (L
FW
U L
51
GEOMETRY
E
A TTx I S
N
N HI ~P
a Eb
T
D O, ,S M,
E
N
y O dI
Fig. 12-10
E
1
C
2
w
I
L
W Y
.
a I a
sh
b = a
i b
A I xL A X
I
2
,
f
A
L
o re ee
ac
c
ST
I X PI
H N I S
T
D S I
6
fs e l
t e c
mr
r
e l
io r c y u
rf
ie
o
c i
a o l .
c
-
s
t p
d mi
u
a
i
s
i t
en
l
x
u
sd
Fig. 12-11
E
1
2
C
.
L
W
AO
2
L A I z A XN
J ST
X IE
P
H
I S
T
S
7
Fig. 12-12
H
1
2
$
.
Y O
z+
1
2
$
O
S
P F
8
_
N
H
E
E
$
Fig. 12-13
E
R
E
B
I
5
2
FORMULAS
FROM
SOLID
H
Note
orientation
of axes
ANALYTIC
YO
in Fig.
T
GEOMETRY
S
IF
W
H
’
O
E
E
E
12-14.
Fig. 12-14
E
1
2
P
.
L
3
A
L
R
I
A
P
0
Fig. 12-15
H
1
2
Note
xz
--a2
orientation
y2
b2
of
axes
=
.
PY
_z
3
AP
RE
AR
1
C
in Fig.
12-16.
/
-
Fig. 12-16
X
D
If y = f(z),
OE
A D
FF
E
t
R
N
t
~
lim f(X+ ‘) - f(X)
=
d
+h
hX
=
G
R
a
O
where h = AZ. The derivative is also denoted by y’, dfldx
called di#e~eAiatiotz.
E
O
D
f
+ A
or f(x).
l
- fi
(
~
(r
~
)
~
F E
A
Ax
Ax-.O
The process of taking a derivative
N
F
t
t
E
F k
R
is
In the following, U, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828. . .
is the natural base of logarithms; In IL is the natural logarithm of u [i.e. the logarithm to the base e] where
it is assumed that u > 0 and a11 angles are in radians.
1
g(e) =3
1
&x)
0
=
3
c
.
2
.
3
1
3
.
4
1
3
.
5
c
u
1
&
3
1
&
3
1
$-(uvw)
3 =
1
1
1
1
=
=
du
dx
-H
v
3
du
_
ijii
-
du
-=-
2
dv
3
du
du dx
1
dyidu
3
-
dxfdu
z n
c
6
.
u
7
dv
+
dx
vw-
u(dv/dx)
V
&
.
uw.
+
v(duldx)
dxfdu
=
v
-
3
dx
dy
z
uv-
_
3z &
-
V
the derivative of y or f(x) with respect to z is defined as
13.1
1
l
$
-(Chai?
.
.
Z
du
dx
gu
gv g
8
9
1
0
. rule)
1
1
.
1
2
.
1
3
j
5
3
)
)
+
E S
54
DERIVATIVES
AL”>. 1
_. .i
”
.,
13.14
-sinu
d
dx
=
du
cos YG
13.17
&cotu
=
-csck&
13.15
$cosu
=
-sinu$
13.18
&swu
=
secu tanus
13.16
&tanu
=
sec2u$
13.19
-&cscu
=
-cscucotug
13.20
-& sin-1
13.21
&OS-~,
13.22
u
-%<
=$=$
=
&tan-lu
-1du
qciz
=
13.23
&cot-‘u
=
13.24
&sec-‘u
=
&
csc-124
[O < cos-lu
dx
< i
C
+&
[O < cot-1 u < Tr]
1
du
zi
-I
1
< z-1
LJ!!+
1 + u2 dx
ju/&zi
13.25
sin-‘u
< tan-lu
1
< t
if 0 < set-lu
d
-log,u
dx
-
=
13.27
&lnu
13.28
$a~
=
13.29
feu
=
~l’Xae
u
=
=
-du
if 0 < csc-l
=
I
u < 42
< csc-1 u < 0
1
ig
aulna;<
TG
d"
fPlnu-&[v
13.31
gsinhu
=
eoshu::
13.32
&oshu
=
13.33
$
=
tanh u
< r
a#O,l
dx
-&log,u
< 7712
if 7712 < see-lu
=
+ if --r/2
13.26
du
lnu]
=
vuv-l~
du
+ uv lnu-
dv
dx
13.34
2
cothu
=
- cschzu ;j
sinh u dx
13.35
f
sech u
=
- sech u tanh u 5
sech2 u 2
13.36
=
- csch u coth u 5
du
A!- cschu
dx
dx
dx
DERIVATIVES
13.37
d
- sinh-1
dx
13.38
-dx cash-lu
13.39
-tanh-1
13.40
-coth-lu
d
u
d
d
dx
u
dx
13.41
=
~
=
~
=
--
=
-- 1
+ if cash-1 u > 0, u > 1
if cash-1 u < 0, u > 1
-
du
1
[-1
1 - u2 dx
1 _
-&sech-lu
55
du
dx
u2
71
=
- d csch-‘u
if sech-1 u > 0, 0 < u < 1
+ if sech-lu<O,
O<u<l
[ -
du
-1
=
du
[-
dx
HIGHER
The second, third
and higher
derivatives
13.43
Second derivative
=
d
dy
ZTz 0
13.44
Third
=
&
13.45
nth derivative
derivative
13.46
DERtVATlVES
are defined
=a
if u > 0, + if u < 0]
d’y
=
as follows.
f”(x)
=
y”
f’“‘(x)
LEIBNIPI’S
Let Dp stand
< u < 11
[u > 1 or u < -11
u-z
13.42
RULE FOR H26HER
for the operator
&
uD%
+
D+.w)
=
so that
II
y(n)
DERIVATIVES
OF PRODUCTS
= :$!& = the pth derivative
D*u
;
(D%)(D”-2~)
of u.
+
...
0
where
0n
1 ’
As special
0n
2
‘...
coefficients
are the binomial
[page
31.
cases we have
13.47
13.48
DlFFERENT1ALS
Let
y = f(x)
and
Ay = f(x i- Ax) - f(x).
13.49
AY
x2=
where
e -+ 0 as Ax + 0. Thus
13.50
If we call
13.51
the differential
Then
f(x + Ax) - f(x)
Ax
AY
Ax = dx
1
=
=
f/(x)
+ e
f’(x) Ax -t rzAx
of x, then we define the differential
dv
=
=
j’(x) dx
of y to be
Then
+ wDnu
1
DERIVATIVES
56
RULES
The rules
for differentials
are exactly
FOR DlFFERENf4ALS
analogous
13.52
d(u 2 v * w -c . ..)
13.53
d(uv)
13.54
d2
0
=
udv
=
-
d(sinu)
=
cos u du
13.57
d(cosu)
=
- sinu
=
du
PARTIAL
x and y. Then
af
lim
az=
derivative
2
13.59
with
derivatives
of higher
order
respect
of f(x,y)
is defined
dx =Ax
Extension
and
x constant,
a2f
of more
than
to be
a df
ax 0 ay 9
a2f
-=ayiG
ayax
and its partial
a af
7~ 0 ay
a
af
0
derivatives
as
=
$dx
+ $dy
dy = Ay.
to functions
is defined
as follows.
a1/2=
df
where
to y, keeping
a af
TGFG'
0
The results in 13.61 will be equal if the function
case the order of differentiation
makes no difference.
The differential
of f(z, y)
AY
can be defined
a2f
-=--axay
13.61
derivative
Ax
AY'O
@f
-=
a22
13.60
the partial
lim fb, Y + AY) - fb, Y)
-
dY
Partial
we define
I
fb + Ax, Y) - f&y)
Ax-.0
of f(x,y)
_^.1
:“” _
i”
DERf,VATIVES
Let f(x, y) be a function
of the two variables
respect to x, keeping y constant, to be
the partial
du?dvkdwe...
nun- 1 du
13.56
Similarly
that
udv
d(e)
13.58
we observe
212
V
13.55
with
As examples
+ vdu
vdu
=
to those for derivatives.
two variables
are exactly
analogous.
are continuous,
i.e. in such
If
2
= f(z),
or the indefinite
then
y is the function
of f(z),
integral
Since the derivative
derivative
is f(z)
and is called
of a definite
integral,
see page
94.
The
process
of finding
In the following,
u, v, w are functions
of x; a, b, p, q, n any constants,
e = 2.71828. . . is the natural
base of logarithms;
In u denotes the natural
logarithm
that u > 0 [in general, to extend formulas
to cases where u < 0 as well, replace
are in radians; all constants of integration
are omitted
but implied.
14.1
14.2
14.3
14.4
S
S
S
S
adz
=
uf(x)
dx
ax
=
a
(ukz)kwk
udv
14.6
14.7
14.8
Sf(m)
S
=
WV -
dx
F{fWl
dx
undu
=
14.10
=
_(‘udx
vdu
S
integration
”
svdx
*
[Integration
by parts,
.(‘wdx
*
by parts]
see 14.48.
aSf(u) du
-
=
F(u)2
S
du
=
F(u)
f’(z)
S
du
where
u =
.&a+1
S
du
-=
S
S
s
n-t
n#-1
1’
In u
U
...
1
=
=
14.9
f(x) dx
. ..)dx
For generalized
14.5
S
if
[For n = -1, see 14.81
u > 0 or In (-u)
if
u < 0
In ]u]
eu du
=
eu
audu
=
S
@Ina&
the anti-derivative
denoted by
if y =
f (4 dx. Similarly
f (4 du, then
s
S
is zero, all indefinite
integrals
differ by an arbitrary
constant.
of a constant
For the definition
integration.
whose
=
eUl”Ll
-=-
In a
au
In a ’
57
a>O,
a#1
f(z)
an integral
of f(s)
$
=
f(u).
is called
restricted
if indicated;
of u where it is assumed
In u by In ]u]]; all angles
INDEFINITE
58
du
=
- cos u
cosu du
=
sin u
tanu
du
=
In secu
14.14
cot u du
=
In sinu
14.15
see u du
=
In (set u + tan u)
=
In tan
csc u du
=
ln(cscu-
=
In tan;
=
#u
-
=
j&u + sin u cos u)
14.11
14.13
14.16
14.17
14.18
14.19
14.20
14.21
14.22
14.23
14.24
14.25
14.26
14.27
14.28
14.29
14.30
sinu
INTEGRALS
I‘
I‘
I‘
.I'
tanu
=
-cotu
tanzudu
=
tanu
cot2udu
=
-cotu
sin2udu
=
- 2
=
;+T
du
=
secu
=
-cscu
S
S
S
s
' co532u du
S
secutanu
s
cscucotudu
S
I‘
I‘
J
U
-
sin 2u
du
=
coshu
coshu
du
=
sinh u
tanhu
du
=
In coshu
coth u du
=
In sinh u
sechu du
=
sin-1
csch u du
=
In tanh;
(tanh u)
J
sechzudu
=
tanhu
14.32
I‘
csch2 u du
=
- coth u
tanh2u
=
u -
s
du
u
sin 2u
4
14.31
14.33
cosu
u
sinhu
S
S
-In
cotu)
=
sec2 u du
* csc2udu
I
=
tanhu
or
or
sin u cos u)
2 tan-l
- coth-1
eU
eU
INDEFINITE
14.34
14.35
14.36
14.37
14.38
S
S
S
S
s
sinheudu
=
sinh 2u
--4
coshs u du
=
sinh 2u
___
i- t
4
59
cothu
u
2
- sech u
csch u coth u du
=
- csch u
=
+(sinh
=
Q(sinh u cash u + U)
u cash u - U)
du
___
=
u’ + CL2
14.42
s
14.43
u -
=
14.41
14.40
=
sech u tanh u du
S
S
S
14.39
cothe u du
INTEGRALS
u2
=
-
du
___
@T7
s
>a2
u2 < a2
=
ln(u+&Zi?)
01‘
sinh-1
t
14.44
14.45
14.46
14.47
14.48
S
f(n)g dx
This
=
is called
f(n-l,g
-
generalized
f(n-2)gJ
+
integration
f(n--3)gfr
-
. . .
(-1)”
s
by parts.
Often in practice an integral
can be simplified
by using an appropriate
and formula
14.6, page 57. The following
list gives some transformations
14.49
14.50
14.51
14.52
14.53
S
S
S
S
S
F(ax+
b)dx
F(ds)dx
F(qs)
1
a
=
=
dx
=
F(d=)dx
=
F(dm)dx
=
S
S
S
S
S
fgcn) dx
F(u) du
transformation
and their effects.
where
u = ax + b
i
u F(u) du
where
u = da
f
u-1
where
u = qs
F(u) du
a
F(a cos u) cos u du
where
x = a sin u
a
F(a set u) sec2 u du
where
x = atanu
or substitution
INDEFINITE
14.54
14.55
14.56
14.57
F(d=)
I‘
F(eax)
dx
F(lnx)
s
=
dz
a
F(u)
s
=
apply
x, cosx)
tan u) set u tan u du
F(a
where
x = a set u
where
u = In 5
where
u = sin-i:
s
dx
results
F(sin
s
$
=
F (sin-l:)
Similar
14.58
=
s
I‘
s
dx
INTEGRALS
dx
e” du
oJ
F(u)
for other
=
cosu
inverse
du
trigonometric
2
functions.
-
du
1 + u?
where
u = tan:
Pages 60 through
93 provide a table of integrals
classified under special types. The remarks
page 5’7 apply here as well. It is assumed in all cases that division
by zero is excluded.
14.59
dx
s
‘, In (ax + a)
as=
xdx
ax + b
14.60
X
=
-
a
dx
x3
14.62
S i&T-%$-
14.63
S z(az
14.64
S x2(ax
14.65
I‘
14.66
S ~(ax
14.67
S ~(ax
14.68
Sm
b
- ;E- In (ax + 5)
(ax + b)2
--ix---
2b(az3+
(ax + b)s
---m----
3b(ax + b)2 + 3b2(ax + b) _ b3
2 In (ax + b)
2a4
a4
b, + $ In (ax + b)
dx
=
dx
+ b)
=
b)
=
dx
x3(ax+
dx
-1
+ b)2
=
a(ux + b)
x dx
+ b)2
=
a2(af+
=
ax + b
--- a3
=
(ax + b)2 _
2a4
x2 dx
x3
dx
14.69
~
S (ax
+ b)2
14.70
S x(ax
dx
+ b)2
14.71
S
xqax
dX
+ by
b)+ $ In(ax+ b)
a3(ax
b2
+ b)
3b(ax + b) +
a4
$
In (ax + b)
bs
+ z
aJ(ax + b)
In (ax + b)
given
on
INDEFINITE
14.72
14.73
14.74
14.75
14.76
dx
s
x3(az+
s
dx
~(ax + b)3
s
14.79
14.80
14.81
S
S
S
S
14.82
2b
a3(az+
=
b)2
+ +3 In (as + b)
b3
2u4(ax+
6x2
=
2b2(u;a+
=
2b5(ux + b)2 -
b)ndx
n = -1, -2,
b)2 -
b3(ux + b)
a4x2
4u3x
b5(ux + b) -
(ax + b)n+l
=
(n+l)a
=
If
*
(ax + b)n+2
~-
+ b)n dx
(n
(ax + b)n+3
+ 3)a3
=
_
'
2b(ux +. b)n+2
(n+ 2)u3
nZ-1*--2
t b)” dx
+
b2(ux + b)n+’
(nfl)u3
see 14.61, 14.68, 14.75.
+ b)n
=
xm(ux
(m +
nb
+
m+n+l
x”‘(ux
see 14.59.
b(ux + b)n+l
(n+l)u2
-
(n + 2)u2
n = -1,
see 14.60, 14.67.
n = -l,-2,-3,
S
In (ax + b)
b3(ax + b)
xm+l(ax
14.83
- 2
by
2ux
2b3(ux + b)2 -
=
x(ux + b)ndx
If
b2
2a3(ax+
b) -
3b2
u4(ux + 6) +
5-
dx
x3(ux + bJ3
SX~(UX
b)
2u
+ bJ3
(ax+
b4(:3c+
b
dx
x2@
If
b4x
a2(as + b) + 2a2(ax + b)2
=
dx
+ bJ3
x(ax
3
3a(az + b) _
-1
=
x3 dx
~(ax + b)3
61
-1
2(as+ b)2
=
x dx
(ax + b)3
dx
~
(ax + b)a
(ax + b)2 +
-2b4X2
=
~
S x2
S
14.77
14.78
b)2
INTEGRALS
S
xm(ux
mfnfl
n
+ b)n+’
+ 1)~
-xm+l(ux+b)n+l
(n + 1)b
_
mb
(m + n + 1)~ .f
+
xm--l(ux
(n S
m+n+2
+ 1)b
+ b)n-1
xm(ux
dx
+ b)“dx
+ b)“+’
dx
62
INDEFINITE
14.89
14.90
14.93
=
xd-6
s
14.91
14.92
dzbdx
s
dx
x%/G
s
“7
=
dx
2(3a;z;
14.96
14.98
dx
=
&dx
2d&3
X
&T”
s
X+GT3
=
dx
=
t2;$8,,
.(‘
Xm
x(ax
-
=
-(ax
+ b)3/2
(m - l)bxm-’
+ b)““z
(as +xbP”2
(ax + b)m’z
x(ux
-
x-l:=
X~-QL-TTdX
c2;“+b3,a
s
=
dcv
dx
X2
dx
+ b)m/2
1) s
x--l:LTT
&GTT
gm--1
_ (2m - 5)a
(2m - 2)b s
2b(ux
-
2(ax + b)(m+s)lz
u3(m -I- 6)
~(CLX + b)““z
m
=
=
2(mf
2(ax + b)(“‘+Q/z
a2(mf4)
=
=
dx
+
dx
-
4b(ux
+ b)(m+4)/2
4)
+
2b2(ax
a3(m+
(ax + b)(m-2)/2
+ b
s
(ax + b)(m+2)‘2
bx
(m - Z)b(ax
_
+ b)(m+z)/z
aym + 2)
+ b)(“‘+2)‘2
2)
u3(m+
dx
X
(ax + b)m’2
+z
2
+ b)(m-2)/2
’
INVOLVfNC
S
S
1
5
X
x(ax
dx
dx
+ b)(“‘--2)/z
c&z + b AND
p;z! + q
>:“:
dx
14.105
14’109
b)3’2
dx
2(ax + b)(“‘+z)lz
a(m + 2)
INTEGRALS
14.108
+
(m-l)xm-’
=
z2(ax + b)m’2 dx
S
S
_ (2m - 3)a
(2m - 2)b s
(as
=
c (ax + b)m’2 dx
s
dXGb
&&x5
dx
Xm
l/zT-ii
-----dx
s
[See 14.871
X&iZT
2mb
(2m + 1)a s
-
\/azfb
-
s
14.107
s
(m - l)bxm-1
xmd=
s
(2m + 1)~
=
dx
14.102
14.106
dx
+;
2LlFqz
s
[See 14.871
x&zz
&zTT
=
s
14.104
dx
+ b
x2
14.100
14.103
+ 8b2) ,,m3
s
s
14.99
14.101
;$a;bx
‘&zT
J
&iTx
14.97
2b’ l&a@
2(15a’x2
=
14.94
14.95
INTEGRALS
(ax + b)(w
x dx
. (‘ (ax + b)(px
S
S
j-
+ d
+ d
dx
(ax + b)2bx
+ d
xdx
(ax + b)2(px
+ 4
x2 ds
(ax + b)z(px
+ q)
=
=
&
g In (ax+
(bp - aq;&ux+
b) -
b) +
(b-
% In (px+
’ ad2
q)
b(bp ,Z 2uq) In (uz + b)
INDEFINITE
14.110
dx
(ax + bpqpx
I’
+ qp
-1
(Yz - l)(bp
=
INTEGRALS
63
1
- aq)
(ax + b)+l(pz
1
+ q)“-’
+ a(m+n-2)
ax + b
-dds
s PX + Q
14.111
=
7
dx
(ax + bpqpx
s
-1
(N
-
l)(bp
-
(ax
(px
uq)
+
+
bp+’
q)“-l
+
(x-m
-
va
s
1
(ax
14.112
+
bp
(px+
s
q)n dx
-1
=
(m
I
14.113
S
14.114
s
-E!C&.Y
d&zT
dx
=
+ q)n-1
+ yh(px+q)
-
m
(ax + bp
-
(n--:)p
l)p
i
{
(px
+
q)n-l
(ax + ap
(pxtqy-1
+
-
m@p
-
aq)
s
,E++q;!Tl
dx
>
(ax + b)m- 1
(px+
4”
dx
>
(ax + by-1
\
(px + qy- l dx1
S
mu
2(apx+3aq-2bp)Gb
3u2
dx
(Px + 9) &ii-G
14.115
14.116
14.117
Jgdx
=
(px + q)” dn~
s
=
S
S
=
=
(n - l)(aq
=
S,
+ q)n-l
+ 2n(aq - W
(2n + 1)a
(2n + 1)u
-&m
(n -
+
(Px + q)”
dn
2(n ‘“^I),;)”
+ qy-
l + 2(n ” 1)p s
INVOLVING
ds
bp) s
* (px + q)“s
l dx
&ii%
1
l)p(pz
INTEBRAES
14.120
- bp)(px
2(px + q)n &iTT
da
14.119
b - aq
(2n + 3)P s
daxi-b
-bx + dn dx
&zTiT
Smdx
I
(2n + 3)P
dx
(px + 9)” &z-i
14.118
2(px + q)n+ l d&T?
dx
dx
(px + qp-’
AND
~GzT
J/K
&ln(dGFG+~)
dx
ZI
(ax + b)(w + q')
i
14.121
xdx
(ax + b)(px
=
+ q)
dbx
+ b)(px
UP
+ 4
b + w
--x&T-
dx
(ax + b)(w
+ q)
dx
dx
(px + q)n-1
&-TT
INDEFINITE
64
INTEGRALS
dx
14.122
(ax + b)(px + q) dx
.
