MATHEMATICS TABLES
Department of Applied Mathematics
Naval Postgraduate School
TABLE OF CONTENTS
Derivatives and Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Integrals of Elementary Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Integrals Involving au + b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Integrals Involving √
u 2 ± a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Integrals Involving √u2 ± a2 , a > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Integrals Involving a2 − u2 , a > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Integrals Involving Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Integrals Involving Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Miscellaneous Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Wallis’ Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Probability and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Discrete Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Standard Normal CDF and Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Continuous Probability Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Legendre Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Trigonometric Functions in a Right Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Trigonometric Functions of Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Complex Exponential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Relations among the Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Sums and Multiples of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Miscellaneous Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Side-Angle Relations in Plane Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
First-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Linear Second-Order Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . 24
Vector Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Fundamentals of Signals and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Version 2.7, July 2010
Initial typesetting:
Graphics:
Editing:
Carroll Wilde
David Canright
Elle Zimmerman, David Canright
Cover: in spherical coordinates
(ρ, φ, θ), shell surface is ρ(φ, θ) = ekθ sin φ,
√
1
1+ 5
where k = π ln G and G = 2 is the Golden Ratio.
1
FOREWORD
This booklet provides convenient access to formulas and other data that are
frequently used in mathematics courses. If a more comprehensive reference
is needed, see, for example, the STANDARD MATHEMATICAL TABLES
published by the Chemical Rubber Company, Cleveland, Ohio.
DIFFERENTIALS AND DERIVATIVES
A. Letters u and v denote independent variables or functions of an independent variable; letters a and n denote constants.
B. To obtain a derivative, divide both members of the given formula for the
differential by du or by dx.
1. d(a) = 0
2. d(au) = a du
3. d(u + v) = du + dv
u v du − u dv
5. d
=
v
v2
u
u
7. d(e ) = e du
du
9. d(ln u) =
u
11. d(sin u) = cos u du
4. d(uv) = u dv + v du
6. d(un ) = n un−1 du
8. d(au ) = au ln a du
du
10. d(loga u) =
u ln a
12. d(cos u) = − sin u du
13. d(tan u) = sec2 u du
15. d(sec u) = sec u tan u du
du
17. d(arcsin u) = √
1 − u2
du
19. d(arctan u) =
1 + u2
14. d(cot u) = − csc2 u du
16. d(csc u) = − csc u cot u du
du
18. d(arccos u) = − √
1 − u2
21. d(cosh u) = sinh u du
22. d(arcsinh u) = √
du
,
23. d(arccosh u) = √
u2 − 1
20. d(sinh u) = cosh u du
u>1
24. (Differentiation of Integrals) If f is continuous, then
Z u
d(
f (t) dt) = f (u) du.
a
25. (Chain Rule) If y = f (u) and u = g(x), then
dy
dy du
=
.
dx
du dx
du
u2 + 1
2
INTEGRALS
A. Letters u and v denote functions of an independent variable such as x;
letters a, b, m, and n denote constants.
B. Generally, each formula gives only one antiderivative. To find an expression for all antiderivatives, add a constant of integration.
ELEMENTARY FORMS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Z
Z
a du = a u
Z
f (u) du
Z
Z
(f (u) + g(u)) du = f (u) du + g(u) du
Z
Z
u dv = u v − v du
(Integration by parts)
Z
un+1
,
n 6= −1
un du =
n+1
Z
du
= ln |u|
u
Z
eu du = eu
Z
au
au du =
,
a>0
ln a
Z
cos u du = sin u
Z
sin u du = − cos u
Z
sec2 u du = tan u
Z
csc2 u du = − cot u
Z
sec u tan u du = sec u
Z
csc u cot u du = − csc u
Z
a f (u) du = a
3
15.
16.
17.
18.
19.
20.
21.
Z
Z
Z
Z
tan u du = − ln | cos u|
cot u du = ln | sin u|
sec u du = ln | sec u + tan u|
csc u du = ln | csc u − cot u|
1
u
du
= arctan
u 2 + a2
a
a
Z
du
u
√
= arcsin
a
a2 − u 2
Z
1
du
u
√
= arcsec
2
2
a
a
u u −a
Z
INTEGRALS INVOLVING au + b
22.
23.
24.
25.
26.
27.
28.
29.
30.