14.123
.('
j/sdx
=
=
(ax + b)(px + 4
‘@‘+
y(px+q)
+ vj-
(ax+;(px+q)
2&izi
14.124
(aq - W d%=i
lNTEGRALS
14.125
s--$$
=
$I-'~
14.126
J-$$$
=
+ In (x2 + a2)
14.127
J$$
=
x -
14.128
s&
=
$
‘4-l
J
x2(x?+
14.131
J
x3(x?+a2)
14.132
J
(x2d;Ga2)2
14.137
14.138
($2)
-
$ln(x2+az)
+3 tan-l:
2a2(xf+
S
14.140
S
(~2
14.141
S
dx
x(x2 + a2)”
.
(x2+ a2)"
-~
x
-- 2:5 tan-l:
2a4(x2 + a2)
1
2a4x2
1
2a4(x2 + u2)
2n - 3
+ (2n- 2)a2
X
=
2(n - l)a2(x2 + a2)%-*
xdx
S
dx
(x2 + a2)n-1
-1
2(n - 1)(x2 + a2)n-1
+ a2)n=
dx
1
2(12 - l)a2(x2 + uy--1
=
xm dx
dx
S x9z2+a2)n
+ &3 tan-':
a2)
--- 1
a4x
dx
S x2(x2 + c&2)2 =
dx
+ a2)2 =
S x3(x2
(x2d+za2)n
14.143
1
2a2x2
-=
14.139
14.142
six
=
=
x’ + a2
a tan-13c a
--30
INVOLVtNO
S
=
=
xm--2 dx
(x2
+
a2)n-l
-
a2
+ $
S
dx
1
2 S 33x2 + a2)n--1
S x(x2 + a2)n-1
x*--2 dx
(x2 + a2)"
--
1
a2 S
dx
xme2(x2 + a2)”
INDEFINITE
:INTEORAES
I.
14.144
14.145
s
~
x2 - a2
xdx
14.147
s m--
14.150
14.151
14.156
14.157
14.158
s
x2(x2 - a2) =
s
x3(x2-a2)
s
(x2?a2)2
s
(x2 - a2)2
s
(x2--2)2
dx
dx
14.162
=
__ 1
2a2x2
=
2a2(sta2)
=
-1
2(x2-a2)
=
2(xFTa2)
=
2(x2 - a‘9
xdx
Lln
-
z
~~3
(
>
x2 dx
'
+
-a2
x3dx
(,Zya2)2
&ln
+ i In (x2 - a2)
dx
s
x(x2 - a2)2
s
x2(x2-a2)2
=
dx
S
S
=
dx
-
---
1
xdx
=
dx
u2)n
-
--x
$5'"
2n - 3 s
(2~2 - 2)a2
dx
(x2
-
a2p-
1
-1
2(n - 1)(x2 - a2)n--1
S - =
=S
S --a?)"
S
S
x(x2
+
2(n - 1)u2(x2 - a2)n-1
a2)n
(X2-a2)n
2a4(xi-a2)
2~~4x2
=
dx
(x2 -
--
=
x3(x2-a2)2
s
14.161
+ $ In (x2 - a2)
dx
x(x2 - a2) =
14.159
14.160
$
s
14.154
14.155
;
Jj In (x2 - ~22)
x3 dx
14.152
14.153
z2 > a2
x2 dx
s n--
14.149
=
65
ix2 - a’,
1
- a coth-1
or
m=
14.146
14.148
INVOLVlNO
dx
*
INTEGRALS
-1
2(n - l)dyx2 - dy-1
x77-2 dx
xm dx
-
1
az
S
x(x2-
S
- u2p-S
a?
a2)n--1
dx
xm--2 dx
(x2
(x2-a2)n-1
dx
Xm(X2qp=
1
dx
,z
xm-2(x2
+
a2
(x2-a2)n
1
xm(x2-
dx
u2)n-l
INDEFINITE
66
tNVOLVlNO
IWTEGRALS
14.163
S
~ dx
a2 - x2
14.164
S
__
a2 - x2
14.165
S
g-z-p-
14.166
Sm
14.167
S
x(a2 - 22)
14.168
S
22(d
=
= ---2
dx
S
-
S
$ In (a2 - x2)
=
22)
22
x3(,Ex2)
=
-&+
&lln
(
dx
22>
-
5
2a2(a2 - x2)
=
2(a2--x2)
=
2(Lx2)
-
a2
2(&-x2)
+ i In (a2 - x2)
1
x dx
(a2 -
__
a2
=
x2)2
S
22 dx
(&-x2)2
14.173
S
(CL2- x2)2 =
14.174
S
14.175
S
14.176
S
14.177
S
(a2 -dx x2)n
S
(a2 - x2)n =
14.179
S
x(a2
14.180
S
(,2-x2p
14.181
i tanh-I$
dx
14.172
14.178
xz<aa
- f In (a2 - x2)
x2
x3 dx
(a2-x2)2
14.171
or
u~--~,
x2 dx
14.169J
14.170
=
x dx
INTEGRALS
x3 dx
=
-
5”’ dx
S
(a2
-
dx
=
dx
1
-
x‘p
(2n2n - 2)a2
3
+
l)a2(;2-x2)n-l
1
2(n - l)(a2 - x2)n-1
xdx
dx
qn-
2(n - l)a2(a2
a2
xm -2dx
S
j- xmc,~xp)n
=
+2s
- x2)n-1
(a2 - x2p
S
+f
x(u2 - xy--1
x*-2dx
-
xm(a2?z2)n-~
s
(a2-
x2)n-l
+$f
x--$-x2)n
x2)n-l
INDEFINITE
14.182
In (x + &&?)
INTEGRALS
or
sinh-1s
a
S
x dx
___
~~
S
lfzT-2
S
~I2xz
14.183
14.184
II
x2 dx
-- a2
2 In(x+@Tz)
2
dx
x3
14.185
x 7 2 +a
=
(x2 + a2)3/2
3
=
-
a2&GZ
14.186
S
14.187
S
=
.2&F&i
s
14.189
14.190
14.191
14.192
14.193
14.194
=
dx
14.188
-
J/Xa22
~~
+ k3
-2a2x2
x3~~5
S
S
s
a+&3T2
In
X
+ $l(x+~W)
xdmdx
(x2 + a2)3/2
=
x%jmdx
3
=
ad-g-q
dx
=
S
x(x2 + a2)312
4
(x2 + a2)5/2 _
5
a2x&T2
&T
S-dx
S
a2(x2 + a2)3/2
3
= --&G-G
+ ln(z+drn)
&s-T-z
X
dx
14.196
S
(x2
14.197
s
(%2
14.198
.f
(x2
s
(x2
s
x(x2 +
S
x2(x2
dx
+
x3(x2
dx
+ a2)3/2
14.199
14.200
14.201
14.202
a2)3/2
x dx
+
a2)3/2
x2 dx
+
x3
a2)3/2
dx
+ a2)3/2
=
= &is
= d&
=
+ ln(x
@TTP
1
a2)3/2
a2)3/2
+ d&i7)
im+a2
dx
S
a-l-&372
- $a In
x3
+
a4
sln(x+j/~)
8
=&qgwalIn
s
X2
14.195
>
=
=
-
a+JZ2
In
2
(
~~
- ~ a4x
-1
=
f
a2&SiZ
2a2x2&GT
x
-
a4&FS
-
3
2a4&FiZ
3
+ s5ln
a+&-TS
2
INDEFINITE
68
14.203
14.204
14.205
14.206
14.207
14.208
14.209
14.210
14.211
14.212
14.213
S
S
S
S
S
S
S
(x2
+
x(x2
a~)312
+
dx
dx
u2)3/2
x(x2
=
+
3&q/~
u2)3/2
4
(x2
=
+
+
~2)3/2
ds
=
x3(x2
+
u2)3/2
dx
=
u2)5/2
x(x2
+
u2)5/2
_
+
u2)3/2
dx
(22
=
+
(x2+
ds
=
u2)3’2
~2(~2
+
U2)3’2
x3
dx
=
(x2 +
-
-
--
u4x@TF2
~~ln(~+~2xq
16
~2)5/2
CL+@-TT?
+ u2@T2
-
x
a2)3/2
a3 In
x
>
+ 3a2 ln (x + q-&-T&)
2
U-kdlXS
2x2
x
S
In (x + j/277),
S
u2)3/2
5
2
s
~ x2 dx
&G=z
+
-
_ (x2 + u2)3’2 + 3x-
x2
(x2
+
24
~247’2
3
X
(x2 + UT’2
u2x(x2
6
7
(x2
+~a4ln(x+~2TTq
8
5
x2(x2
’
+
INTEGRALS
5 P--x-a
=
2
x3dx
s G= 1
5
asec-l
x2- u2
X
I
U
I
14.214
14.215
14.216
14.217
14.218
14.219
14.220
@=2
S
=
x3(&
s
dndx
S
Sx2@73
S
=
xda~dx
,“d~
s-dx
+ k3 see-l xU
I I
2u2x2
x
=
dx
=
dx
=
7
x2-a
(x2
_
-$ln(x+dm)
u2)3/2
3
x(x2
cAq/m~
- a2)3/2
+
4
cx2
-
~2)5/2
+
8
~2(~2
-
5
=
dm-
~2)3/2
3
a see-l
I;1-
-- “8” ln(x
+ +2TS)
>
INDEFINITE
INTEGRALS
69
14.224
14.225
14.226
14.227
14.228
14.229
S 22 dx
S
S
S
S
(~2 -
a2)3/2
=
x3 dx
(22 - a2)3/2
=
-~
-1
a2@qp
=
dx
z2(s2
-
lJZ2
a2)3/2
=
-_
x3(x2
-
(~2 -
x
a+iGZ
&)3/z
&
x(x2 - a2)3/2
4
-
z
x(52 - a2)3/2 dx
(x2
=
-
14.236
x2(99 -
a2)3/2
x3(52 -
a2)3/2
S
dx
2(x2 -
=
a2)5/2
14.238
dx
(22 -
=
a2)7/2
a2x(x2
+
-
14.240
az(x2
-
In (5 + &372)
a4x&FS
16
-
a2)5/2
@2 _ a2)3/2
S
S
S
X
(x2 _ a2)3/2
dx
=
tx2
-
a2)3'2
-
a2da
+
a3 set-'
c
3
dx
=
-
(x2
,jx
=
_
(x2;$33'2
I
-xa2)3'2
+
3xy
+
"y
_
ia
I
ln (1 + da)
X2
@2 -
a2)3/2
_
ga
sec-'
x3
Sda& =
lNVC)LVlNG
<%=??
sin-l:
xdx
____
= -dGi
___x2 dx
).lm
x3 dx
____
S
sjlzz
[El
a
x 7 a-x
=
-
=
(a2 - x2j312 _
3
2
a2dpz3i
a+&KG
X
14.243
:
Ia I
5
7
14.241
14.242
a2)3/2
24
@G?
14.239
see-l
+ :a4
8
+
6
1NtEORAtS
14.237
3a2x&iF2
2a5
a!2)5/2
S
14.235
3
--
5
14.233
14.234
3
=
*
I
2
IaI
-
a4x
1
a2)3/2
S
14.232
1
-- a3
set-1
dx
14.230
14.231
- dx2aLa2
GTZ-
dx
4x2 - a2P2
+ ln(x+&272)
&z
dx
Sx743x5
-~
2a2x2
-
&3 In
a + I/-X;
5
+ $
In (z + $X2 - a2 )
INDEFINITE
70
14.244
14.245
+
xqTF-2
s
dx
x+s-?5
s
14.249
&AT
s ~ x2
14.250
S~
14.252
14.253
14.254
14.255
14.256
14.257
14.258
14.259
14.260
14.261
dx
-
x(a2-
=
_
a2(a2-
=
x2 dx
(a2 ex2)3/2
=
*
x3 dx
(a2-x2)3/2
=
daz_,Z+d&
g
a
a+@=-2
(
1
_sin-1:
x
a
+
2x2
xdx
(,2mx2)3/2
sin-l
8
x2)3/2
&GT2
-~
x3
+ g
3
~~-CLln
=
a+@=2
&In
(
X
>
X
>
X
.3Lz2
&A?
-
x2)3/2
-
a2&z
=
2
sin-l-
dx
a
a+&GS
i31n
(
diFT1
dx
x2(a2-x2)3/2
S
a2xF
8
dx= _~
x(a2-
+
5
dx
@2ex2)3/2
s
x2)3/2
4
(a2 - x2)5/2
=
Wdx=
S
S
S
S
S
-x2)3/2
3
=
@=z
-dx
S
-ta2
=
x3dmdx
s
14.248
14.251
sin-l:
s
14.246
14.247
$f
INTEGRALS
=
x
614x
dx
a4&iGz
-1
x3(a2-x2)3/2
=
3
+
x(a2
-
&51n
2a4&FG
2a2x2@T2
S($2 - x2)3/2 dx=
Sx(&-43/2& = Sx2(& - &)3/2 ,&= S x2)3/2 dx=
-
+
3a2x&Ci3
8
x2)3/2
4
a+@?
(
X
>
ia4 sin-l:
+
(a2-x2)5/2
s
(a2 -xx2)3'2
14.263
S
14.264
s
(a2-
x2)3/2
-
x2)5/2
+
a2x(a2--2)3/2
6
x2)7/2
=
(a2 -3x2)3'2
dx
=
-(a2-x2)3/2
+
_
a2(a2-
=
+
x2)5/2
a2dm
3x&z%
-
2
a3 ln
_
(a
+ y)
;a2sin-1~
a
_
“7
+ gain
a+&PZ
X
.
+ igsin-l;
5
_
_ ta2 ;x;2)3’2
a6
16
X
dx
a4xjliGlF
24
7
dx
x2
la2 -x;2)3’2
x(a2
(a2 -
x3(&2 -
14.262
5
>
x
INDEFINITE
INTEOiRALS
INTEGRALS
71
ax2 f bz + c
LNVULWNG
2
14.265
s
&LFiP
dx
bx + c
ax2+
=
2ax + b - \/b2--4ac
$-z
If
results
14.268
14.269
14.270
14.271
14.272
14.273
14.274
14.275
s
xdx
ax2 + bx + c
=
&
s
x2 dx
ax2 + bx + c
=
--X
a
s
ax2-t
x”’ dx
bx+c
S
s
dx
+ bx + c)
xz(ax2
S
S
S
S
S
xn(ax2
ax2 + bx + c
(
dx
+ bx + c)
14.277
14.278
14.279
X2
1
=
-(n
- l)cxn-l
-- b
c
b
2c
--
( ax2 + bx + c )
&ln
2ac
~“-1
dx
ax2 + bx + c
I b2 - 2ac
23
x”-l(ax2
(4ac -
x dx
(ax2 + bx + ~$2
=
- (4ac -
=
2c
(b2 - 2ac)x + bc
f4ac - b2
a(4ac - b2)(ax2 + bx + c)
=
- (2n - m - l)a(ax2
2ax + 6
2a
+b2)(ax2 + bx + c)
4ac - b2, f
x”’ dx
+ bx + c)n--l
(n - m)b
(2n - m - 1)a s
dx
ax2 + bx + c
S
xnp2(ax2
dx
ax2 + bx + c
S
S
dx
ax2 + bx + c
(m - 1)~
(2n-m1)a s
’
~“‘-2 dx (ax2 + bx + c)n
xm-1 dx
(ax2 + bx + c)fl
+bx+c)n= $S(a392f~~3~~)“-I - $S(ax:";;:!+
-iS
S
S
S
S
S
S
.I
S
Sx~-~(ccx~
s
x2n--1 dx
(m2
dx
x(ax2 + bx f
x2n-2
dx
(ax2 + bx -t- c)n
c)~
dx
x2(ax2 f bx + c)~
xn(ax2
dx
+ bx + c)
dx
ax2 + bx + c
b
-4ac
xWL-l
(ax2 + bx f CP
S
dx
-- a
+ bx + c)
c
S
=
$2 dx
use
S
dx
(ax2 + bx + c)2
(ax2 + bx + c)2
b = 0
dx
ax2 + bx + c
J
_ 1
cx
>
bx + 2c
b2)(ax2 + bx + c)
If
dx
ax2 + bx + c
s
x”-2
dx
-- b
ax2 + bx + c
a
s
60-61 can he used.
dx
ax2 + bx + c
b2 -
X2
$1,
=
a
:i
on pages
+ T
C
--
(m-l)a
=
s
&ln(ax2+bx+c)
x?T-l
=
dx
+ bx + c)
x(ax2
In (ax2 + bx + c) - $
-
14.276
i( 2ax + b + dn
b2 = 4ac, ax2 + bx + c = a(z + b/2a)2 and the results
on page 64. If a or c = 0 use results on pages 60-61.
14.266
14.267
In
dx
f bx $
1
-2c(ax2 + bx + c)
=
1
=
- cx(ax2
+ bx + C)
b
2c
-- 3a
c
dx
+$
(ax2 + bx + c)2
dx
-- 2b
(ax2 + bx + c)2
c
1
c)~
=
-(m
- l)cxm-l(ax2
_ (m+n-2)b
(m - 1)~
+ bx + c)n--l
-
(m+2n-3)a
(m - 1)c
dx
+ bx + c)n
dx
x(ax2 + bx + c)
x(6x2
dx
+ bx + c)2
x-~(ux~
dx
+ bx + c)”
72
INDEFINITE
INTEGRALS
In the following
results if b2 = 4ac, \/ ax2 + bx + c = fi(z
+ b/2a) and the results
be used. lf b = 0 use the results on pages 67-70. If a = 0 or c = d use the results
$
ax
14.280
=
ax2+bx+c
a
In (2&dax2
-&sin-l
on uaaes 60-61 can
on pages 61-62.
+ bx + e + 2ax + b)
(J;rT4ic)
or
&
sinh-l(~~~c~~2)
14.281
14.282
x2 dx
s,
ax2+bx+c
14.283
dx
14.284
=
-
ax2 + bx + c
14.285
ax2+bx+cdx
(2ax+
=
14.286
b)
ax2+
4a
bx+c
+4ac-b2
16a2
14.288
14.289
14.290
14.291
14.292
14.293
=
6az4a25b
bx+c
(ax2 + bx + c)~/~ +
“““,,,“”
J
d ax2f
bx+c
dx
ax2+bx+c
S“
X
ax2+bx+c
X2
S
S
ax2
Scax2 x2
S+x2+%+c)3’2 = cdax2 : bx+e+: SJ
s
S,
S
dx
(ax2 + bx + c)~‘~
2(2ax + b)
(4ac - b2)
x dx
(ax2 + bx +
dx
+ bx +
x2(aX2
ax2 + bx + c
2(bx + 2c)
~)3’~
(b2 - 4ac) \/
43’2
a(4ac - b2)
+ bx + c
(2b2 - 4ac)x
+ 2bc
dx
+ bx +
c)~‘~
=
ax2 + 2bx + c
- &?xdax2 + bx + c +
2c2
(ax2 + bx + c)n+1/2dx
=
dx
1~x2 + bx + c
-- 3b
14.295
ax2 + bx + c
axz+bx+c
x
14.294
.
(ax2 + bx + c)3/2
b(2ax + b) dp
~
ax2+
3a
8a2
dx
- b(4ac - b2)
=
14.287
dx
8a
ax2+bx+c
S +ifif+
dx
(QX~
axz+bx+c
b2 -
26
2ac
Scax2
dx
+ bx +
43’3
dx
x
ax2+bx+c
(2ax + b)(ax2 + bx + c)n+ 1~2
4a(nf 1)
+ (2% + 1)(4acb2)
(a&+
8a(n+ 1)
S
bx + c)n-1’2dx
4312
.
INDEFINITE
14.296
14.297
S
s’(ax2-t
x(uxz + bx + C)n+l/z dx
=
(ax2 + bx + C)n+3'2
cq2n+ 3)
2(2ax
dX
bx + ~)n+l’~
=
dx
+ bx + ++I’2
x(ux2
s
73
.
_ $
(ax2 + bx + ~)~+l’zdx
s
+ b)
(2~2 - 1)(41x - b2)(ax2 + bx + +--1/z
8a(n1)
dx
(2~2 - 1)(4ac - b2). (‘ (61.x2 + bx + c)n--1E
+
14.298
INTEGRALS
1
=
(2~2 - l)c(ux’J
+ bx + c)n--1’2
dx
JPJTEORALS
Note
14.299
14.300
14.301
14.303
14.304
14.305
that
14.308
14.309
14.310
dx
~
s
involving
=
+
X3
x2 - ax + c-9
+ 1
(x + c-42
=
__
=
x3 + CL3
$ In (x3 +
ClX
s
.(
s
x2(x3
u3)
=
'(z3yu3)2
'
1
-+
u3)2
%(X3
+ a3)2
s
x2(x3
dx
+
+ &In
14.312
=
1 -
--
=
CL62
xdx
+ u4
=
x4
S
x3
dx
~
x4 + a4
3u5fi
(x + a)2
x2
xm-3
-
m-2
a3
~
x3
-1
1)x+-’
c&3@-
-
1
In
4u3fi
&
-L
4ufi
14.314
2x-u
a \r 3
x2 - ax + a2
2x +
=
tan-l
In
-4-.---
3u6 s
3a6(x3 + u3)
xm-2
-
=
a3)
=
~
tan-l
3utfi3 tan-’
3
&,3(x3 + as)
=
u3)2
x4 + a4
S
2
-
1
u3
dx
+
F
- 3(x3 + US)
dX
I'
+3tanP1
x2 - ax + a2
+ &n
a3)
x2 + axfi
x2 - uxfi
x3
x dx
+ u3
[See 14.3001
dx
+ a3
dx
-2
JNTEORALS
14.311
-
(x + a)2
=
x(x3+u3)
(xfcp
x2
3a3(x3 +
=
x-’ dx
s
G-4 In
3u3(s3 +a3)
dx
x9x3+
dx
s
1
s
s
43
x2 - ax + u2
1
-
x2 dx
(x3+
+
2x-u
7
tan-l
X
=
s
x3
a by --a.
14.302
~3)
a32
xdx
(x3 + c&3)2 =
~
(ax2 + bx + c),+ l/i
3ea+ a3
a\/3
x2 dx
s
x3 - u3 replace
2”~ s
x2 - ax + cl2
u3
~ x dx
x3 + a3
s
14.306
14.307
for formulas
JNVOLVING
dx
--
x(ux2 + bx + c)“-~‘~
s
u3
s
+
xn-3(x3
INVOLYJNG
+ a2
c?+* a*
1
--
u3)
tan-1
2aqi
+ c&2
-!!tC-LT
22 - CL2
$
x2 - axfi
+ a2
x2 + ax&
+ u2
$ In (x4 + a4)
--
1
2ckJr2
tan-1
-!!G!- 6
x2 - a2
a
74
14.315
14.316
INDEFINITE
INTEGRALS
dx
x(x4 + d)
s
s
dx
x2(x4
14.317
+ u4)
+-
=
1
2a5&
dx
x3(x4
.
+ a4)
=
14.322
14.323
14.324
14.325
14.326
14.327
14.328
14.332
dx
.I’ x(xn+an)
fs
=
S
xm dx
(x”+
c&y
I’
dx
xm(xn+
an)’
xn + an
‘, In (29 + an)
=
s
xm--n dx
(xn + (yy-l
1
=
2
x”’ dx
s-- (xn - an)’
14.333
14.334
&nlnz
=
=
an S
s
-
an s
x”’ --n dx
(xn + an)T
dx
xm(xn + IP)~--~
xm--n dx
(~“-a~)~
1
an
-s
xm--n dx
+ s
(xn-an)r-l
=
S
dx = m..?wcos-~
!qfzGG
m/z
dx
xmpn(xn + an)r
tan-l
CiXfi
___
x2 -
a2
INDEFINITE
14.335
xp-1 dx
INTEGRALS
75
x + a cos [(2k - l)d2m]
1
ma2m-P
I‘----=xzm + azm
a sin [(2k - l)r/2m]
x2 + 2ax cosv
where
14.336
xv- 1 dx
X2m
s
-
m-1
1
a2m
=
2ma2m-P
PI2
cos kp7T In
km
sin m
x
(’
2*
ka
x -
tan-l
+ a2
a cos (krlm)
a sin (krlm)
k=l
+
.
2ax ~0s;
m-1
1
where
x2 -
m
k=l
-&pFz
14.337
+ a$!
0 < p 5 2m.
{In (x - 4
+ (-lJp
>
ln (x + 4)
0 < p 5 2m.
x2m+l
xP-ldX
+ a2m+l
2(-l)P--1
(2m + l)a2m-P+1k?l
=
m
sin&l
x + a cos [2kJ(2m
a sin [2krl(2m
+ l)]
+ l)]
m
(-1p-1
(2m + l)az”-“+‘k?l
-
tan-l
cossl
In
x2 + 2ax cos -$$$+a2
+ (-l)p-l
In (x + a)
(2m + l)a2m-P+
l
where
14.338
O<pSim+l.
xp-1
s
x2m+l
-
dx
a2m+l
77,
1
(zrn+
x -
2kpr
kzlSin
l)a22m-P+l
2m + 1 Iian-’
a cos [2krl(2m
a sin [2k7;/(2m
+ l)]
+ l)]
>
m
+
s
14.340
sinaxdx
=
=
%sinax+
14.342
=
(T-
siyxdx
=
14.345
14.346
14.347
=
-$)sinax
sin ax
s
dx
+ a
S Ydx
=
=
sin2 ax dx
+ (f-f&--$)
cosax
5*5!
X
S sin ax
xdx
S sin ax
s
cos ax
3*3!
dx
sin ax
ax-(aX)3+(a2)5-...
s
sinx;x
lNVOLVlNC3
x cos ax
___
a
y-
14.341
14.344
2ax cos
-- cos ax
a
=
‘ssinaxdx
14.343
x2 -
O<pS2m+l.
INTEGRALS
14.339
In
In (x-a)
(2m + l)a2m-n+1
+
where
cos&
(2m + 1)ta2m-p+ ‘,li,
=
: _ sin 2ax
2
4a
[see 14.3731
a2
76
INDEFINITE
14.348
14.349
x sin2 ax dx
sin3 ax dx
s
X2
=
-
x sin 2az
4a
-
4
=
_ cos ax
-+-
=
3x
8
14.351
~
=
- 1. cot ax
a
__ dx
sin3 ax
=
-
14.352
s
14.353
14.354
1 -
dx
sin ax
sin 4ax
32a
cos ax
2a sin2 ax
sin px sin qx dx
s
3a
sin 2ax
-+-t
4a
sin4 ax dx
cos 2ax
8a2
--
cos3 ax
a
14.350
INTEGRALS
sin (p - q)x
2(P - 4)
=
=
_ sin (p + q)x
[If
2(P + (I)
p = *q,
see 14.368.1
‘, tan
14.355
14.356
14.357
14.358
p tan *ax
14.360
14.361
14.362
dx
p + q sin ax
I‘
ad&2
tan-’
a&2
In
+ q
@q
=
ptan+ax+q--
If
p = *q
see 14.354
s
dx
(p + q sin ax)2
If
p = *q
( p tan +ax
-
+ q + dm
and 14.356.
q cos ax
a(p2 - q2)(p + q sin ax)
=
see 14.358
t--J---p2
and 14.359.
dx
p” + q” sin2 ax
p2 -
s
14.364
14.367
14.368
xmsinaxdx
1
.I’
s
=
sijlnuxdx = sinn ax dx
s
=
-’
In
m cos ax
mxm--l
+
a
sin ax
(n - 1)xn-l
_ sinn--l
tanax
tan
ap&2
dx
q2 sin2 ax
_ 2wdF7z
14.366
dx
p + q sin ax
s
1
14.365
1
q2
-1 dm
14.363
)
+a
- dx
sinn ax
=
- cos ax
a(n - 1) sin”-’
ax
~ xdx
sinn ax
=
-x cos ax
a(72 - 1) sinn--l
ax -
dn
tan ax +
( dm
tan ax -
sin ax
a2
=$
n-1
ax cos ax
an
P
m(m - 1)
-7
dx
s
xmp-2 sin ax dx
[see 14.3951
s
+-
72-l
n
s
sinnp-2 ax dx
dx
sin”-”
ax
az(n - l)(n
1
- 2) sinnez
ax
+-
n-2
n-1
xdx
sinnP2 ax
.
INDEFINITE
14.369
14.370
' cosax
*
xcosaxdx
s
- xzcosaxdx
14.372
'
cosax
x3
dx
Fdx
a
=
$,,,a.
=
(T---$)cosax
";,'
dx
+
(axY
--GE-=
14.376
=
- cos ax _ a
2*2!
14.377
x dx
cos ax
-
14.378
14.379
14.380
14.381
s
=
=
cos4
ax dx
=
dx
=
14.383
cos ax cos px dx
14.384
14.385
dx
1 - cosax
s
x dx
1 - cos ax
=
14.387
xdx
1 + cos ax
=
14’389
S
dx
cos ax)2
dx
(1 + cosax)2
=
sin3 ax
3a
sin (a - p)x
2(a - P)
-- x cot E
a
2
=
dx
1 + cosax
_
...
cos 2ax
8cG
+
sin (a + p)x
2(a + P)
=
14.386
JtI
+
=
s
14.388
+2
tan
ax
a
-
ax
x sin 2ax
-+4a
-sin ax a
14.382
cos3
=
sin 2ax
4a
f+-
=
ax dx
COG ax
[See 14.3433
En(ax)2n
cos3
___dx
dx
...
(2n-k2)(2n)!
x co9 ax dx
s
'y
S'
=
co532ax dx
s
-+(axF
6*6!
+ 4*4!
$ In (see ax + tan ax)
cos ax
S
--kd4
Ins--
X
14.375
sin ax
+ ($-$)sinax
=
s
14.374
77
-cos ax
~x sin ax
a2
+
a
=
14.371
14.373
=
dx
INTEGRALS
=
=
2
+ - In sin ax
a2
2
[If
a = *p,
see 14.377.1
78
INDEFINITE
I
INTEGRALS
ad-2tan-’
14.390
dx
p+qcosax
s
=
dt/(p - Mp
tan *ax
&j&2
14.391
dx
(p + q cos ax)2
s
=
a($
dx
14.392
=
! tan &ax -
tan-l
s
tan-l
p2-
ptanax-dm
( ptanax+dv
14.395
14.396
14.397
14.398
14.399
14.400
14.401
14.402
14.403
14.404
14.405
14.406
14.407
ydx
=
-
s
s
co@ ax dx
S
S
S
s
S
S
co@ ax
xdx
COP ax
-
sinax
a
cos ax
-n-1
(n - 1)x*- 1
cos ax -
sin ax cosn--I ax +?Z-1
n
an
=
=
x sin ux
a(n - 1) COP--I ax
=
-
sin px cos qx dx
=
_ cos (P - q)x
VP - 4
sinn ax cos ax dx
=
COP ax sin ax dx
=
dx
S sin ax cos ax
s
xm-2
S
sinn + 1 ax
(n + 1)~
-cosnflax
(n + 1)a
=
co@-2ax
dx
dx
COP-2 ax
1
- l)(n - 2) cosnP2 ax
a2(n
+n-2
cos (P + q)x
VP + 9)
_
[If
n = -1,
[If
n = -1,
sin 4ax
32a
X
- 8
dx
sin2 ax cos ax
=
-
S
dx
sin ax ~052 ux
=1 ;lntan
1
a sin ax
A In tan
a
-2cot2ax
a
y
+ &
cos ax dx
[See 14.3651
=1 a In tan ax
=
a2
2a
S
dx
S sin2 ax cos2 ax
mtm - 1)
sin2 ax
cosax dx
sin2 ax cos2 ax dx
>
sdx
S’
sin ax
+n-2
a(n - 1) co@--I ax
n-l
-s
.-AL=
s
xm sin ax
mxm--l
+a
a2
=
[If p = *q see
14.388 and 14389.1
E
In
1
xm cos ax dx
s
dx
p + q cos ux
P
42 - P2 s
q2
I WdFT2
14.394
- P)
dn7
ap
=
-PI
d(q + dl(q
--
[If p = *q see
14.384 and 14.386.1
P tan ax
w/FS
+
dx
p2 - q2 cos2 ax
+ d(q + dl(q
q sin ax
- $)($I + q cos ax)
1
s p2 + q2 cos2 ax
14.393
In
+ 4 tan ?px
see 14.440.1
see 14.429.1
n-1
-s
xdx
cosn-2 ax
INDEFINITE
14.408
INTEGRALS
79
s
14.409
s
cos ax(1
dx
C sin ax)
=
.
sinax(1
dx
2 cosax)
-
S
dx
sin ax rfr cos ax
14.410
14.411
14.412
14.413
14.414
s
L
a&
sin ax dx
sin ax * cos ax
=
I
cos ax dx
sin ax f cos ax
=
2:
sin ax dx
p+qcosax
=
14.416
cos ax dx
p+qsinax
=
14.417
14.4 18
1
f sin ax)
2a(l
1
* cos ax)
k
=
14.415
2a(l
i
- $
$
In tan
T $a In (sin ax * cos ax)
+ +a In (sin ax C cos ax)
In (p + q cos ax)
In (p + q sin ax)
S
sin ax dx
(p + q cos axy
=
aq(n - l)(p
1
+ q cos axy-1
s
cos ax dx
(p + q sin UX)~
=
aq(n - l)(p
-1
+ q sin UX)~--~
dx
p sin ax + q cos ax
14.4 19
=
adi+
q2 In tan
ax + tan-l
2
(q/p)
2
dx
p sin ax + q cos ax + T
14.420
p + (r - q) tan (ax/z)
a&2-p2-q2tan-1
=
T2
1
ln
aVp2 + q2 - ~-2
If
14.421
I‘
r = q see 14.421.
If
q + p tan 5
=
ax + tan-’
2
=
1
In
2apq
- sinmP1
14.425
I‘
(q/p)
dx
p2 sin2 ax + q2 cos2 ux
dx
p2 sin2 ax - q2 COG ax
14.424
q2
dp2 + q2 - r2 + (r - q) tan (ax/2)p + dp2 + q2 - r2 + (T - q) tan (ax/2)
dx
S
-
p -
psinax+qcosax*~~
14.423
p2
~~ = p2 i- q2 see 14.422.
dx
p sin ax + q(1 + cos ax)
14.422
-
sinm uz COP ax dx
p tanax
- q
p tan ax + q
ax co@+ l ax
m-l
+a(m + n)
mfn
sinm-2
ux cosn ax dx
=
sin”
+ l ax cosnwl
a(m + n)
ax +
n-l
m+n
s
sinm ax COS”-~
ux
dx
80
14.426
INDEFINITE
_r’s
dx
=
INTEGRALS
sinm-l ax
a(n - 1) co??--1 ax -
m-l
-n-l
sinm + 1 ax
a(n - 1) cosn--1 ax
m--n+2
n-l
- sinme ax
I a(m - n) cosnel ax
- cosn-l
ax
a(n - 1) sinn--l
14.427
S
Ed,
ax
a(n - 1) sinn--l
COP-~
14.428
S
ax
14.430
14.431
14.432
14.433
14.434
14.435
14.436
14.437
14.438
14.439
S
S
S
S
tan ax dx
1
=
-ilncosax
tan3 ax dx
=
tan2 ax
2a
+ $ In co9 ax
=
edx
dz=
tan ax
ydx
s;;;”
2”zx dx
S
‘;?&l,az
dx
m+n-2
n-1
S
S
dx
sinm ax cosnw2 ax
sinm-2
dx
ax COP ux
tamuzc
ax
x
tarP + 1 ax
(n + 1)a
=
1 (ax)3
;Ei 1 3
=
I 2(ax)7
105
-2 tan ax + $ In cos uz - f
a
=
p2 + 42
PX
tan”-’
ax
(n _ l)a
+
Q
ah2 + q2)
-
S
I . . . + 22922n-
l)B,(ax)*~+'
(2n + 1) !