Z
Z
Z
Z
Z
Z
Z
Z
Z
(au + b)n du =
1 (au + b)n+1
,
a
n+1
n 6= −1
1
du
= ln |au + b|
au + b
a
u
b
u du
= − 2 ln |au + b|
au + b
a a
u du
b
1
= 2
ln |au + b|
+
(au + b)2
a (au + b) a2
1 u du
= ln
u(au + b)
b
au + b √
3
2(3au − 2b)
(au + b) 2
u au + b du =
15a2
u du
2(au − 2b) √
√
au + b
=
3a2
au + b
√
3
2(8b2 − 12abu + 15a2 u2 )
u2 au + b du =
(au + b) 2
3
105a
u2 du
2(8b2 − 4abu + 3a2 u2 ) √
√
=
au + b
15a3
au + b
4
INTEGRALS INVOLVING u2 ± a2
31.
32.
33.
34.
35.
36.
37.
Z
Z
Z
Z
Z
Z
Z
du
1
u
= arctan
[See 19]
2
+a
a
a
du
1 u − a =
ln
2
2
u −a
2a u + a u du
1
= ln |u2 ± a2 |
u 2 ± a2
2
2
a u − a u du
= u + ln
u 2 − a2
2
u + a
u2 du
u
= u − a arctan
u 2 + a2
a
2
1
du
u
ln
=
±
2
2
2
2
2
u(u ± a )
2a
u ±a u du
1
=−
,
(u2 ± a2 )n+1
2n(u2 ± a2 )n
u2
INTEGRALS INVOLVING
38.
39.
40.
41.
42.
43.
44.
45.
Z
Z
Z
Z
Z
Z
Z
Z
√
u 2 ± a2 ,
n 6= 0
a>0
p
u du
= u 2 ± a2
[See 5]
u 2 ± a2
p
3
1
[See 5]
u u2 ± a2 du = (u2 ± a2 ) 2
3
p
du
√
= ln |u + u2 ± a2 |
u 2 ± a2
p
p
up 2
a2
u2 ± a2 du =
u ± a2 ±
ln |u + u2 ± a2 |
2
2
du
1
u
√
√
= ln
a
u u 2 + a2
a + u 2 + a2 du
1
u
√
= arcsec
[See 21]
2
2
a
a
u u −a
√
p
u 2 − a2
u
du = u2 − a2 − a arcsec
u
a
√
p
2
2
u +a
u
√
du = u2 + a2 + a ln
u
a + u 2 + a2 √
5
INTEGRALS INVOLVING
√
u 2 ± a2 ,
p
u2 du
up 2
a2
√
u ± a2 ∓
=
ln |u + u2 ± a2 |
2
2
u 2 ± a2
Z
p
3
a2 u p 2
u
u ± a2 −
u2 u2 ± a2 du = (u2 ± a2 ) 2 ∓
4
8
p
a4
ln |u + u2 ± a2 |
8
47.
INTEGRALS INVOLVING
49.
50.
51.
52.
53.
54.
55.
56.
57.
(Continued)
Z
46.
48.
a>0
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
√
a2 − u 2 ,
a>0
du
u
√
= arcsin
[See 20]
a
a2 − u 2
p
u
up 2
a2
a2 − u2 du =
a − u2 +
arcsin
2
2
a
p
u du
√
= − a2 − u 2
[See 5]
a2 − u 2
p
3
1
[See 5]
u a2 − u2 du = − (a2 − u2 ) 2
3
√
p
a2 − u 2
u
2
2
√
du = a − u + ln
u
a + a2 − u 2 1 du
u
√
√
= ln
a
u a2 − u 2
a + a2 − u 2 √
du
a2 − u 2
√
=−
a2 u
u 2 a2 − u 2
√
√
u
a2 − u 2
a2 − u 2
du = −
− arcsin
2
u
u
a
2
u2 du
a
up 2
u
√
a − u2 +
=−
arcsin
2
2
2
2
a
a −u
2 p
p
a
u
u
a2 − u 2 +
u2 a2 − u2 du = − (a2 − u2 )3/2 +
4
8
u
a4
arcsin
8
a
6
INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
71.
72.
73.
74.