2*n(22n - 1)B,(ax)2n-1
(2n- 1)(2n)!
~(cLx)~
=
dx
+ q tanax
I (ax)5
15
(ad3
a~+~+~+-+
=
tann ax dx
s
dx
ilntanax
=
xtanzaxdx
Sp
z;;:;;z
i In sin ax
xtanaxdx
s
‘-, lnsec
tan ax
a
S
S
S
s
1
=
=
tann ax sec2 ax dx
s
INVOLVING
dx
tanzax
m-l
m-n
dx
-1
mtn-2
+
1 a(m - 1) sinm--l ax ~0.9~~~ ux
m-l
INTtkRAlS
14.429
+-
dx
sic”;;z;x
n-l
ax
c;;:;;x
S
S
1
+
~(72 - 1) sinmP1 ax cosn--l ax
=
sinm ux co@ a5
-- m-l
72-l
ax
I a(m - n) sinn--l
dx
m-l
m-n
.s
_ m-n+2
-coSm+lax
=
f-
sinme ax
cos”-!2ax
dx
S
In (q sin ux + p cos ax)
tann--2 ax dx
+ *”
+
...
INDEFINITE
14.440
14.441
14.442
14.443
14.444
14.445
14.446
14.447
14.44%
14.449
14.450
14.451
14.452
14.453
14.454
14.455
14.456
14.457
14.458
cot ax dx
s
s
s
=
=
-- cot ax a
cot? ax dx
=
- -
=
dx
cot ax
=
+%dx
qcotax
set ax dx
S
S
ax
(n-1)a
-
cotn--2 ax dx
i In (set ax + tan ax)
=
tan ax
sec3 ax dx
=
set ux tan ax
+ & In (set a2 + tan ax)
2a
se@ ax tan ax dx
x secax
=
S
x sec2 ax dx
a
=
se@ ux
na
-sin ax
a
dx
ydx
...
cot--l S
-
dx
-
Q
In (p sin ax + q cos ax)
a(p2 + 92)
p2’Tq2
-
.**
sin ax - g
=
S set ax
S
+ -$ln
--
=
(2n-1)(2n)!
sec2 ax dx
S
S
a
x cot ax
=
=
135
- -
-
22nBn(ax)2n--1
ax
=
+1
(2n+l)!
-~-!$%-i!?%..,-
dx
cotn ax dx
2w3n(ux)~~
ax
a2
dx
p+
ax
ax
1
=
x cot2ax
S
-iIncot
=
1 In sin ax
a
-cotnflax
(n + 1)~
=
--a Incas
zcotaxdx
x
cots ax
2a
cotn ax csc2 ax dx
sdx
81
i In sin ax
cotzaxdx
S
S
S
S
S
S
S
S
INTEGRALS
=
(ax)2 + (ax)4 + 5(ax)6
8
- 144
W2
lnx+T+-gg-f-
=
=
5(ax)4
E tan ax + 5 In cos ax
E,(ax)2n +2
+
Gl(ax)s
4320
**.
+ (2n+2)(2n)!
+
. . . + E,(ax)2”
2n(2n)!
+
.”
+
**’
INDEFINITE
82
14.459
14.460
dx
S
s
dz
P
Q s p + q cos ax
=x ---
q + p set ax
Q
se@ ax dx
INTEGRALS
secne2 ax tan ax
n-2
+n-1
a(n - 1)
=
se@--2 ax dx
s
;
1NTEQRALS
14.461
14.462
14.463
14.464
14.465
s
s
csc ax dx
csc2ax
S
csc3
s
s
14.466
=
k In (csc ax -
--
ax dx
=
- csc CL5
2a
csc
cot
+ z
1
UX
In tan T
_ cscn ax
na
=
a
,jx
=
$
ax
.l
14.467
c&x
-- cos ax
=
ar
$ In tan 7
a
CSC” ax cot ax dx
- x
=
ax
=
dx
csc r&x
cot ax)
cm az
cot
dx
-
INVOLVING
+
k$
+
!k$
+
. . .
+
2(22n-’
-
S
S csc2
S
?%!!?
dx
=
1)B,(ax)2n+’
+
. . .
(2n + 1) !
f
_ &
+ $? + !&I?$
+
... +
2’22’;;n-m1$$;‘2’-
’ +
...
5
14.460
14.469
14.470
x
ax dx
=
- ~x
=
E-I?
dx
q + p csc ax
s
Q
CSC” ax dx
=
S
sin-1
14.473
14.474
Ed%
=
s
sin-l
(x/a)
z &
a
dx
=
=
=
[See 14.3601
n-2
n-1
S
csc”-2
ax dx
IRZVRREiZ TR100NQMETRfC
a
x3
j- sin-l
z+-
z +
a
X&Z?
4
(x2 + 2a2) &K2
z +
9
(x/aj3
- sin-1
+
(x/u)
X
1 * 3(x/a)5
2.4.5.5
1 3 5(x/a)7
+ 2*4*6*7*‘7
l
l
a-kdG2
-
$l
-
2x + 2dm
X
2
14.476
fl&CtlONS
ZZ + dm
2*3*3
dx
+-
sin-l
5
14.475
dx
p + q sin ax
=
39 sin-1
In sin ax
S
P
5 sin-l
U
S
+ $
lNVotVlN@
‘xsin-lzdx
14.472
ax
a
CSC~-~ ax cot ax
a(n - 1)
-
INTEORALS
14.471
cot
sin-l
z
+
...
“’
INDEFINITE
14.477
cos-1 :dx
a
.(‘
=
zc,,-l~&
x cos-1%
-
INTEGRALS
@?2
=
cos-ls
_
xr a
a
39 cos-l
14.479
:
a
,&
cos-1 (x/a)
14.480
x
cos-;;xln)
14.481
=
cos-1
fj
;lnx
-
dx
=
_ cos-1 (x/a)
tan-1Edx
14.484
x tan-1
14.486
14.487
=
Edx
=
x2 tan-1
z dx
=
tan-~(xiu)
dx
=
-
a
&(x2+
(x/a)
x
&i72
-
dx
[See 14.4741
a+~~~
+ iln
z ( cos-1 xa)2
xtan-1E
2a2)
+
9
x
=
-5
4
sin-1
=
s
(x2
-
dx
ds
X
2x -
cot-‘?dx
14.489
x cot-’
=
a
zdx
2dz&os-'~
zIn(xzfa2)
a2) tan-1
(x/u)3
; _ 32
cot-* (x/u)
14.491
X
(x/a)
cot-1
14.492
x2
s
see-*z
a
x cot-l
=
52 cot-’ ; dz
x a
7
+ ~(xla)5
_ -(x/a)772
+ *.*
z + % In (x2 + a2)
4(x”
a2) cot-1 E + 7
+
=
;
dx
=
g In x -
dx
=
_
tan-’
(x/a)
dx
[See 14.4861
X
(x/a)
cot-'
X
dx
!
=
2 set-l
z -
a In (x + &?C3)
x set-*
z + a In (x + dm)
o<sec-*:<;
5 < set-*
2 see-lE - a 7x-a
14.494
>
.
14.488
14.493
i?3
83
S
x set-1
z dx
x2
see-* f +
x3
,secelz
s
x2 see-1:
a
ds
0 < set-1 z < i
=
z
14.495
2 < i7
-
=
X3
i
ysec-1
z +
t < set-*
2
ax&F2
6
ax&2G3
6
-
$In(x
-t $ln(x+da)
t
< T
+ dZ72)
0 < see-1
i
< set-11
i
< g
< T
INDEFINITE
84
14.496
set-l
(x/a)
X
.I’
dx
=
;1nx
dx
=
+ ; + w3
_ see-l
14.497
set-l
s
(da)
X2
14.499
s
s
* csc-1
2 dx
a
x csc-1:
a
x2 csc-1
5 < set-1
dx
dx
(x/a)
xcsc-1:
-
-5
E +
a 7 x-a
22
y csc-l
% -
2
csc-l
X
;
csc-1
a -
E ,
(dx)3
2-3-3
-5
14.506
s
=
_
dx
(x/u)
- csc-1
5
a
z < 0
< ;
dx
-5
I 1 ’ 3(a/x)5
204-5.5
+
1 ’ 3 5(a/x)7
2*4*6*7-T
l
(x/u)
=
< csc-1;
...
0 < w-1 z < ;
+
; < csc-1:< 0
___
mt1
s
Stan-l:
-
xm cot-1 f dx
=
-$+eot-l~
+ -&.I'=""
set-l
mfl
&Jsdz
(x/u)
0 < s,1:
< 5
i
< set-l%
< T
0 < csc-1
E < ;
=
xm+l
see-1 (x/a)
+ -
a
m+l
mS1
xm+l csc-1(x/u)
m+l
dx
+
-
=
xm see-1 z dx
< 0
X
x dx
a
' x"'tzsc-1:
< ;
+
=
xm+l
14.508
< ,se-1;
X
(x/a)
xm tan-1
< T
< csc-1
0 < csc-1;
Xlnfl
I'
t
=
X
14.505
< i
0 < csc-1;
uln(x+~~)
X2
2 csc-1
X
sin--l
...
=
f dx
X2
xm
+
ax
_ csc-1
.s
1*3*5(a/2)7
2-4.6-7-7
0 < set-lz
&ikS
+ aIn(x+@=2)
X
14.502
+ &GFG
x csc -1:
3
CSC-~
+
1~3(cLlX)5
2-4-5-5
=
X3
s * w-1
+
.
X
x3
3
14.501
(x/u)
.
_ sec-lx(xiu)
1
14.498
INTEGRALS
xm+l
csc-1
mfl
a
m+1
=
i
I
(x/a)
~ xm dx
s d=
S
xm dx
@qr
-;<cse-$<O
< 0
INDEFINITE
14.509
14.510
14.511
14.512
eaz dx
=
a
xeaz dx
=
e””
s
dx
Z2eaz
=
""
a
xneaz dx
$dx
=-
eaz
xn---+
a (
Inx
~
P + waz
S
dx
(p + qeaz)2
14.516
14.517
dx
S
n
a
I taxJ2
.
X
- P
S
14.519
S
14.520
. . . ~(-l)%!
an
S
14.522
&
1
+ WY
a&
-
$2
t ...
w
1
In (p + qeaZ)
2?em
Q
>
tan-l
eaz - jLjFp
In
eaz + &G&
e” sin bx ds
=
eaz(a sin bx - b cos bx)
a2 + b2
eaz cos bx dx
=
eQz(a cos bx + b sin bx)
a2 + b2
xem sin bx &
=
xeaz(a si~2b~~2b
xeax cos bx dx
=
xeax(a cos bx + b sin bx)
a2 + b2
eaz In x dx
=
14.523
S
14.524
S
integer
In (p + qeaz)
=
e”lnx
--a
S
n = positive
3*3!
‘OS bx)
_ ea((a2
S
14.521
if
a
___
1 2&G
14.518
-
--ssdx
n-l
1
adiG
peaz + qe-a.%
I taxj3
Z-2!
+
;+
=
dx
n(n - 1)xn-2
a2
nxnel
(n - 1)x”-’
=
>
xn--leaz
a S
+ la;,
-
dx
S
a2
-eaz
z
S
14.515
a
Pea2
--a
S
14.514
%2-&+Z
(
=
=
Fdx
1
a>
X--
a (
s
14.513
85
e""
s
s
INTEGRALS
1
5
a S
- b2) sin bx - 2ab cos bx}
(a2 + b2)2
_ eaz((a2 - b2) cos bx + 2ab sin bx}
(a2 + b2)2
dx
eu sinn bx dx
=
e”,2s~~2’,~
eaz co@ bx dx
=
em COP--~ bx
(a cos bx + nb sin bx)
a2 + n2b2
in sin bx - nb cos bx)
+
+
n(n
- l)b2
a2 + n2b2 S
n(n - l)b2
a2 + n2b2
S
eu sin”-2
em
bx dx
cosn--2 bx dx
86
INDEFINITE
INTEGRALS
HWEOiRA1S 1NVOLVfNO
14.525
14.526
14.527
14.528
14.529
14.530
14.531
14.532
14.533
14.534
14.535
14.536
s
14.538
14.539
=
S
S
S$Qx
xlnx
xlnxdx
=
xm lnx
dx
-
$1
=
2
nx-4)
--$ti
1
m+1
-
lnx
(
=
14.541
14.542
see 14.528.1
;lnzx
P
1+x dx
J
=
x ln2x
~Inn x dx
=
-lP+lx
s
dx
xln
=
x
-
lnnx
[If
In (lnx)
=
In x
+ 2x
n = -1
+ lnx
ln(lnx)
dx
xlnnx
=
+ $$
+ s
* .
-
n
m = -1
see 14.531.
S
S
S
Inn-1
=
x ln(x2+&)
In (x2 - ~2) dx
=
x In (x2 - u2) xm+l
=
sinh ax dx
x sinh ux dx
x2 sinh ax dx
+ (m+3!)~~x
x dx
In (x2 + ~2) dx
xm In (x2 f a9 dx
.*a
.
+ (m+2t)Iyx
n
xm+l Inn x -m+l
m+1
=
+
l
+ (m+l)lnx
xmlnnxdx
S
S
S
see 14.532.1
In (lnx)
=
xm dx
2x lnx
nfl
X
S
Sf&
S
S
S
-
-
In (x2*
m+l
INTEGRALS
14.540
[If m = -1
s
If
14.537
lnxdx
Inx
xm Inn-1
s
2x + 2a tan-1
&)
--
2
m+1
!NVOLVlNO
~
=
x cash ax -- sinh ax
U
=
u2
coshax
z
2x + a In
cash ax
a
=
x dx
-
$sinhax
S
Y$gz
sinh (cx
c-lx
+ a**
INDEFINITE
14.543
'14.544
14.545
14.546
14.547
14.548
14.549
14.550
sinLard
14.552
14.553
14.554
sinizax
*
i In tanh 7
xdx
sinh ax
=
1
az
ax
sinhz ax dx
=
sinh ax cash ax
2a
-
s
s
x sinha ax dx
,I'
dx
sinh2 ax
~
I‘
I
.(
cash 2ax
8a2
x2
4
a
px dx
sinh (a + p)x
%a+p)
=
p)x
aa - P)
sinh (a -
' sinh ax sin px dx
=
a cash ax sin px -
' sinh ax cos px dx
=
a cash ax cos px +
p sinh ax sin pz
a2 + p2
ax+p--m
qeaz + p + dm
1
s
(p +
=
ad~2
dx
S
p +
q sinh ax
=
dx
-
’ sinh” ax dx
dx
S sinhn ax
~ x dx
.I’ sinhn ax
xrn
cash
a
ux
--
m
a
+ -n-l
=
- cash ax
a(n - 1) sinhnP1
ax
=
- x cash ax
a(n - 1) sinhn--l
ax -
dx
=
I’
p + dm
tanh ax
p - dm
tanh ax
xm--l
sinhn--l ax coshax _ -n-1
n
an
- sinh ax
(n _ l)xn-’
sinh ax
Xn
2apGP
=
=
In
1
=
q2 sinh2 ax
xm sinh ax dx
~
a(p2 +
dx
q2 sinh2 ax
p2 +
-
=
>
- q cash ax
+”
q2)(p + q sinh ax)
P2 + 92
dx
q sinh ax)2
p sinh ax cos px
c&2+ p2
dx
p + q sinhax
S
2
a = *p see 14.547.
s
14.558
-~--
X
--
-- coth ax
=
sinh ax sinh
.I'
x sinh 2ax
4a
=
[See 14.5651
=Fdx
s
=
87
,. . . .
I a
dx
sinh ax
I‘ p”
S
14.561
=
-
S
14.556
14.560
dx
x
S
14.559
I jJ$:
/ 05
* .
5*5!
s
14.555
14.557
ax
s
For
14.551
=
INTEGRALS
cash ax dx
S
sinhnP2
cash ax
a
S QFr
-- n-2
92-l
[See 14.5851
ax dx
[See 14.5871
dx
dx
S sinh*--2
as(n - l)(n
ax
1
- 2) sinhnP2
ax
-- n-2
n-l
~- x dx
S sinhnP2 ax
88
INDEFINITE
INTEGRALS
INTEGRALS
14.562
cash ax dx
14.563
14.564
cash
-& ax
14.565
s
a
x sinh ax -- cash ax
a
a2
=
- 22 cash ax
a2
=
z
*
dx
=
- dx
cash ax
14.570
14.571
14.572
14.573
14.574
14.575
14.576
14.577
14.580
14.581
=
xcosh2axdz
s
dx
cosh2 ax
s
=
s
4+
P)
%a + P)
=
a sinh ax sin px - p cash ux cos px
a2 + p2
cash ax cos px dx
=
a sinh ax cos px + p cash ax sin px
a2 + p2
dx
s
dx
cash ax - 1
s
cash ax + 1
=
$tanhy
=
-+cothy
=
!? tanh
a
xdx
x dx
cash ax - 1
--$coth
=
dx
(cash ax + 1)2
dx
(cash ax - 1)2
7
7
-$lncosh
+ -$lnsinh
-
&tanh3y
=
& coth 7
-
&
coths y
tan-’
ln
s
war + p - fi2
( qP
s
7
&tanhy
p + q cash ax
dx
(p + q cash ax)2
f
=
S dx =
14.582
+
sinh (a - p)z + sinh (a + p)x
=
cash ax sin px dx
cash ax + 1
s
. . . + (-UnE,@42n+2
(2%+2)(272)!
~tanh ax
a
2(a -
s
S
5(ax)6 +
144
x sinh 2ax
cash 2ax
4a
-8a2
X2
=
+
(ad4
8
sinh ax cash ux
2a
;+
cash ax cash px dx
s
[See 14.5431
s
-
S
14.570
14.579
cosh2 ax dx
s
. . .
=
-
14.569
+
; a
X
s
(axP
6*6!
4*4!
.
cash ax
s
14.567
+
lnz+$!!@+@+-
X
cos&ax
14.566
-
x2 cash ax dz
.
cash ax
sinh ax
=
x cash ax dx
.