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
Z
u sin 2au
−
2
4a
u
sin
2au
cos2 au du = +
2
4a
3
cos au
cos
au
−
sin3 au du =
3a
a
3
sin
au
sin
au
−
cos3 au du =
a
3a
Z
n−1
sin
au
cos au n − 1
sinn au du = −
+
sinn−2 au du,
na
n
Z
cosn−1 au sin au n − 1
n
+
cosn−2 au du,
cos au du =
na
n
u sin 4au
sin2 au cos2 au du = −
8
32a
du
1
= − cot au
a
sin2 au
1
du
= tan au
cos2 au
a
1
tan2 au du = tan au − u
a
1
cot2 au du = − cot au − u
a
1
1
sec au tan au +
ln | sec au + tan au|
sec3 au du =
2a
2a
1
1
csc3 au du = − csc au cot au +
ln | csc au − cot au|
2a
2a
u
1
u sin au du = 2 sin au − cos au
a
a
1
u
u cos au du = 2 cos au + sin au
a
a
a2 u 2 − 2
2u
cos au
u2 sin au du = 2 sin au −
a
a3
a2 u 2 − 2
2u
sin au
u2 cos au du = 2 cos au +
a
a3
sin2 au du =
7
INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS
75.
76.
77.
78.
79.
80.
eau
(au − 1)
a2
eau
u2 eau du = 3 (a2 u2 − 2au + 2)
a
Z
un eau
n
un eau du =
−
un−1 eau du, n > 1
a
a
eau
eau sin bu du = 2
(a sin bu − b cos bu)
a + b2
eau
(a cos bu + b sin bu)
eau cos bu du = 2
a + b2
1
du
= u − ln(1 + eau )
au
1+e
a
Z
ueau du =
Z
Z
Z
Z
Z
MISCELLANEOUS INTEGRALS
81.
82.
83.
84.
85.
86.
87.
√
1 − a2 u 2
arcsin au du = u arcsin au +
a
√
Z
1 − a2 u 2
arccos au du = u arccos au −
a
Z
1
arctan au du = u arctan au −
ln(1 + a2 u2 )
2a
Z
1
ln(1 + a2 u2 )
arccot au du = u arccot au +
2a
Z
ln au du = u ln au − u
Z
ln au
1
un ln au du = un+1
−
n + 1 (n + 1)2
√
Z ∞
2 2
π
e−a u du =
2a
0
Z
8
WALLIS’ FORMULAS
88.
Z
π
2
sinm u du =
 (m−1)(m−3)···(3)(1) π
 (m)(m−2)···(4)(2) 2 , m even;
cosm u du =
Z
0
89.
Z
(m)(m−2)···(5)(3)
π
2
0
90.
Z
 (m−1)(m−3)···(4)(2)
π
2
π
2
,
m odd
sinm u du
0
sinm u cosn u du =
0
 (m−1)(m−3)···(3)(1)(n−1)(n−3)···(4)(2)
,

(m+n)(m+n−2)···(3)(1)






(m−1)(m−3)···(4)(2)(n−1)(n−3)···(3)(1)

,

(m+n)(m+n−2)···(3)(1)

(m−1)(m−3)···(4)(2)(n−1)(n−3)···(4)(2)

,

(m+n)(m+n−2)···(4)(2)





 (m−1)(m−3)···(3)(1)(n−1)(n−3)···(3)(1) π
2,
(m+n)(m+n−2)···(4)(2)
m even, n odd;
m odd, n even;
m odd, n odd;
m even, n even
GAMMA FUNCTION
Definition :
Γ(x) =
Z
∞
e−t tx−1 dt,
x>0
0
Shift Property :
Γ(x + 1) = xΓ(x),
x>0
Definition :
Γ(x) =
Γ(x + 1)
,
x
x < 0, x 6= −1, −2, . . .
Special Values :
√
1
Γ( ) = π
2
Γ(n) = (n − 1)!,
n = 1, 2, . . .
9
LAPLACE TRANSFORMS
1.
f (t)
2.
af (t) + bg(t)
3.
f ′ (t)
′′
4.
5.
f (t)
tn f (t),
n = 1, 2, 3, . . .
6.
eat f (t)
7.
f (t + P ) ≡ f (t)
8.
f (t)H(t − a)
9.