INVOLVING
=
+ p + @GF
q sinh ax
-a(q2 - p2)(p + q cash as)
)
P
42 -
P2
dx
p + q coshas
S
***
INDEFINITE
In
1
14.583
p2 -
s
dx
q2 cosh2 ax
INTEGRALS
p tanh ax + dKz
p tanh ax -
2apllF3
=
89
I
14.584
dx
!
p tanh ax + dn
In
p tanh ax - dni
2wdFW
=
s p2 + q2 cosh2 ax
1
1
tan
--1 p tanhax
dF2
14.585
14.586
xm cash ax dx
.
coshn ax dx
s
coshnax
14.587
dx
coshn--l
=
=
ax sinh ax
14.591
14.592
14.593
s
s
s
14.594
s
14.595
I
ax +
(n-
=
sinh2 ax
~
2a
sinh px cash qx dx
=
cash (p + q)x
2(P + 9)
sinhn ax cash ax dx
=
sinhn + 1 ax
(n + 1)a
coshn ax sinh ax dx
=
coshn+ l ax
(n + 1)a
sinh 4ax
~
32a
dx
sinh2 ax cash ax
=
14.597
S
______ dx
sinh ax cosh2 ax
zz -sech a2 + klntanhy
S
14.600
S
14.601
S
z
dx
=
sinh
;s,hh2;;
dx
=
cash ax + ilntanhy
a
dx
cash ax (1 + sinh ax)
[See 14.5591
i tan-1
ax
- 2,‘a2 coshn--2
ax ’
cash (p - q)x
[If
n = -1,
see 14.615.1
[If
n = -1,
see 14.604.1
ax _
- 2 coth 2ax
a
-
,jx
n-2
-n-l
sinh ax AND c&t USG
a
a
ax dx
-- x
8
S
=
coshn--2
ax
_ t tan - 1 sinh
[See 14.5571
2(P - 9)
14.596
14.599
1 In tanh
a
+
dx
sinh ax cash ax
dx
sinh2 ax cosh2 ax
l)(n
INVOLVCNG
S
14.598
S
dx
coshnPz
sinh ax cash ax dx
=
n-1
n
?$!?
x sinh ax
a(n - 1) coshn--l
=
sinh ax dx
s
ax
sinh2 ax cosh2 ax dx
f-
xn--l
a
n-1
sinh ax
a(n - 1) coshn--l
INTEGRALS
,('
_ m
a s
an
-cash
ax
(n - l)xn-1
s
14.590
l.h=7
xm sinh ax
a
=
>
sinh ax
csch ax
a
J
~- xdx
coshn--l:
ax
.:,".'
INDEFINITE
90
14.602
14.603
S
S
dX
sinh ux (cash ax + 1)
dX
sinh ax (cash
14.604
14.605
14.606
14.607
14.608
14.609
14.610
14.611
14.612
14.613
14.614
14.615
14.616
14.617
14.618
14.619
14.620
S
S
S
S
S
S
S
S
S
S
S
tanhax
dx
x
=
tanhs ax dx
=
=
=
ax
tanhn + 1
(72 + 1)a
1
2
1
X2
=
=
(ax)3
3
- 2
-
bxJ5
+
-
-2k47
105
15
ax _ k!$
dx
=
+ ?k$
tanhn ax dx
cothax
dx
=
- PX
P2 -
_
dP2 - q2)
- tanhn--l ax +
a(?2 - 1)
=
x -
coths ax dx
=
i In sinh ax -
cothn ax csch2 ax dx
- dx
coth ax
dx
=
S
=
-
=
-
-coth2 ax
2a
cothn + 1 ax
(n + 1)a
- i In coth ax
$ In cash ax
...
...
(-l)n--122n(22n
- l)B,(ax)2n+
(2n + 1) !
(-l)n--122n(22n
- l)B,(ax)2n-’
(2% - 1)(2?2) !
In (q sinh ax + p cash as)
tanhnw2 ax dx
coth ax
a
coth2 ax dx
s
Q
-
42
i In sinh ax
=
-
x tanh ax + -$ In cash ax
a
X
S
S
1
2a(cosh ux - 1)
tanh2 ax
7
k In cash ax -
=
p+qtanhax
S
S
S
-
‘, In sinh ax
xtanhzaxdx
s
-&lntanhy
ilntanhax
xtanhaxdx
tanh ax dx
___
=
1
2a(cosh ux + 1)
+
tanhax
a
tanhn ax sech2 ax dx
=
7
i In cash ax
tanhe ax dx
~ dx
tanh ax
klntanh
- 1)
ux
=
edx
=
INTEGRALS
-t . . .
1
+
...
>
INDEFINITE
14.621
14.622
14.623
s
s
x coth ax dx
1
i-2
=
x coth2 ax dx
cothaxdx
1
ax
x2 -
=
-
2
x coth ax
+ +2 In sinh ax
a
b-d3
135
-$+7-v
X
14.624
14.625
14.626
14.627
14.628
14.629
14.630
14.631
14.632
14.633
14.634
14.635
14.636
14.637
14.638
14.639
S
S
dx
p+
qcothax
cothn ax dx
S
S
S
S
S
S
S
S
Sq + p
S
- PX
=
sech ax dx
cothn--l ax
+
a(n - 1)
-
=
+
i tan-l
. . . (-l)n22nBn(ux)2n--1
(2n- 1)(2n)!
9
In
a(P2 - q2)
-
P2 - !I2
=
cothn-2
tanh ax
___
a
sech3 ax dx
=
sech ax tanh ux
+ &tan-lsinhax
2a
xsechaxdx
na
+ 5(ax)s +
144
=
x sech2 ux da
x tanh ax
a
=
=
sechn ax dx
=
=
“-2
9
9
S
Gus
4320
dx
i In tanh y
coth ux
a
csch2 ax dx
=
- -
csch3 ax dx
=
- csch ax coth ax 2a
=
cschn ax
na
- -
+
. . (-lP~,kP
2n(2?2)!
[See 14.5811
p+qcoshax
sechnP2 ax tanh ax + n-2
a(n - 1)
m-1
cschn ax coth ax dx
. . . (-1)n~&X)2”+2
(2n + 2)(2n)!
+
...
$ In cash ux
5(ax)4
lnx--m++-- (ad2
=
dx
sechas
csch ax dx
- ~sechn ax
=
sinh ax
a
.A!-=
sech ax
S
S
S
S
ax dx
eaz
=
sechn ax tanh ax dx
+ ---
(p sinh ax + q cash ax)
sech2 ax dx
“e”h”“,-jx
91
INTEGRALS
$lntanhy
ssechnm2
ax dx
+
** *
INDEFINITE
92
14.640
14.641
14.642
14.643
S
S
ds=
csch ax
i cash ax
x csch ax dx
S
Sq + p
S
csch*xdx
1
2
=
x csch2 ax dx
s
=
=
X
14.644
14.645
14.646
14.647
14.648
dx
csch ax
cschnax
S
S
S
sinh-1
=
S
a
-
$+f
=
sinh-1
S
(x/a)
dx
I
14.650
14.651
14.652
14.653
14.654
sinh;~W*)
dx
S
S
E dx
S
; dx
S
S
14.656
14.657
14.658
cash;:
S
S
S
r
(u/x)2
2.2.2
--
- ln2 (-2x/a)
2
-
1 3 5(a/xY
2*4*6*6*6
l
+
1x1 < a
+
l
...
l-3 * 5(alx)6
2*4*6*6*6
_
x>a
...
*Jr&F2
:In
X
(
)
(x/a)
-
d=,
cash-1
(x/a)
> 0
i x cash-1
(x/a)
+ d=,
cash-1
(x/a)
< 0
&(2x2 - a2) cash-1
(x/a)
-
i a(222 - a2) cash-1
(x/a)
+ $xdm,
=
f
(x/a) > 0,
dx
E dx
a
=
=
x tanh-19
dx
x2 tanh-1
z dx
Il.
ix@??,
4x3
cash-1 (x/a)
-
$x3
cash-1
+ Q(x2 + 2a2) dm,
-
C
f
ln2(2x/a)
if cash-1
_ cash-1
(x/a)
X
tanh-1
...
cash-1
(x/a)
> 0
cash-1
(x/a)
< 0
3(x2 + 2~2) dm,
cash-1 (x/a)
> 0
cash-1
< 0
=
dx
(da)
_
1*3(a/x)4
2*4*4*4
-
+
l
+ 1. 3(a/x)4
2.4.4.4
+ __
(a/~)~
2.2.2
(x/a)
1.3
5(x/a)’
2*4*6*7*7
x cash-1
i
cosh-;W*)
_
l
=
x2 cash-1 E dx
+ if cash-1
14.655
+ 1 3(x/a)5
2.4~505
(xlaJ3
2.3.3
_ sinh-1
=
&FT2
9
=
a
x cash-’
cschn--2 ax dx
x m x 4 +a
-
X
cash-1
S
(2a2 - x2)
z +
ln2 (2x/a)
2
=
X
...
[See 14.5531
)
g sinh-1
+
dm~
sinh-1;
(
f dx
dX
p + q sinhax
xsinh-1:
=
a
Q
cschnm2 ax coth ax -- n-2
a(n - 1)
n-l
-
z dx
x2 sinh-1
E-P
Q
=
g dx
a
x sinh-1
x coth ax
+ -$ In sinh ax
a
- 1)B,(ax)2n-1
v*x)3 + . . . (-l)n2(22n-1
e&-y+1080
(272 - 1)(2n) !
-
=
dx
ax
X
--a
14.649
INTEGRALS
x tanh-1
=
=
7
F
(x/a)
+(a/5)2
+
1. 3(a/x)4
2-4-4-4
292.2
+ 1.3 * 5(a/x)6
2*4*6*6*6
+
...
1
(x/a) < 0
r
1 ln
a + v
a
X
(
z + % In (a2 - x2)
+ # x2 - ~2) tanh-1:
+ $tanh-1:
(x/a)
a
+ $ln(a2-x2)
[- if cash-1 (x/a) > 0,
+ if coshk1 (x/a) < 0]
x < -a
INDEFINITE
tanh-1
14.659
14.660
14.661
14.662
14.663
14.664
14.665
S
S
S
S
S
S
tanhi:
14.669
14.670
=
“+@$+&f$+...
a
(z/u)
dx
=
_ tanh-1
!! dx
a
x coth-’
'Oth-i
(x/u)
=
7
a
(xia)
' sech-'2
a
x sech-1
+ +(x2 - ~2) coth-’
dx
=
F
+ fcoth-1:
dx
=
_
;
dx
=
_ coth-1
dx
J? dx
(x/a)
(x/u)
+ a sin-l
(x/u),
sech-1
(x/u)
> 0
r x sech-1
(z/u)
-
(x/u),
sech-1
(x/u)
< 0
=
dx
(x/u)
-
+a~~,
sech-1
(x/u)
> 0
+x2 sech-1
(x/u)
+ +ada,
sech-1
(x/u)
< 0
-4
=
14.674
14.675
4 In (a/x)
S
S
csch-1
” dz
=
x csch-1
U
x ds
a
x csch-’
S
csch-;
(x/u) dx
S
xm sinh-15
s
xm cash-’
S
S
a
x”’ coth-’
dx
E
U
U
T
=
=
14.677
xm sech-1
S
1 * 3Wu)4
_
...
2.4.4.4
’
sech--1
(s/u)
z+--
=
xm csch-’
: dx
a
U
5
> 0
if x > 0, -
if x < 0]
[+
if z > 0, -
if
1. 3(d44
+
-
sech-1
(x/u)
x < 0]
...
O<x<a
2-4.4-4
+
...
1x1 > a
-
cash-’
E -
cash-’
i
--&s$=+
xmfl
coth-’
+
?
U
-
dx
~
a
mt1
E -
-J?m+l
+ am
xm+1
m+lswh-‘s
U
cash-1
(x/a)
> 0
cash-1
(x/u)
< 10
S x2
SCL2- x2
S
Zm+l
dx
u2 -
Zm+l
+ 1
xm dx
~~
dx
seckl
(da)
> 0
sech-1
(s/a)
< 0
xm+l
m+l
csch-1:
c
a
< 0
- $T$$
+ ' '3(x/u)4
-.... -u<x<O
-
a
tanh-1
mS-l
=
=
f. .,
@+l
i
: dx
+
[+
1. 3W45
U
U
3(x’u)4
2.4.5.5
Xmfl
m-tl
5 dx
”
2.4.4.4
2.404.4
(a/xl3
nz+lSinh-lE
+
+ +@$.$
. .
ln (-x/4u)
2.3.3
=
=
2
In (4alx)
+ In (-x/a)
U
dx
a&FTS
k
U
s
s dx
xm tanh-15
S
-
.
+ -$$$
. .
z k a sinh-1
x2 csch-‘z
=
In (4ulx)
~sp&l%
14.676
- a.
In (4ulx)
In (u/x)
X
i
14.673
a sin-1
&x2 sech-1
--
14.672
+ $In(xZ--2)
(x/a)
4 In (x/u)
14.671
x
x sech-1
=
U
sech-1
+ tIn(xz-u2)
a
coth;~(xlu)
S
xcoth-lx
U
x2 coth-1:
S
=
” dx
.(
14.668
dx
93
X
coth-’
14.666
14.667
(z/a)
x
s
INTEGRALS
[+
if
x > 0, -
if x < 0]
15
DEFINITE
DEFINITION
OF A DEFINITE
INTEGRAL
a)/n.
Let f(x)
(b -
INTEGRALS
the interval
into n equal parts
be defined in an interval
a 5 x 5 b. Divide
Then the definite integral
of f(x) between
z = a and x = b is defined as
of length
Ax
=
b
15.1
f(x)dx
s a
The limit
If
will
f(x)
=
certainly
S
if f(x)
f(x)dx
S
dx
=
lim
dx
S
S
S
S
f(x)
b-tm
dx
=
b-m
continuous.
theorem
=
g(x)
calculus
the above
definite
integral
a
=
c/(b)
-
in the interval,
the definite
limiting
procedures.
For
integral
example,
s(a)
dx
f(x)
dx
dx
=
lim
t-0
f(x)
dx
=
lim
f(x)
a
dx
if b is a singular
point
if a is a singular
point
b
f(x)
c-0
dx
a+E
F6RMULAS
INVOLVING
b
DEFINITE
INTEGRALS
b
{f(x)“g(s)*h(s)*...}dx
S
=
a
f(x)
dx *
a
b g(x) dx *
s a
Sb h(x) dx
a
2
* **
b
b
cf(x)dx
=
c
S
where
f (4 dx
c is any constant
cl
a
15.9
of the integral
a
GENERAL
15.8
f(a + (n - 1) Ax) Ax}
b--c
f(x)
a
S
S
Sa
S
Sb
Sb
. . . +
a
iim
n-r--m
b
15.7
+
b
Cc f(x)
-m
a
15.6
Ax
or if f(x) has a singularity
at some point
and can be defined by using appropriate
b
15.5
+ 2Ax)
b
m f(x)
a
15.4
f(a
b
b d
-g(x)
(I dx
=
If the interval
is infinite
is called an improper
integral
S
S
S
S
is piecewise
f
the result
a
15.3
Ax + f(a + Ax) Ax
then by the fundamental
by using
b
15.2
{f(u)
exist
= &g(s),
can be evaluated
lim
n-m
f(x)
dz
=
0
=
-
a
b
15.10
f(x)dx
a
15.11
a f(x)dx
b
f(x)dx
=
a
15.12
S
f(z)dx
=
SC f(x)
a
dx + jb
(b - 4 f(c)
f(x)
dx
c
where
c is between
a and b
a
This
aSxSb.
is called
the mearL vulzce theorem
for
94
definite
integrals
and is valid
if f(x)
is continuous
in
DEFINITE
b
s
15.13
f(x) 0)
dx
=
$
This is a generalization
g(x) 2 0.
of 15.12 and is valid
LEIBNITZ’S
RULE FOR DIFFERENTIATION
S
a
a and b
* a
dlz(a)
15.14
95
where c is between
f(c) fb g(x) dx
a
and
INTEGRALS
if j(x)
and g(x)
are continuous
in
a 5 x Z b
OF lNTEGRAlS
m,(a) aF
F(x,a)
dx
S
=
xdx
f
F($2,~)
2
-
F(+,,aY)
2
m,(a)
6,(a)
APPROXIMATE
FORMULAS
FOR DEFINITE
INTEGRALS
In the following
the interval
from x = a to x = b is subdivided
into n equal parts by the points
a = ~0,
. . ., yn = j(x,),
h = (b - a)/%.
Xl, 22, . . ., X,-l, x, = b and we let y. = f(xo), y1 = f(z,), yz = j(@,
Rectangular
formula
b
S(I f (xl dx
15.15
Trapezoidal
=
h(Y, + Yl +
i=
$(Y,
Yz
+ ..*+
Yn-1)
formula
b
S
15.16
j(x)
dx
+ 2yi
+
ZY,
+
...
+
%,-l-t
Y?J
a
Simpson’s
15.17
formula
(or
b
I‘ a
f(z)
dz
DEFINITE
15.18
15.19
15.21
INTEGRALS
xp-ldx
1+x
=
-
o
for
--?i
sin p7r ’
=
RATiONAl
+ 4Yn-l f
Yn)
OR IRRATIONAL
O<p<l
,an+l-n
n sin [(m + 1),/n]
cos p + x2
=
sin
’
o<m+1<n
mi7
77
xm dx
1 + 2x
n even
INVOLVING
xmdx
~ + an
xn
o
formula)
; (y. + 4y, + 2Y, + 4Y, + . . . + 2Y,-2
=
dx
x2 + a2
Som---z-g
y;
S~ =
S
S
0
15.20
parabolic
sin m/3
sin /3
15.22
15.23
15.24
a ,,mdX
s
=
?$
0
S
a xm(an - xn)p dx
0
am+*+n~l?[(m+l)ln]~(p+l)
=
nl’[(m
+ 1)/n
+ p + l]
(-l)r--17ram+1-nrr[(m
15.25
n sin \(m + l)nln](r-
l)! l’[(m
+ 1)/n]
+ 1)/n - T + l] ’
o<m+1<nr
EXPRISS!ONS
DEFINITE
96
DEFINITE
All letters
15.26
15.27
s
IM’fEGRdiLS~JNVdLVSNO
are considered
positive
0
S
mxcos nx
S TTsin
and m f n
i r/2
m, n integers
and m = n
0
m, n integers
and m # n
sin2 x dx
0
m, n integers
and m + n odd
m, n integers
and m + n even
1.3.5...2rn-l1
2-4-6..*
2m
??I2
a/2
sin2mx
dx
=
cos2”‘x
S
0
dx
n/2
n/2
si$m+l
s
15.32
jr12
x dx
co+“+12
=
0
s
sin2P-1
"-dx
=
2 r(P
0
p
m sin p:;in
qx dx
<
0
0
15.37
15.38
0
S
0
15.39
S
0 < p < q
i iTI4
p = q > 0
i uql2
m sin2px
-
dx
=
X2
x2
m cos px -
cos qx dx
dx
5 q
p 2 q > 0
15.41
15.42
2
2
=
=
m sinmx
dx
s o X(x2+ a2)
15.43
49 2- P)
15.44
S
cosmx
0
277
S
:e-ma
ike-ma
15.45
S
0
=
s(l-e-ma)l
=
cos-1
dx
dx
a + b cos x
ii/2
=
=
a + b sin x
0
* ___
o x2 + u2 dx
S
m -dx
x sin mx
211
ln 9
P
0
15.40
S
0
2
m~o~p~-/sq~
0 <p
9
2
=
"l--osPxdx
s
p>q>o
d2
apl2
=
S
S
9)
X
0
15.36
+
p=o
=
m sin px cos qx dx
0
15.35
m=l,2,...