12.
f (t − a)H(t − a)
Rt
f (u) du
0
Rt
f (u)g(t − u) du
0
13.
1, H(t) , u(t)
10.
11.
14.
f (t)
t
∗
tn ,
F (s) =
R∞
0
e−st f (t) dt
aF (s) + bG(s)
2
sF (s) − f (0)
s F (s) − sf (0) − f ′ (0)
(−1)n F (n) (s)
F (s − a)
R P −st
0
e
f (t) dt
1−e−sP
e−as L{f (t + a)}
e−as F (s)
F (s)
s
F (s)G(s)
R∞
F (v) dv
s
1
s
n!
n = 1, 2, 3, . . .
16.
eat
17.
sin ωt
sn+1
Γ(a+1)
sa+1
1
s−a
ω
s2 +ω 2
18.
cos ωt
s
s2 +ω 2
19.
1 at
a (e − 1)
1
at
bt
a−b (e − e )
at
1
s(s−a)
15.
20.
21.
22.
23.
a
t ,
a > −1
te
tn eat ,
n = 1, 2, 3, . . .
sinh at
24.
cosh at
25.
1
a2 (1 − cos at)
1
a3 (at − sin at)
26.
1
(s−a)(s−b) ,
1
(s−a)2
* The Heaviside step function H is defined by
0, if t < 0
H(t) =
1, if t ≥ 0.
a 6= b
n!
(s−a)n+1
a
s2 −a2
s
s2 −a2
1
s(s2 +a2 )
1
s2 (s2 +a2 )
10
LAPLACE TRANSFORMS (Continued)
28.
1
2a3 (sin at − at cos at)
1
2a t sin at
1
(s2 +a2 )2
s
(s2 +a2 )2
29.
1
2a (sin at
s2
(s2 +a2 )2
27.
+ at cos at)
s2 −a2
(s2 +a2 )2
30.
t cos at
31.
b sin at−a sin bt
ab(b2 −a2 )
1
(s2 +a2 )(s2 +b2 ) ,
a sin at−b sin bt
(a2 −b2 )
s2
(s2 +a2 )(s2 +b2 ) ,
cos at−cos bt
(b2 −a2 )
32.
33.
s
(s2 +a2 )(s2 +b2 ) ,
34.
e−bt sin ωt
ω
(s+b)2 +ω 2
35.
e−bt cos ωt
s+b
(s+b)2 +ω 2
36.
1 − cos ωt
ω2
s(s2 +ω 2 )
37.
δ(t − a)
√
(a−b)2 +ω 2
ω
38.
a 6= b
a 6= b
a 6= b
e−sa
e−bt sin(ωt + Ψ)
s+a
(s+b)2 +ω 2
where
ω
Ψ = arctan a−b
39.
1+
where
√
b2 +ω 2 −bt
e
ω
sin(ωt − Ψ)
ω
Ψ = arctan −b
√
−act
1 + √e 1−c2 sin( 1 − c2 at − Ψ)
40.
where
41.
−1 < c < 1
Ψ = arctan
√
(a−b)2 +ω 2 −bt
a
e sin(ωt − Ψ)
c2 +
cω
where
√
1−c2
−c ,
Ψ = arctan c2aω
−ab ,
c2 = b 2 + ω 2
b2 +ω 2
s[(s+b)2 +ω 2 ]
a2
s[s2 +2acs+a2 ]
s+a
s[(s+b)2 +ω 2 ]
11
PROBABILITY AND STATISTICS
Formulas for counting permutations and combinations:
n!
n
n!
(b) n Ck =
=
(a) n Pk =
(n − k)!
k!(n − k)!
k
Let A and B be subsets (events) of the same sample space.
(a) P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
(b) The conditional probability of B given A, P (B|A), is defined by
P (A ∩ B)
,
P (A) 6= 0.
P (A)
(c) A and B are statistically independent if and only if
P (B|A) =
P (A ∩ B) = P (A)P (B).
Let X be a discrete random variable with probability function f (x).
(a) P (X = x) = f (x),
where x ∈ S ≡ range of X
X
(b) E(X) =
xf (x) = µ = mean of X
S
X
(c)
var(X) = E((X − µ)2 ) = E(X 2 ) − µ2 =
(d)
= variance of X
p
σ = var(X) = standard deviation of X.