X
0
S
2*4*6..*2m
... 2m+l’
1.3.5
)...
p > 0
-%-I2
15.34
=
m=1,2
2’
UP) r(4)
=
x cos29--1z dx
xl2
0
dx
0
0
s
=
0
15.31
15.33
= ;
cot325 dx
0
15.30
s
n
0
a/2
S
=
and m =
2mf (m2 - 4)
II
mx cos nx dx
FUNCTIONS
indicated.
i 7~12 m, n integers
T/2
s
otherwise
m, n integers
0
15.29
TR10ONOMETRIC
0
=
dx
0
15.28
unless
=
ii sin mx sin nx dx
D cos
INTEGRALS
dx
a + b cosx
$2-3
(ala)
DEFINITE
15.46
2r;
S
S
S
S
o
(a + b sin x)2
257
15.47
0
15.48
S
=
o
dX
o
dx
l-2acosx+az
=
cos mx dx
l-2acosx+a2
Ial
ram
l-a2
> 1
a2 < 1,
m = 0, 1,2, . .
r
sin ax2 dx
S
naYn
cos ax2 dx
=
=
i
S
S
S
S
S
S
w sinaxn
=
-
m cos axn dx
=
---&
0
15.52
1
dx
2
II-
0
15.51
r(lln)
sin & ,
rfl/n)
cos
jc sin
0
15.54
S
m cos x
dx
6
dx=
-@/dx
=
0
15.55
-!?$i?dx
=
0
15.56
=
0
6
2Iyp)
Sk (pn/2)
’
2l3p)
c,“, (pa/2)
’
m sin ax2 cos 2bx dx
=
k
=
i
O<p<l
O<p<l
0
15.57
S
m cos ax2 cos 2bx dx
0
15.58
S
* sin3 x
-
15.60
S
S
S
S
=
* -tanxdx
0
$f
=
T
z
x
VT/2
15.61
&y
x3
0
0
dx
1 + tarP
=T
4
2
?r/z
15.62
0
1
15.63
tan-'
1
S
sin-'x
dx
ll-cosxdx
S
15.66
s:
15.67
S
X
(h
5, tan-l
0
'+$A+...
_ 32
$
=
;ln2
_
S
X
0
15.65
=
X
0
15.64
x dx
-
cosx)'$
px - tan-lqx
X
m cos x
-dx
=
X
=
dx
y
=
p
n>l
n>l
2,
0
15.53
b’)312
laj < 1
77 In (1 + l/a)
=
(az-
O<a<l
(57/a) In (1 + a)
=i
97
227-a
(a Jr b cos x)2
27r
1--’
=
x sin x dx
iT
o 1 - 2a cos x + a2
Tr
15.49
27r
dX
INTEGRALS
y
e-axcosbx
15.69
15.70
s
=
a2 + b2
m e-az sin bx dx
=
b
~
a2 + b2
0
S
S
S
S
S
m e-az
sin
mC-az-
dx
=
tan-l
k
e-bz dx
=
In!!
a
X
0
15.72
bx
X
0
15.71
a
dx
0
15.73
m ecaz2 cos bx dx
1
=
15.74
-
b2/4a
5
0
e-(az2tbz+c)
dz
erfc - b
=
2fi
0
where
co
15.75
S
S
,-&tbztc)
ds
=
--m
cc
15.76
0
xne-azdx
Iyn + 1)
=
an+1
cc
Xme-azz dx
15.77
r[(m
=
s 0
15.78
S
S
S
m e-k&+b/z2)
+ 1)/2]
2a(mfl)/Z
dx
=
;
15.79
"-g+
=
;e2'6
a
d-
0
A+$+$+$+
***
=
f
0
15.80
-
xn-l
s
dx
=
L+&+$+
ln
l'(n)
. . .
>
(
0
For even n this can be summed
15.81
15.82
15.83
= -12
S-
1
m xdx
ez + 1
0
m
S
o
xn-l
dx
eZ+l
=
$+$-$+
r(n)
For some positive
integer
S
=
“cdl:
$ -&+
S
co e-z2-e-*dx
0
15.86
&-
values
+coth;
X
=
&
of Bernoulli
..*
(
0
15.85
in terms
***
=
numbers
9
12
>
of n the series can be summed
-
&
[see pages 108-109 and 114-1151.
[see pages 108-109 and 114-1151.
DEFINITE
m e-az
15.87
15.88
_
m e-~x
s
@-bs
_
e-bz
dx
x csc px
0
m e-“‘(lx;
s
‘OS
‘)
,jx
1
xm(ln x)” dx
i -
cot-l
=
tan-l%
a
(--l)%!
(m + l)n+l
=
s 0
If
n#0,1,2,...
replace
S
o
l - lnx
1+x
dx
=
-$
&
=
-$
’ In (1 + x) dx
S
S
S
S
S
2
0
15.94
tan-1
=
-
;
’ ln(l-x)
x
0
(a2
+
dx
=
$
=
-?
m > -1,
1)
n = 0, 1,2, . . .
n! by r(n. + 1).
6
1
15.95
In
0
15.90
15.93
99
x set px
15.89
15.91
INTEGRALS
In x In (1 + x) dx
=
2-2ln2-12
In x In (l-x)
=
2 -
572
0
1
15.96
dx
c
0
15.97
- 772WC pn cot pa
O<p<l
0
’ F
dx
=
In s
=
-y
m
e-xlnxdx
=
-5(-y
dx
=
S
S
In sin x dx
=
RI2
lncosx
=
-l
(In cos x)2 dx
=
dx
In2
a/2
(ln sin x)2 dx
=
0
15.104
S
S
0
0
15.103
$
n/z
n/2
15.102
+ 2 ln2)
0
srxlnsin
x dx
=
-$ln2
0
S
7712
15.105
sin x In sin x dx
In 2 -
=
1
0
2a
15.106
S
0
2n
In (a + b sin x) dx
=
S
0
In(a+bcosz)dx
=
2rrIn(a+dn)
DEFINITE
100
7r
15.107
ln(a
s
+ b cosx)dx
U+@=G
T In
=
(
0
2
7i
15.108
INTEGRALS
2~ In a,
a 2 b > 0
2~ In b,
b 2 a > 0
=
In (a2 - 2ab cos x + b2) dx
.(‘ 0
)
T/4
15.109
S
In (1 + tan x) dx
=
i In2
0
dx
=
+{(cos-~u)~
sin 2a
y
+ T+
-
(cos-1
sin 3a
32
b)2}
+ ...
(’
See also 15.102.
“.
:
DEFiNlTi
ti!tThRAl.S
1NVOLVlNG
S
m - sinaz
dx
sinh bx
=
$ tanh $
15.113
p -cos ax
dx
s o cash bx
=
&
15.114
S
15.112
0
-6
=
$
=
Sr(n+
NYPERBQLIC
FUNCTtC?NS
a7
sech%
0
15.115
m xndx
o sinh az
S
If n is an odd positive
m ___
sinh ax dx
ebz + 1
0
15.116
S
15.117
S
* sinh ux
ebz
dx
m ftux)
i ftbx)
1)
integer,
=
2
csc $
=
&
-
5
the series can be summed
-
[see page 1081.
1
2a
cot %
0
15.118
S
&
=
{f(O) - f(m)}
ln i
0
This
15.119
is called
’ dx
S
-
Ia
(u+x)m-l(a--x)-l&
0
15.120
Frulluni’s
--a
It holds
integral.
=
22
=
(2a)m+n-1;;'f;;
if f’(x)
is continuous
and
s
- f(x) - f(m) dx converges.
1
x
16
THE GAMMA
DEFINITION
OF THE GAMMA
16.1
r(n)
FUNCTION
FUNCTION
r(n)
cc
S
tn-le-tdt
=
FOR n > 0
n>O
0
RECURSiON
FORMULA
16.2
lT(n + 1)
=
nr(n)
16.3
r(n+l)
=
n!
THE GAMMA
For
n < 0 the gamma
function
r(n)
GRAPH
by using
=
n=0,1,2,...
where
O!=l
FOR n < 0
FUNCTION
can be defined
16.4
if
16.2, i.e.
lyn + 1)
___
n
OF THE GAMMA
FU
CTION
Fig. 16-1
SPECIAL
VALUES
FOR THE GAMMA
r(a)
16.5
16.6
16.7
r(m++)
r(-m
FUNCTION
= 6
= 1’3’5’im * em - 1) 6
+ 22
=
(-1p2mG
1. 3. 5 . . . (2m
101
m = 1,2,3,
... _
m = 1,2,3,
...
k&n\!
MI
- 1)
Y-
ti
6
THE
102
GAMMA
RELAT4ONSHIPS
FUNCTION
AMONG
GAMMA
16.8
r(P)r(l--pP)
16.9
22x-1 IT(X) r(~ + +)
This is called
the duplication
=
*
=
Gr(2x)
formula.
r(x)r(x+J-)r(x+JJ-)...r(..+)
16.10
For
m = 2 this reduces
OTHER
=
r(s+
OF THE QAMMA
.
1) =
JE
-=1
r(x)
16.12
This
is an infinite
mM--mz(2a)(m-l)‘2r(rnz)
to 16.9.
DEflNIflONS
16.11
FUNCTIONS
product
. ..k
(x + 1:(x”+ 2”, . . . (x + k) kZ
xeY+il
representation
DERWATIVES
.
FUNCTION
{(1+;)r.‘m)
for the gamma
Of
THL
function
GAMMA
where
y is Euler’s
constant.
FUNCTION
m
16.13
r’(1)
=
e-xlnxdx
.(’
m4
r(x)
16.14
_
-
-y
+ (p)
ASYMPTOTIC
r(x+l)
16.15
+ (;-A)
+
EXPANSIONS
=
&iixZe-Z
=
-y
0
.**
+ (;-
FOR THE OAMMA
..t,_,>
If we let
[e.g. n > lo]
is called
Stirling’s
x = n a positive
is given by Stirling’s
asymptotic
integer
._
t
16.17
that
where
>
series.
in 16.15,
n!
- is used to indicate
n!
1+&+&-a+...
then
a useful
&n
nne-n
approximation
for
formula
16.16
where
-.*
FUNCTION
-i
This
+
the ratio
-
of the terms
MISCELi.ANEOUS
Ir(ix)p
=
on each side approaches
RESUltS
i7
x sinh TX
1 as n + m.
n is large
17
THE BETA
FUNCTION
7
DEFINITION
OF THE BETA
FUNCTION
B(m,n)
=s1
17.1
P-1
(1 - t)n--l
dt
B(m,n)
m>O,
n>O
0
RELATIONSHIP
OF BETA
17.2
FUNCTION
Extensions
of B(m,n)
to
m < 0, n < 0 is provided
SOME
by using
IMPORTANT
17.3
B(m,n)
=
B(n,m)
17.4
B(m,n)
=
2
17.5
B(m,n)
=
17.6
B(m,n)
=
FUNCTION
r(m) r(n)
r(m + n)
=
B(m,n)
TO GAMMA
16.4, page 101.
RESULTS
n/2
s0
sinzmp-1 e COF?-1 e de
T~(T-+ l)m
103
.ltm-l(l-
.(
0
Ql-1
(T + tp+n
dt
,,
fiASlC
18
tWFERfNtfAL
18.1
EQUATION
Separation
Linear
SOfJJTfON
of variables
fl(x) BI(Y) dx + f&d C&(Y) dy
18.2
difF’ERENTIA1
EQUATIONS
and -SOLUTIONS
first
order
=
s
g)dx
0
equation
Bernoulli’s
ye.!-J-‘dz
= I‘
equation
P(x)Y
J-P&
where
Q(x)Y”
=
=
Exact
M(x,
QeefPdxdx
U-4
If
v = ylen.
lny
18.4
=
c
-t- c
I
2)e(l--n)
2 +
Sz(Y)
-dy
g,(y)
s
I
2 + P(x)y = Q(x)
18.3
+
=
f
Qe (1-n)
jPdz&
n = 1, the solution
.
(Q-P)dx
+
c
is
+ c
equation
y) dx + N(x,
where
aivflay = m/ax.
18.5
Homogeneous
y) dy
=
~iV~x+j+‘-$L3x)dy
0
=
where ax indicates that the integration
with respect to x keeping
y constant.
equation
c
is to be performed
I
dy
z = F:0
S
lnx=
where
104
v = y/x.
-
F(v)
If
F(w)
dw
- w
fc
= V, the solution
is y =
CX.
BASIC
DIFFERENTIAL
DIFFERENTIAL
EQUATIONS
AND
105
SOLUTIONS
EQUATION
SOLUTION
18.6
y F(xy)
dx + x G(xy)
dy
=
0
lnz
where
Linear,
second
18.7
homogeneous
order equation
$$+ag+by
w = xy.
If
=
Case 1.
and distinct:
real
y =
0
m,,me
real
constants.
y
where
nonhomogeneous
order equation
the solution
is
:cy = c.
m2 + am + 6 = 0.
Then
+ c2em2J
and equal:
=
clemP
+ e2xemlz
m2=p-qi:
=
epz(cl cos qx + c2 sin qx)
p = -a&
There
above.
of
clemP
m,=p+qi,
y
Linear,
second
= F(v),
roots
Case 3.
18.8
G(v)
m2
mi,m,
+ c
S wCG(4
Let m,,
be the
there are 3 cases.
Case 2.
a, b are real
G(v) dv
- F(v))
=
q = dm.
are 3 cases
corresponding
to those
of entry
18.7
Case 1.
$$+a$+
a, b are
by
real
=
=
Y
R(x)
cleWx
+ c2em2z
emP
+----ml - m2
constants.
S c-ml%
em9
+-
m2 - 9
R(x)
dx
S e-%x
R(x)
dx
Case 2.
=
Y
cleniz
+ c2xenG
+ xernlz
-
s
e-ml= R(s)
emP
S
dx
xe-mlx
R(x)
dx
Case 3.
Y
=
ePz(cl cos qs + c2 sin qx)
+
18.9
Euler
or Cauchy
equation
Putting
x2d2Y
dx”
+ ,,dy
dx
+ by
epx sin qx
e-c; R(x) cos qx dx
S
P
- epz cos qx
c-pz R(x) sin qx dx
S
P
=
S(x)
x = et, the equation
3
and can then
+
(a-l)%
be solved
+ by
as in entries
becomes
=
S(et)
18.7 and 18.8 above.
BASIC
106
18.10
Bessel’s
DIFFERENTIAL
EQUATIONS
AND
SOLUTIONS
equation
x2=d2y + Z&dy + (A‘%-n2)y
=
0
Y
=
C,J,(XX)
+ czY,(x)
See pages 136-137.
18.11
22%
dx2
Transformed
+ (2 +1)x&
’
dx
Bessel’s
equation
--
+ (a%Pf~2)y
0
Y = x-’ {CLJo (@
where
18.12
(l-zs’)$$
Legendre’s
-
2x2
+ c2ypls (;c)}
q = dm~.
equation
+ n(n$-1)y
=
0
Y
See pages 146-148.
=
cup,
+ czQn(4
19
SERIES
of CONSTANTS
ARlTHMEtlC
19.1
a + (a+d)
where
Some special
+ (u+2d)
I = a + (n - 1)d
+
**.
SERIES
+ {a + (n-
l)d}
=
dn{2u
+ (n-
l)d}
=
+z(a+
I)
is the last term.
cases are
19.2
1+2+3+**.
19.3
1+3+5+*.*+(2n-1)
+ n
=
+z(n + 1)
=
GEOMETRIC
n2
SERIES
19.4
where
If
-1
1 = urn-1
is the last term
and
r # 1.
< r < 1, then
19.5
a + ur + ur2 -I- a13 +
...
=
ARITHMETIC-GEOMETRIC
a + (a+@.
19.6
where
If
+ (a+2d)r2
SERIES
**a + {a+(n-l)d}rrt-1
=
!G$+Tfl
+ rd{l-nr"-'+(n-lPnl
(1 - r)2
r P 1.
-1 < r < 1, then
19.7
a + (a+
SUMS
19.8
+
lnr
1p + 2p + 3* +
...
d)r + (a+
2d)r2 +
OF POWERS
...
=
OF POSITIVE
*
+ (1 ?r),
INTEGERS
+ ?zp =
where the series terminates
numbers
[see page 1141.
at n2 or n according
107
as p is odd or even, and
B, are the Bernoulli
108
SERIES
Some special
OF
CONSTANTS
cases are
19.9
1+2+3+...+n
19.10
12
+
22
+ 32 +
... +
%2
=
n(n+1g2n+1)
19.11
13
+
23
+
33
+
... +
n3
=
n2(n4+ ‘I2
=
19.12
14 + 24 +
34
+
... +
%4
=
n(n+
+iA(3n2
If
19.13
Sk = lkf
(“+
=
2k+
3k+
...
+ (“;‘)S2
dy
+ nk where
+
*..
lNzn
(1 + 2 + 3 + * * * + 72)s
+ 3n-
k and n are positive
+ (“:‘)Sk
=
l)
integers,
(n+l)k+‘-
then
(n+l)
SERIES
OF
CONSTANTS
~'P-'~~PB
P
(2P)!
19.36
&
+ &
+ &
+ &
+
...
(22~ - 1)&B
=
P
2(2P)!
- l)&‘B
(22~-'
P
(2P)!
19.38
&
-
-!-
+
32~+1
1
__ 1
-
52~+1
+...
79 + ‘E,
=
72p+1
22Pf2(2p)!
MlSCEI.LANEOUS
1
2
19.39
-+cosa+cos2a+~*~+cosna
19.40
sina
19.41
1 + ?-cos(u
19.42
r sina
19.43
1 + rcosa
19.44
rsincu
+
sin2a
+
+ r2cos2a
...
+ r3cos3a
+ r2cos2a
+
+
...
2 sin (a/2)
+ sinna
+
+ r2 sin 2a + + sin 3a +
+ r2sin2n
sin (n + +)a
=
+ sin3a
***
a**
SERIES
sin [*(n
=
..*
sin &na
=
1-‘2,,‘,‘,“,“;r2,
ITI < 1
r sin (Y
l-22rcosafr2’
=
+ r”cos?za.
+ msinm
+ l)]a
sin (a/2)
b-1 < 1
m+2COSnLu-?-r”+1cos(n+l)a-~rosa+1
1 - 2r cos a + ?-2
=
-
rsincu-V+1sin(n+l)cu+rn+2sinncu
1 - 2r cosa + r2
=
THE EULER-MACLAURIN
SUMMATION
FORMULA
n-1
19.45
&
F(k)
=
j-&k)
dk
-
f P’(O)
+ F(n)1
0
+
& {F’(n)
+
- F’(O)}
&{F(v)(n)
,
+
THE POISSON
19.46
,=iii,
F(k)
-
...
&
- F(v)(o)}
(--lF1
SUMMATION
=
,J--,
{F”‘(n)
3
{F
(Zp)
!
- F”‘(O)}
-
&
t
?
(ZP-~)(~)
-
FORMULA
{S” --m eznimzF(x)
dx
>
{F(vii)(n)
F(~P-l,(O)}
- F(vii)(O))
+
. . .
20
TAYLOR
TAYLOR
f(x)
20.1
=
SERIES
FOR
f@&) + f’(a)(x-
SERIES
FUNCTIONS
a) + f”(4(2z,-
OF
42
+
ONE
1
.
VARIABLE
. . . + P-“(4(x
-4n-’
+ R,
(n-l)!
where R,, the remainder
20.2
20.3
Lagrange’s
Cauchy’s
after
n terms,
form
form
R,
=
R,
=
is given
by either
f’W(x
of the following
forms:
- 4n
n!
f’“‘([)(X
-p-y2
- a)
(n-l)!
The value 5, which may be different
in the two forms,
continuous
derivatives
of order n at least.
lies between
a and x.
The result
holds
if f(z)
has
If
lim R, = 0, the infinite
series obtained
is called the Taylor
series for f(z) about
x = a. If
tl-c-3
a = 0 the series is often called a Maclaurin
series.
These series, often called power series, generally
converge for all values of z in some interval
called the interval
of convergence
and diverge for all x outside
this interval.
BINOMIAL
20.4
(a+xp
=
&I
+
nan-lx
=
an
+
(3
Ek$a
an--15
.
Special
+
+
20.5
(c&+x)2
=
a2 + 2ax + x2
20.6
(a+%)3
=
a3 +
3a2x
+
3ax2
20.7
(a+x)4
=
a4 +
4a3x
+
6a2x2
20.8
(1 + x)-i
=
1 -
x + x2 -
x3 + 24 -
20.9
(1+x)-2
=
1 -
2x
-
20.10
(1+x)-3
=
1 -
3x + 6x3 -
20.11
(l$
20.12
(1 fx)i’3
20.13
(1 +x)-l'3
20.14
(l+z)'/3
=
x)-l'2
=
=
+
dn--
1,‘,‘”
an--2z2
+
(‘;)
@--3X3
23
+
+
4ax3
+
x4
...
4x3 + 5x4 -
-l<x<l
.**
10x3 + 15x4 -
-1<2<1
* *a
1-;x+~z2-~x3+.
1 + 2”
1
=
3x2
(3
an-2x2
. I
I
cases are
+
SERIES
-
2x31
-l<sSl
+ 2.4.6
l-3
x3 -
..,
l-;x+~x2-~x3+.~
1 + 3x
1
-
&x2
-l<x<l
-l<xZl
-l<xCzl
+ $&x3
-
110
***
-l<xSl
-
2)
an-3z3
+
. **
+
. . .
TAYLOR
SERIES FOR EXPONENTIAL
20.15
e= =
l+x+~+~+*.*
20.16
a~ =
@Ina
20.17
ln(l+x)
20.18
$
=
=
=
(
AND
LOGARITHMIC
23
+ k$.d!+
+ _“3” -
$
-“4” +
5 + g + f
k-!&d!
...
Ins
=
2{(~)+;(++;(~)5+
20.20
Inx
=
(s?+)
+ q + . ..
+ ~(~)”
x-2”+“-sc’+
3!
--m<2<m
-l<x<l
. ..)-
+$(z$!)“+
SERIES
=
+ **.
-l<xzzl
)
20.19
sin 2
20.22
cosx
20.23
tanx
=
20.24
cotx
= 1 _ : _ f
5
20.25
secx = l+g+g+!g+...+-
20.26
cscx = ;+~+~+A!??+
20.27
sin-l
20.28
cos-lx
= T2
20.29
tan-lx
=
5!
2>0
...
X2+
FOR TRIGONOMETRIC
...
20.21
FUNCTIONS
--m<x<m
1 + xlna
x -
‘2
ln
22
111
SERIES
FUNCTIONS
--m<x<m
7!
--m<x<m
2922n
- 1)&x+-1
(2n) !
+
- . . . _ 22n~~2inp1 - . . .
im
x+$+z+E+.*.+
_ g
...
0 < 1x1< P
E,x2"
(2n)!
. .. +
15,120
1.3.5
~-
x’
7 +
2.4.6
=
1p,x2n--1
(2n) !
x
sin-lx
1x1<R2
+ ."
2(2+‘-
x+2y+=5+
1 x3
T2
x-$+$-$+
.**
x5
3x3
5x5
...
0 < 1x1 < ?r
I4 < 1
I4 < 1
. ..
*E-1+1-L+
2
x
+
/xl < 1
*.*
1.3
I4 < ;
[+ if 5 2 1, -
...
if 5 zZ -11
1x1< 1
20.30
cot-lx
= 9 - tan-12
=
2
[p = 0 if x > 1, p = 1 if x < -11
0:
20.31
see-l x
=
cos-‘(l/x)
=
E2 -
20.32
csc-1 x
=
sin-1
=
k+‘-
(l/x)
I4 > 1
2-3x3
+
2
*l-3
4 * 5x5
+
...
14 > 1
/
TAYLOR
112
SERIES
SERIES FOR HYPERBOLIC
20.33
sinh x
=
x+g+g+g+
20.34
cash x
=
l+$+e+e+...
20.35
tanh x
=
x-if+z&rg+...
20.36
cothx
=
~+fA+E+
20.37
sechx
=
l-~+~x&+
20.38
cschx
=
1 -
FUNCTIONS
-m<x<m
***
--m<x<m
(-l)n-l22n(22n
...
(-I)*-
122nBnx2n-
+
. . . (-l)nEnx2n
(2n) !
; + g
X
-
E.
sinh-lx
-
G
+
2.4.5
1
+
,
l
+
1x1 <x
l)B,Gn--1
...
+
20.40
cash-1x
20.41
tanh-1~
=
20.42
coth-1s
=
=
-r-
lnj2xl
+ A--
1x1 < 1.
‘*’
k{In(2x)-
l-3*5
+ 2.4.6.6~6
1*3
204.4~~
(
esinz =
20.44
ecosz
[‘ii
L-
1
E~~~I:~~~:
:::I
I4 < 1
x2
SERtES
x5 + . . .
x4
--m<X<m
l-$+x!pz!+...
--m<x<m
(
)
20.45
etanz
20.46
ez sin x
20.47
e2
20.48
In lsin xl
=
In(x(
20.49
~nlcosxl
=
-$
20.50
In ltan x1
=
x2
7&
In 1x1 + -py + g-
20.51
In
- (1 +x)
1+x
=
x -
x
+ifxZl
if x 5 -1
>
1x1 > 1
e
=
’ ‘.
x+$+g+$+...
1+x+;i--s-z
=
-
(&+&+,.::“,Y”,x6+.**))
MlSCELLAN(KMJS
20.43
2
0 < 1x1 < x
=
1
2
0 < /xl < a
(2n) !
1.3 ’ 5x7
+
2.4.6.7
3x5
...
...
(-l)n2(22”-l-
-0.
1x1 <f
. . .
+
1
(2n) !
x3
cos
1px2n-1
(2n) !
’
20.39
-
1+.+;+g+y+...
=
~+x2++3+~+
=
1+x-+$+...+
1x1
. . .
2nf2 sin (m/4)
?Z!
+
2ni2 cos (m/4)
n!
22n-
-
-
f
-
go
$
-
$
(1 +&)x2
-
&
-
17x*
- 2520
62x6
+ 2835
... -
**.
+
+ (1 + & + #a+
+
22922n--1n(S)
* **
Jr
2
--m<x<m
--m-LX-Cm
...
22n- 1(22” - l)B,xz”
n(k) !
** * +
...
...
1Bn52n
n(2n) !
-
-
xn +
xn +
<
l)B,xzn
!
0 < 1x1 < ?r
+
...
+
...
I4 < ;
0 < 1x1 < ;
I4 < l
If
20.52
y = qx + c‘@ +
+
c323
+
c424
c525
+ I+?9 + . * *
then
20.53
x = c,y +
+
C2Y2
+
c3y3
cqy4
+ C5y” + Csy6 + * * -
where
20.54
c,cl = I
20.55
c;C,
20.56
c;C3 =
20.57
c;C4 = Sc,c,c, -
20.50
c;C, =
20.59
c;'C, = 7cfc2c5+
20.60
= -c2
2~;
- clc3
6cfc,c,
5$
+
- c1c4
2
3cFc,2
- $c5 +
84qc~c,
14~24 - 21c,c~c3
+ 7cfc3c4-
fb, Y) = f@, b) + (z - dfzb, b) +
(?I
- W&
+ $ {(x - 4‘Vi,b, b) + 2(x
where
fz(a, b), f,(a, b), . . . denote
partial
- ctc6 -
28cfc2ci
derivatives
-
a)(~
with
28cfo~c4
-
42~;
b)
-
bYi&,
respect
b) +
(Y -
Wfyy(%
b))
to 5, y, . . . evaluated
+
. **
at z = a, y =
b.
21
BERNOlJtLI
and
&
DEFINITION
The Bernoulli
numbers
21.1
- x
ez -
1
21.2
1 -
: cot 5
2
2
=
1 -
OF BERNOULLI
B,, B,, B,, . . . are defined
f
+ A?!$
_ B;r’
\ B;;”
B,x2
B2x4
~+~+-y-+*-
=
numbers
E,x2
sechx
=
l--
21.4
set x
=
1+
2!
E,x6
E,x4
+-G---
6!
E1x2
E,x4
F+qr+F+*-
TABLE
OF FIRST
Bernoulli
by the series
-
...
OF EULER
El, E,, E,, . . . are defined
21.3
NUMBERS
B,x6
DEFINJTION
The Euler
EULER NUMBERS
+
NUMBERS
by the series
1x1 < 9
2
*.-
E,x6
1x1 -cE
FEW BERNOUttl
AND
numbers
Euler
2
EULER NUMBERS
numbers
Bl
=
l/6
El
=l
B2
=
l/30
E,
=
5
B3
=
l/42
~93
=
61
B4
=
l/30
E4
=
1385
B5
=
5/66
E5
=
50,521
B6
=
691/2’730
E6
=
2,702,‘765
B7
=
716
E?
=
199,360,981
63
=
3617/510
E3
=
19,391,512,145
B,
=
43,867/798
E,
=
2,404,879,675,441
ho
=
174,611/330
EIO
=
370,371,188,237,525
41
=
854,513/138
El1
=
69,348,874,393,137,901
B12
=
236,364,091/2730
E12
=
15,514,534,163,557,086,905
114
BERNOULLI
21.6
E,
=
('2")Enm1
21.7
B,
=
22.($m1,{(2n.+,
21.12
- (y)E,-,
AND
EULER
+ (;)E,-,
- (‘3Env2
-
NUMBERS
115
. ..(-l)n
+ (2n;1)Ene,
-
... (-l)n-1)
FORMULAS
from
VECTOR ‘ANALYSIS
22
VECTORS
AND
Various quantities
in physics such as temperature,
Such quantities
are called scalars.
SCALARS
volume
and speed can be specified
by a real number.
Other quantities
such as force, velocity
and momentum
require
for their specification
a direction
as
by an arrow or directed
well as magnitude.
Such quantities
are called vectors.~ A vector is represented
The magnitude
of the vector is determined
by the length of the arrow,
line segment indicating
direction.
using an appropriate
unit.
NOTATION
A.
FOR VECTORS
A vector is denoted by a bold faced letter such as A [Fig. 22-l].
The magnitude
is denoted by IAl or
The tail end of the arrow is called the initial
point while the head is called the terminal
point.
FUNDAMENTAL
1.
2.
3.
Equality
of vectors. Two vectors
magnitude
and direction.
Thus
Multiplication
(scalar), then
magnitude
of
to A according
called the zero
DEFINITIONS
are equal if they have the same
A = B in Fig. 22-l.
If m is any real number
of a vector by a scalar.
mA is a vector whose magnitude
is ]m] times the
A and whose direction
is the same as or opposite
as m > 0 or m < 0. If m = 0, then mA = 0 is
or null vector.
A
B
/
/
Fig. 22-l
Sums of vectors. The sum or resultant
of A and B is a vector
C = A+ B formed
by placing
the
initial
point of B on the terminal
point of A and joining
the initial
point of A to the terminal
point
of B [Fig. 22-2(b)].
This definition
is equivalent
to the parallelogram
law for vector addition
as inThe vector A - B is defined as A + (-B).
dicated in Fig. 22-2(c).
Fig. 22-2
116
FORMULAS
FROM
VECTOR
Extensions
to sums of more than two vectors
the sum E of the vectors A, B, C and D.
117
ANALYSIS
are immediate.
Thus
Fig.
22-3 shows how to obtain
B
I
D
Y\
(b)
(4
Fig. 22-3
4.
Unit vectors.
the direction
A unit vector is a vector with unit
of A is a = AfA &here A > 0.
magnitude.
LAWS OF VECTOR
If A, B, C are vectors
and m, n are scalars,
22.2
A+(B+C)
22.3
m(nA)
22.4
(m+n)A
=
mA+nA
Distributive
law
22.5
m(A+B)
= mA+mB
Distributive
law
=
=
(mu)A
Commutative
(A+B)+C
= n(mA)
a unit
vector
in
then
A+B
B+A
then
ALGEBRA
22.1
=
If A is a vector,
law for addition
Associative
law for addition
Associative
law for scalar
COMPONENTS
multiplication
OF A VECTOR
A vector A can be represented
with initial
point at the
origin of a rectangular
coordinate
system.
If i, j, k are unit
vectors in the directions
of the positive
x, y, z axes, then
22.6
A
where A,i, Aj,
i, j, k directions
= A,i
Y
+ A2j + Ask
A,k are called component
and Al, A,, A3 are called
vectors
of A in the
the components
of A.
Fig. 22-4
DOT
22.7
where
A-B
B is the angle
between
A and B.
OR SCALAR
=
ABcose
PRODUCT
059Sn
Index of Special Symbols and Notations
The following
list shows special symbols and notations
used in this book together
with pages on which
Cases where a symbol has more than one meaning
will be clear from
they are defined or first appear.
the context.
Symbole
Berri (x), Bein
(xj
B(m, n)
4l
(34
Ci(x)
e
elp e2, e3
natural
Euler
7, T-l
Fourier
elliptic
Hermite
in curvilinear
unit vectors
In(x)
modified
Jr, (4
Bessel
in rectangular
Bessel
function
function
complete
kind, 138
coordinates,
117
integral
136
of first kind, 179
140
Bessel
or loge x
natural
logarithm
common
function
polynomials,
and inverse
pn (4
Legendre
f%4
associated
Qn (4
Legendre
Qt’b)
associated
Legendre
cylindrical
coordinate,
polynomials,
functions
kind, 148
functions
of second
22, 36
sine integral,
50
184
183
polynomials
of first kind, 157
Chebyshev
polynomials
of second
function
kind,
49
Chebyshev
Bessel
transform,
of first kind, 149
of second
coordinate,
sine integral,
155
Laplace
146
Legendre
functions
coordinate,
Fresnel
153
Laguerre
transform
spherical
kind, 139
.of x, 23
polynomials,
Laplace
polar
of second
of x, 24
logarithm
Laguerre
associated
r
175, 176
124
of first kind, 138
of first kind,
elliptic
modified
L?(x)
transform,
151
of first and second
Wr)
<,-Cl
Fourier
eoordinates,
unit, 21
i, i, k
J%(r)
of first kind, 179
and inverse
functions
imaginary
160
integral
polynomials,
Hankel
kind, 1’79
114
transform
HA)
kind, 179
of second
183
function,
elliptic
scale factors
124
183
of second
integral
integral,
&Y h
or logl”x
function,
integral
numbers,
incomplete
Kern (x), Kein (x)
errer
hypergeometric
F(k, @)
eoordinates,
183
elliptic
exponential
1
in curvilinear
function,
Ei(x)
K = F(k, 742)
184
184
base of logarithms,
unit vectors
incomplete
H’;‘(x)
114
integral,
integral,
E(k, $)
i
logx
cosine
complete
En
F(u, b; c; x)
lnx
Fresnel
cosine
103
numbers,
complementary
erfc (x)
E = E(k, J2)
H’;‘(x),
Bernoulli
errer
erf (x)
h
140
beta function,
of second
263
kind, 158
kind, 136
150
161
INDEX
264
OF SPECIAL
SYMBOLS
AND NOTATIONS
Greek Sym bols
Euler’s constant, 1
6
spherical coordinate, 50
lW
gamma function, 1, 101
77
1
Hr)
Riemann zeta function, 184
ti
spherical coordinate, 50
Y
e
cylindrieal coordinate, 49
e(P)
the sum 1 + i + i + - *. +;,
e
polar coordinate, 22, 36
@(xl
probability distribution function, 189
-a(O)=O, 137
Notations
A=B
A equals B or A is equal to B
A>B
A is greater than B [or B is less than A]
A<B
A is less than B [or B is greater than A]
AZB
A is greater than or equal to B
ASB
A is less than or equal to B
A-B
A is approximately
A-B
A is asymptotic to B or A/B approaches 1, 102
absolute value of
equal to B
A
=
AifA
-A
if A 5 0
factorial n, 3
binomial coefficients, 3
Y
,, --
d2Y
D
derivatives of y or f(x) with respect to x, 53, 55
=
etc.
f’(x),
pth derivative with respect to x, 55
differential of y, 55
partial derivatives,
56
Jacobian, 125
s-
1(x) ch
lJ
J a f(x) dx
A * dr
line integral of A along C, 121
dot product of A and B, 11’7
AXB
cross product of A and B, 118
vs=v-v
=
definite integral, 94
A-B
V
v4
indefinite integral, 57
V(V2)
del operator, 119
Laplacian operator, 120
biharmonic operator, 120
I
N
Addition formulas,
for Bessel functions,
145
for elliptic functions,
180
for Hermite polynomials,
152
for hyperbolic
functions,
27
for trigonometric
functions,
15
Agnesi, witch of, 43
Algebraic
equations,
solutions of, 32, 33
Amplitude,
of complex number, 22
of elliptic integral, 179
Analytic
geometry,
plane [sec Plane analytic
geometry] ; solid [see Solid analytic geometry]
Angle between lines, in a plane, 35
in space, 47
Annuity,
amount of, 201, 242
present value of, 243
Anti-derivative,
57
Antilogarithms,
common, 23, 195, 204, 205
natural or Napierian,
24, 226, 227
Archimedes,
spiral of, 45
Area integrals,
122
Argand
diagram, 22
Arithmetic-geometric
series, 10’7
Arithmetic
mean, 185
Arithmetic
series, 107
Associated
Laguerre
polynomials,
155, 156
[sec uZs0 Laguerre
polynomials]
generating
funetion for, 155
orthogonal
series for, 156
orthogonality
of, 156
reeurrence
formulas
for, 156
special, 155
special results involving,
156
Associated
Legendre
functions,
149, 150 [sec also
Legendre functions]
generating
function
for, 149
of the first kind, 149
of the second kind, 150
orthogonal
series for, 150
orthogonality
of, 150
recurrence
formulas
for, 149
special, 149
Associative
law, 117
Asymptotes
of hyperbola,
39
Asymptotic
expansions
or formulas,
numbers, 115
for Bessel functions,
143
for gamma function,
102
Base of logarithms,
23
change of, 24
Ber and Bei functions,
140,141
definition of, 140
differential
equation for, 141
graphs of, 141
Bernoulli
numbers, 98,107,114,
asymptotic
formula
for, 115
definition of, 114
relationship
to Euler numbers,
series involving,
115
table of first few, 114
for
D
E
Bernoulli’s
differential
equation,
104
Bessel functions,
136-145
addition formulas
for, 145
asymptotic
expansions
of, 143
definite integrals
involving,
142, 143
generating
functions
for, 137,139
graphs of, 141
indefinite integrals
involving,
142
infmite products for, 188
integral representations
for, 143
modified [see Modified Bessel functions]
of first kind of order n, 136, 137
of order half an odd integer, 138
of second kind of order n, 136, 137
orthogonal
series for, 144, 145
recurrence
formulas
for, 137
tables of, 244-249
zeros of, 250
Bessel’s differential
equation, 106, 136
general solution of, 106, 137
transformed,
106
Bessel’s modified differential
equation, 138
general solution of, 139
Beta funetion,
103
relationship
of to gamma function,
103
Biharmonic
operator,
120
in curvilinear
coordinates,
125
Binomial coefficients,
3
properties
of, 4
table of values for, 236, 237
Binomial distribution,
189
Binomial formula,
2
Binomial series, 2, 110
Bipolar coordinates,
128, 129
Laplaeian
in, 128
Branch, principal,
17
Briggsian
logarithms,
23
Cardioid, 41, 42, 44
Cassini, ovals of, 44
Catalan’s
constant, 181
Catenary,
41
Cauchy or Euler differential
equation, 105
Cauchy-Sehwarz
inequality,
185
for integrals,
186
Cauchy’s form of remainder
in Taylor series,
Chain rule for. derivatives,
53
Characteristic,
194
Chebyshev
polynomials,
157-159
generating
functions
for, 157, 158
of first kind, 157
of second kind, 158
orthogonality
of, 158, 159
orthogonal
series for, 158, 159
recursion
formulas
for, 158, 159
relationships
involving,
159
special, 157, 158
special values of, 157, 159
Chebyshev’s
differential
equation, 157
general solution of, 159
Bernoulli
115
115
2
6
5
110
2
6
Chebyshev’s
inequality,
Chi square distribution,
percentile
values for,
6
186
189
259
Circle, area of, 6
equation of, 37
involute of, 43
perimeter
of, 6
sector of [sec Sector of circle]
segment of [sec Segment of cirele]
Cissoid of Diocles, 45
Common antilogarithms,
23, 195, 204, 205
sample problems involving,
195
table of, 204, 205
Common logarithms,
23, 194, 202, 203
computations
using, 196
sample problems involving,
194
table of, 202, 203
Commutative
law, for dot products,
118
for vector addition, 117
Complement,
20
Complementary
error function,
183
Complex conjugate,
21
Complex inversion formula,
161
Complex numbers, 21, 22, 25
addition of, 21
amplitude
of, 22
conjugate,
21
definitions
involving,
21
division of, 21, 25
graphs of, 22
imaginary
part of, 21
logarithms
of, 25
modulus of, 22
multiplication
of, 21, 25
polar form of, 22, 25
real part of, 21
roots of, 22, 25
subtraction
of, 21
vector representation
of, 22
Components
of a veetor, 117
Component
vectors, 117
Compound
amount, table of, 240
Cone, elliptic, 51
right circular
[sec Right circular cane]
Confocal ellipses, 127
ellipsoidal
coordinates,
130
hyperbolas,
127
parabolas,
126
paraboloidal
coordinates,
130
Conical coordinates,
129
Laplacian
in, 129
Conics, 3’7 [see aZso Ellipse, Parabola,
Hyperbola]
Conjugate,
complex, 21
Constant of integration,
57
Convergence,
interval of, 110
of Fourier series, 131
Convergence
faetors, table of, 192
Coordinate
curves, 124
system, 11
Coordinates,
curvilinear,
cylindrical,
49, 126
polar, 22, 36
rectangular,
36, 117
124-130
INDEX
Coordinates,
curvilinear
(cent.)