S
x2 f (x) − µ2 = σ 2
Let X be a continuous random variable with and probability density f (x).
Z b
(a) P (a ≤ X ≤ b) =
f (x) dx.
a
Z
xf (x) dx = µ = mean of X
(b) E(X) =
S
Z
2
2
2
x2 f (x) dx − µ2 = σ 2
(c) var(X) = E((X − µ) ) = E(X ) − µ =
S
(d)
= variance of X
p
σ = var(X) = standard deviation of X.
Let X1 , X2 , . . . , Xn be n statistically independent random variables, each having the same expected value, µ, and the same variance, σ 2 . Let
Y =
X1 + X2 + · · · + Xn
.
n
Then Y is a random variable for which E(Y ) = µ and var(Y ) =
σ2
n .
12
PROBABILITY FUNCTIONS (Discrete Random Variables)
Binomial
f (x; n, p) =
n x
p (1 − p)n−x ,
x
µ = np,
σ 2 = np(1 − p).
Poisson
f (x; a) =
e−a ax
,
x!
(x = 0, 1, 2, . . . , n),
(x = 0, 1, 2, . . .),
µ = σ 2 = a.
STANDARD NORMAL CUMULATIVE DISTRIBUTION FUNCTION
N(0,1)
1
F (z) = √
2π
0
z
Z
z
−∞
t
TABLE OF NORMAL AREAS
z
F (z)
z
F (z)
z
F (z)
z
F (z)
0.0
0.500
0.8
0.788
1.6
0.945
2.4
0.992
0.1
0.540
0.9
0.816
1.7
0.955
2.5
0.994
0.2
0.579
1.0
0.841
1.8
0.964
2.6
0.995
0.3
0.618
1.1
0.864
1.9
0.971
2.7
0.997
0.4
0.5
0.655
0.691
1.2
1.3
0.885
0.903
2.0
2.1
0.977
0.982
2.8
2.9
0.997
0.998
0.6
0.726
1.4
0.919
2.2
0.986
3.0
0.999
0.7
0.758
1.5
0.933
2.3
0.989
t2
e− 2 dt
13
PROBABILITY FUNCTIONS (Continuous Random Variables)
Uniform
f (x; a, b) =
1
b−a
a+b
;
2
Exponential
σ2 =
µ=
f (x; b) = be−bx
µ=
1
;
b
(a ≤ x ≤ b).
(b − a)2
.
12
(x ≥ 0, b > 0).
σ2 =
1
.
b2
Normal
(x−a)2
1
f (x; a, b) = √ e− 2b2
b 2π
µ = a;
(−∞ < x < ∞).
σ 2 = b2 .
FOURIER SERIES
The Fourier series expansion of a function f (t) is defined by:
∞
X
f (t) = a0 +
(an cos
n=1
where
1
a0 =
2L
an =
bn =
Z
nπt
nπt
+ bn sin
),
L
L
L
f (t) dt
−L
1
L
Z
L
1
L
Z
L
f (t) cos
nπt
dt,
L
n≥1
f (t) sin
nπt
dt,
L
n≥1
−L
−L
(Note: for the complex Fourier Series, see page 29)
14
FOURIER SERIES (Continued)
Examples of Fourier Series
f(t)
f (t) =
1,
0 < t < L;
−1, −L < t < 0.
1
L
t
πt 1
3πt 1
5πt
4
sin
+ sin
+ sin
+ · · ·)
π
L
3
L
5
L
1
f (t) =
f(t)
0, L2 < |t| < L
1, 0 < |t| < L2 .
L
t
πt 1
2
3πt 1
5πt
1
cos
+
− cos
+ cos
− + · · ·)
2 π
L
3
L
5
L
L
f (t) = t,
f(t)
−L < t < L
L
t
πt 1
2πt 1
3πt
2L
sin
− sin
+ sin
− + · · ·)
π
L
2
L
3
L
f(t)
f (t) = |t|,
−L < t < L
L
L
L 4L
πt
3πt
5πt
1
1
− 2 cos
+ 2 cos
+ 2 cos
+ · · ·)
2
π
L
3
L
5
L
t
15
SEPARATION OF VARIABLES
Eigenvalues and Eigenfunctions for the differential equation
ϕ′′ + λϕ = 0.