rotation of, 36, 49
special orthogonal,
126-130
spherical,
50, 126
transformation
of, 36, 48, 49
translation
of, 36, 49
Cosine integral,
184
Fresnel, 184
table of values for, 251
Cosines, law of for plane triangles,
19
law of for spherical triangles,
19
Counterclockwise,
11
Cross or vector product, 118
Cube, duplication
of, 45
Cube roots, table of, 238, 239
Cubes, table of, 238, 239
Cubic equation, solution of, 32
Curl, 120
in curvilinear
coordinates,
125
Curtate cycloid, 42
Curves, coordinate,
124
special plane, 40-45
Curvilinear
coordinates,
124, 125
orthogonal,
124-130
Cyeloid, 40, 42
curtate, 42
prolate, 42
Cylinder, elliptic, 51
lateral surface area of, 8, 9
volume of, 8, 9
Cylindrical
coordinates,
49, 126
Laplacian
in, 126
Definite integrals,
94-100
approximate
formulas
for, 95
definition of, 94
general formulas
involving,
94, 95
table of, 95-100
Degrees, 1, 199, 200
conversion
of to radians, 199, 200, 223
relationship
of to radians, 12, 199, 200
Del operator,
119
miscellaneous
formulas
involving,
120
Delta function,
170
DeMoivre’s
theorem, 22, 25
Derivatives,
53-56 [sec aZso Differentiation]
anti-, 57
chain rule for, 53
definition of, 53
higher, 55
of elliptic functions,
181
of exponential
and logarithmie
functions,
of hyperbolic
and inverse hyperbolic
functions,
54, 55
of trigonometrie
and irlverse trigonometric
functions,
54
of vectors, 119
partial, 56
Descartes,
folium of, 43
Differential
equations,
solutions of basic,
Differentials,
55
rules for, 56
Differentiation,
53 [sec aZso Derivatives]
64
104-106
INDEX
Differentiation
(cent.)
general rules for, 53
of integrals,
95
Diocles, cissoid of, 45
Direction
cosines, 46, 47
numbers, 46, 48
Directrix,
37
Discriminant,
32
Distance, between two points in a plane, 34
between two points in space, 46
from a point to a line, 35
from a point to a plane, 48
Distributions,
probability,
189
Distributive
law, 117
for dot products,
118
Divergence,
119
in curvilinear
coordinates,
125
Divergence
theorem, 123
Dot or scalar .product, 117, 118
Double angle formulas,
for hyperbolic
functions,
for trigonometric
functions,
16
Double integrals,
122
Duplication
formula
for gamma functions,
102
Duplication
of cube, 45
Envelope,
Epicycloid,
27
Eccentricity,
definition of, 37
of ellipse, 38
of hyperbola,
39
of parabola,
37
Ellipse, 7, 37, 38
area of, 7
eccentricity
of, 38
equation of, 37, 38
evolute of, 44
focus of, 38
perimeter
of, 7
semi-major
and-minor
axes of, 7, 38
Ellipses, confocal,
127
Ellipsoid,
equation of, 51
volume of, 10
Elliptic cane, 51
cylinder,
51
paraboloid,
52
Elliptic cylindrical
coordinates,
127
Laplacian
in, 127
Elliptic functions,
179-182 [sec uZso Elliptic
integrals]
addition formulas
for, 180
derivatives
of, 181
identities involving,
181
integrals
of, 182
Jacobi’s, 180
periods of, 181
series expansions
for, 181
special values of, 182
Elliptic integrals, 179,180 [see aZso Elliptie
amplitude of, 179
Landen’s transformation
for, 180
Legendre’s
relation for, 182
267
Equation
of line, 34
general, 35
in parametric
form, 47
in standard form, 47
intercept form for, 34
normal form for, 35
perpendicular
to plane, 48
Equation
of plane, general, 47
intercept
form for, 47
normal form for, 48
passing through three points, 47
Errer function,
183
complementary,
183
table of values of, 257
Euler numbers, 114, 115
definition of, 114
relationship
of, to Bernoulli
numbers, 115
series involving,
115
table of first few, 114
Euler or Cauchy differential
equation, 105
Euler-Maclaurin
summation
formula,
109
Euler’s constant, 1
Euler’s identities,
24
Evolute of an ellipse, 44
Exact differential
equation, 104
Exponential
functions,
23-25, 200
periodicity
of, 24
relationship
of to trigonometric
functions,
24
sample problems involving
calculation
of, 200
series for, 111
table of, 226, 227
Exponential
integral,
table of values for,
Exponents,
189
95th and 99th percentile
Factorial
n, 3
table of values
Factors,
values
for,
for, 234
2
Focus, of conic, 37
of ellipse, 38
of hyperbola,
39
of parabola,
38
Folium
of Descartes,
43
Fourier series, 131-135
complex form of, 131
convergence
of, 131
definition of, 131
I’arseval’s
identity for,
special,
functions]
131
132-135
Fourier transforms,
174-178
convolution
theorem for, 175
cosine, 176
definition of, 175
I’arseval’s
identity
sine, 175
table of, 176-178
Fourier’s
table of values
Fresnel
254, 255
183
251
23
F distribution,
of the first kind, 179
of the second kind, 179
of the third kind, 179, 180
for,
44
42
integral
for,
theorem,
sine and cosine
175
174
integrals,
184
260, 261
268
Frullani’s
integral,
Frustrum
of right
area of, 9
volume of, 9
INDEX
100
circular
cane, lateral
surface
Gamma function,
1, 101, 102
asymptotic
expansions
for, 102
definition of, 101, 102
derivatives
of, 102
duplication
formula for, 102
for negative values, 101
graph of, 101
infinite product for, 102, 188
recursion
formula
for, 101
relationship
of to beta function,
103
relationships
involving,
102
special values for, 101
table of values for, 235
Gaussian plane, 22
Gauss’ theorem, 123
Generalized
integration
by parts, 59
Generating
functions,
13’7, 139, 146, 149, 151, 153,
155,157,158
Geometric
formulas,
5-10
Geometric
mean, 185
Geometric
series, 107
arithmetic-,
107
Gradient,
119
in curvilinear
coordinates,
125
Green’s first and second identities,
124
Green’s theorem, 123
Half angle formulas,
for hyperbolic
functions,
for trigonometric
functions,
16
Half rectified sine wave function,
172
Hankel functions,
138
Harmonie
mean, 185
Heaviside’s
unit function,
173
Hermite polynomials,
151, 152
addition formulas
for, 152
generating
function
for, 151
orthogonal
series for, 152
orthogonality
of, 152
recurrence’formulas
for, 151
Rodrigue’s
formula for, 151
special, 151
special results involving,
152
Hermite’s
differential
equation, 151
Higher derivatives,
55
Leibnitz rule for, 55
Holder’s inequality,
185
for integrals,
186
Homogeneous
differential
equation, 104
linear second order, 105
Hyperbola,
37, 39
asymptotes
of, 39
eccentricity
of, 39
equation of, 37
focus of, 39
length of major and minor
Hyperbolas,
confocal,
127
Hyperbolic
functions,
26-31
addition formulas
for, 27
axes of, 39
27
Hyperbolic
functions
(cont.)
definition of, 26
double angle formulas
for, 27
graphs of, 29
half angle formulas
for, 27
inverse [sec Inverse hyperbolic
functions]
multiple angle formulas
for, 27
of negative arguments,
26
periodicity
of, 31
powers of, 28
relationship
of to trigonometric
functions,
31
relationships
among, 26, 28
sample problems for calculation
of, 200, 201
series for, 112
sum, difference
and product of, 28
table of values for, 228-233
Hyperbolic
paraboloid,
52
Hyperboloid,
of one sheet, 51
of two sheets, 52
Hypergeometric
differential
equation, 160
distribution,
189
Hypergeometric
functions,
160
miscellaneous
properties
of, 160
special cases of, 160
Hypocycloid,
general, 42
with four cusps, 40
Imaginary
part of a complex
Imaginary
unit, 21
Improper
integrals,
Indefinite integrals,
definition of, 57
number,
21
94
57-93
table of, 60-93
transformation
of, 59, 60
Inequalities,
185, 186
Infinite products,
102, 188
series [sec Series]
Initial point of a vector, 116
Integral calculus, fundamental
theorem of, 94
Integrals,
definite [SM Definite integrals]
double, 122
improper,
94
indefinite
[SW Indefinite integrals]
involving
vectors, 121
line [sec Line integrals]
multiple, 122, 125
Integration,
57 [SM also Integrals]
constants
of, 57
general rules of, 57-59
Integration
by parts, 57
generalized,
59
Intercepts,
34, 47
lnterest, 201, 240-243
Interpolation,
195
Interval of convergence,
110
Inverse hyperbolic
functions,
29-31
definition of, 29
expressed
in terms of logarithmic
functions,
graphs of, 30
principal
values for, 29
relationship
of to inverse trigonometric
functions,
relationships
31
between,
30
29
INDEX
Inverse Laplace transforms, 161
Inverse trigonometric functions, 17-19
definition of, 17
graphs of, 18,19
principal values for, 17
relations between, 18
relationship of to inverse hyperbolic
functions, 31
Involute of a circle, 43
Jacobian, 125
Jacobi’s elliptic functions, 180
Ker and Kei functions, 140, 141
definition of, 140
differential equation for, 141
graphs of, 141
Lagrange form of remainder in Taylor series, 110
Laguerre polynomials, 153, 154
associated [sec Associated Laguerre polynomials]
generating function for, 153
orthogonal series for, 154
orthogonality of, 154
recurrence formulas for, 153
Rodrigue’s formula for, 153
special, 153
Laguerre’s associated differential equation, 155
Laguerre’s differential equation, 153
Landen’s transformation, 180
Laplace transforms, 161-173
complex inversion formula for, 161
definition of, 161
inverse, 161
table of, 162-173
Laplacian, 120
in curvilinear coordinates, 125
Legendre functions, 146-148 [sec uZso Legendre
polynomials]
associated [sec Associated Legendre functions]
of the second kind, 148
Legendre poiynomials, 146, 147 [sec uZso
Legendre functions]
generating function for, 146
orthogonal series of, 147
orthogonality of, 147
recurrence formulas for, 147
Rodrigue’s formula for, 146
special, 146
special results involving, 147
table of values for, 252, 253
Legendre’s associated differential equation, 149
general solution of, 150
Legendre’s differential equation, 106, 146
general solution of, 148
Legendre’s relation for elliptic integrals, 182
Leibnitz’s rule, for differentiation of integrals, 95
for higher derivatives of products, 55
Lemniscate, 40, 44
Limacon of Pascal, 41, 44
Line, equation of [see Equation of line]
integrals [see Line integrals]
slope of, 34
269
Linear first order differential equation, 104
second order differential equation, 105
Line integrals, 121, 122
definition of, 121
independence of path of, 121, 122
properties of, 121
Logarithmic functions, 23-25 [see uZso Logarithms]
series for, 111
Logarithms, 23 [sec aZso Logarithmic functions]
antilogarithms and [see Antilogarithms]
base of, 23
Briggsian, 23
change of base of, 24
characteristic of, 194
common [sec Common logarithms]
mantissa of, 194
natural, 24
of compiex numbers, 25
of trigonometric functions, 216-221
Maclaurin series, 110
Mantissa, 194
Mean value theorem, for definite integrals, 94
generalized, 95
Minkowski’s inequality, 186
for integrals, 186
Modified Bessel functions, 138,139
differential equation for, 138
generating function for, 139
graphs of, 141
of order half an odd integer, 140
recurrence formulas for, 139
Modulus, of a complex number, 22
Moments of inertia, special, 190, 191
Multinomial formula, 4
Multiple angle formulas, for hyperbolic
functions, 27
for trigonometric functions, 16
Multiple integrals, 122
transformation of, 125
Napierian logarithms, 24, 196
tables of, 224, 225
Napier’s rules, 20
Natural logarithms and antilogarithms, 24, 196
tables of, 224-227
Neumann’s function, 136
Nonhomogeneous equation, linear second order, 105
Normal, outward drawn or positive, 123
unit, 122
Normal curve, areas under, 257
ordinates of, 256
Normal distribution, 189
Normal form, equation of line in, 35
equation of plane in, 48
Nul1 function, 170
Nul1 vector, 116
Numbers, complex [sec Complex numbers]
Oblate spheroidal coordinates, 128
Laplacian in, 128
Orthogonal curvilinear coordinates, 124-i30
formulas involving, 125
2
7
Orthogonality
and orthogonal
series,
14’7, 150, 152, 154,156,158,159
Ovals of Cassini, 44
0
INDEX
144, 145,
Parabola,
37, 38
eccentricity
of, 37
equation of, 37, 38
focus of, 38
segment of [sec Segment of parabola]
Parabolas,
confocal,
126
Parabolic
cylindrical
coordinates,
126
Laplacian
in, 126
Parabolic
formula
for definite integrals,
95
Paraboloid
elliptic, 52
hyperbolic,
52
Paraboloid
of revolution,
volume of, 10
Paraboloidal
coordinates,
127
Laplaeian
in, 127
Parallel, condition for lines to be, 35
Parallelepiped,
rectangular
[see Rectangular
parallelepiped]
volume of, 8
Parallelogram,
area of, 5
perimeter
of, 5
Parallelogram
law for veetor addition, 116
Parseval’s
identity, for Fourier transforms,
175
for Fourier series, 131
Partial derivatives,
56
Partial fraction
expansions,
187
Pascal, limacon of, 41, 44
Pascal’s triangle, 4, 236
Perpendicular,
condition
for lines to be, 35
Plane, equation of [see Equation
of plane]
Plane analytic geometry,
formulas
from, 34-39
Plane triangle,
area of, 5, 35
law of cosines for, 19
law of sines for, 19
law of tangents for, 19
perimeter
of, 5
radius of circle circumscribing,
6
radius of circle inscribed
in, 6
relationships
between sides and angles of, 19
Poisson distribution,
189
Poisson summation
formula,
109
Polar coordinates,
22, 36
transformation
from rectangular
to, 36
Polar form, expressed
as an exponential,
25
multiplication
and division in, 22
of a complex number, 22, 25
operations
in, 25
Polygon, regular [sec Regular polygon]
Power, 23
Power series, 110
reversion
of, 113
Present value, of an amount, 241
of an annuity, 243
Principal
branch, 17
Principal
values, for inverse hyperbolic
functions,
for inverse trigonometric
functions,
17, 18
Probability
distributions,
189
Products,
infinite, 102, 188
special, 2
Prolate cycloid, 42
Prolate spheroidal
coordinates,
Laplacian
in, 128
Pulse function,
173
Pyramid,
volume of, 9
128
Quadrants,
11
Quadratic
equation, solution of, 32
Quartic equation, solution of, 33
Radians, 1, 12, 199, 200
relationship
of to degrees, 12, 199, 200
table for conversion
of, 222
Random numbers, table of, 262
Real part of a complex number, 21
Reciprocals,
table of, 238, 239
Rectangle,
area of, 5
perimeter
of, 5
Rectangular
coordinate
system, 117
Rectangular
coordinates,
transformation
of to
polar coordinatee
36
Rectangular
formula for definite integrals,
95
Rectangular
parallelepiped,
volume of, 8
surface area of, 8
Rectified sine wave function,
172
half, 172
Recurrence
or recursion
formulas,
101,137,
139,
147,149,
151, 153, 156, 158, 159
Regular polygon, area of, 6
cireumscribing
a circle, 7
inscribed in a cirele, 7
perimeter
of, 6
Reversion
of power series, 113
Riemann zeta function,
184
Right circular cane, frustrum
of
[sec Frustrum
of right circular
cane]
lateral surface area of, 9
volume of, 9
Right-handed
system, 118
Rodrigue’s
formulas,
146, 151, 153
Roots, of complex numbers, 22, 25
table of square and cube, 238, 239
Rose, three- and four-leaved,
41
Rotation of coordinates,
in a plane, 36
in space, 49
Saw tooth wave function,
1’72
Scalar or dot product, 117,118
Scalars, 116
Scale factors,
124
Schwarz inequality
[see Cauchy-Sehwarz
Sector of circle, arc length of, 6
area of, 6
Segment of circle, area of, 7
Segment of parabola,
area of, 7
29
inequality]
arc length of, 7
Separation
of variables,
104
Series, arithmetic,
107
arithmetic-geometric,
107
binomial,
2, 110
Fourier
[sec Fourier series]
geometric,
107
of powers of positive integers, 10’7, 108
of reciprocals
of powers of positive integers,
108, 109
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