1. ϕ(0) = ϕ(L) = 0
2
λn = ( nπ
L ) ,
ϕn = sin nπ
L x,
2. ϕ′ (0) = ϕ(L) = 0
π 2
,
λn = (n − 21 ) L
3. ϕ(0) = ϕ′ (L) = 0
π 2
,
λn = (n − 12 ) L
n = 1, 2, . . .
π
x,
ϕn = cos(n − 21 ) L
n = 1, 2, . . .
π
ϕn = sin(n − 21 ) L
x,
n = 1, 2, . . .
4. ϕ′ (0) = ϕ′ (L) = 0
2
λn = ( nπ
L ) ,
ϕn = cos nπ
L x,
5. ϕ(−L) = ϕ(L),
λ0 = 0,
λn =
n = 0, 1, 2, . . .
ϕ′ (−L) = ϕ′ (L)
ϕ0 = 1,
2
( nπ
L ) ,
ϕn = sin nπ
L x
and
cos nπ
L x,
n = 1, 2, . . .
Solution of inhomogeneous ordinary differential equations
1. u′n + k λn un = hn (t)
Rt
un (t) = an e−kλn t + 0 hn (τ )e−kλn (t−τ ) dτ
2. u′′n + µ2n un = hn (t)
un (t) = an cos µn t + bn sin µn t +
1
µn
Rt
0
hn (τ ) sin µn (t − τ ) dτ
16
BESSEL FUNCTIONS
1. The differential equation
x2 y ′′ + xy ′ + (λ2 x2 − n2 )y = 0
has solutions
y = yn (λx) = c1 Jn (λx) + c2 Yn (λx),
where
Jn (t) =
∞
X
m=0
(−1)m tn+2m
.
+ m + 1)
2n+2m m!Γ(n
2. General properties:
J−n (t) = (−1)n Jn (t);
J0 (0) = 1;
Jn (0) = 0,
n = 1, 2, 3, . . . ;
for fixed n, Jn (t) = 0 has infinitely many solutions.
3. Identities:
(a)
d n
dt [t Jn (t)]
(b)
d −n
Jn (t)] = −t−n Jn+1 (t);
dt [t
Jn′ (t) = 21 [Jn−1 (t) − Jn+1 (t)]
(c)
= tn Jn−1 (t).
= Jn−1 (t) − nt Jn (t) =
(d) Jn+1 (t) =
2n
t Jn (t)
J0′ (t) = −J1 (t)
n
t Jn (t)
− Jn−1 (t)
− Jn+1 (t)
(recurrence).
4. Orthogonality:
Solutions yn (λ0 x), yn (λ1 x), . . . , yn (λi x), . . . of the differential eigensystem
x2 y ′′ + xy ′ + (λ2 x2 − n2 )y = 0
A1 y(x1 ) − B1 y ′ (x1 ) = 0,
A2 y(x2 ) + B2 y ′ (x2 ) = 0
have the property
Z x2
xyn (λi x)yn (λj x) dx = 0
x1
for i 6= j, and
d
2 x2
Z x2
(λ2i x2 − n2 )yn2 (λi x) + x2 dx
yn (λi x) 2
xyn (λi x) dx =
2λ2i
x1
x1
for i = j.
17
5. Integration properties:
Z
(a)
tν Jν−1 (t) dt = tν Jν (t) + C
Z
(b)
t−ν Jν+1 (t) dt = −t−ν Jν (t) + C
Z
(c)
tJ0 (t) dt = tJ1 (t) + C
Z
(d)
t3 J0 (t) dt = (t3 − 4t)J1 (t) + 2t2 J0 (t) + C
Z
(e)
t2 J1 (t) dt = 2tJ1 (t) − t2 J0 (t) + C
Z
(f )
t4 J1 (t) dt = (4t3 − 16t)J1 (t) − (t4 − 8t2 )J0 (t) + C
LEGENDRE POLYNOMIALS
The differential equation
(1 − x2 )y ′′ − 2xy ′ + n(n + 1)y = 0
has solution y = Pn (x), where
[n/2]
2n − 2m n−2m
1 X
m n
(−1)
x
Pn (x) = n
2 m=0
m
n
on the interval [−1, 1]. The Legendre polynomials can also be obtained iteratively from the first two,
P0 (x) = 1
and
P1 (x) = x,
and the recurrence relation,
(n + 1)Pn+1 (x) = (2n + 1)xPn (x) − nPn−1 (x).
The Legendre polynomials are also given by Rodrigues’ formula:
Pn (x) =
(−1)n dn
[(1 − x2 )n ].
2n n! dxn
Orthogonality:
Z
1
−1
Pn (x)Pm (x) dx =
0,
2
2n+1 ,
Standardization:
Pn (1) = 1.
n 6= m;
n = m.
18
TRIGONOMETRIC FUNCTIONS IN A RIGHT TRIANGLE
If A, B, and C are the vertices (C the right angle), and a, b, and r the
sides opposite, respectively, then
b
cosine A = cos A = ,
r
b
cotangent A = cot A = ,
a
r
cosecant A = csc A = ,
a
exsecant A = exsec A = sec A − 1
a
,
r
a
tangent A = tan A = ,
b
r
secant A = sec A = ,
b
sine A = sin A =
B
r
A
versine A = vers A = 1 − cos A
a
coversine A = covers A = 1 − sin A
1
haversine A = hav A = vers A
2
C
b
TRIGONOMETRIC FUNCTIONS OF SPECIAL ANGLES
0
π
6
sin
0
cos
1
1
2
√
3
2
√
3
3
tan
cot
sec
csc
0
∞
1
∞
√
3
√
2 3
3
2
π
4
√
2
2
√
2
2
1
1
√
2
√
2
π
3
√
3
2
π
2
π
3π
2
1
0
−1
1
2
0
−1
0
∞
0
∞
0
∞
0
∞
−1
∞
1
∞
−1
√
3
√
3
3
2
√
2 3
3
COMPLEX EXPONENTIAL FORMS (i2 = −1)
1 ix
(e − e−ix )
2i
1
cos x = (eix + e−ix )
2
1 eix − e−ix
tan x =
i eix + e−ix
sin x =
2i
eix − e−ix
2
sec x = ix
e + e−ix
eix + e−ix
cot x = i ix
e − e−ix
csc x =
19
RELATIONS AMONG THE FUNCTIONS
sin x = csc1 x
cos x = sec1 x
sin x
tan x = cot1 x = cos
x
sin2 x + cos2 x = 1
1 + cot2 x = csc2 x
√
∗
sin x = ± 1 − cos2 x
√
∗
tan x = ± sec2 x − 1
√
∗
cot x = ± csc2 x − 1
sin x = cos( π2 − x) = sin(π − x)
tan x = cot( π2 − x) = − tan(π − x)
csc x
sec x
cot x
1 + tan2 x
∗
cos x
∗
sec x
∗
csc x
cos x
cot x
= sin1 x
= cos1 x
= tan1 x =
= sec2 x
cos x
sin x
p
= ± 1 − sin2 x
√
= ± 1 + tan2 x
√
= ± 1 + cot2 x
= sin( π2 − x) = − cos(π − x)
= tan( π2 − x) = − cot(π − x)
SUMS AND MULTIPLES OF ANGLES
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y ∓ sin x sin y
tan x ± tan y
tan(x ± y) =
1 ∓ tan x tan y
sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x
sin 3x = 3 sin x − 4 sin3 x
cos 3x = 4 cos3 x − 3 cos x
sin 4x = 8 sin x cos3 x − 4 sin x cos x
cos 4x = 8 cos4 x − 8 cos2 x + 1
2 tan x
cot2 x − 1
tan 2x =
cot
2x
=
2 cot x
1 − tan2 x
3 tan x − tan3 x
tan 3x =
1 − 3 tan2 x
r
r
1
1 − cos x
1 + cos x
1
∗
∗
sin x = ±
cos x = ±
2
2
2
2
r
1 − cos x
1 − cos x
sin x
1
∗
=
=
tan x = ±
2
1 + cos x
sin x
1 + cos x
* The choice of the sign in front of the radical depends on the quadrant
in which x, regarded as an angle, falls.
20
MISCELLANEOUS RELATIONS
sin x ± sin y = 2 sin 21 (x ± y) cos 21 (x ∓ y)
cos x + cos y = 2 cos 12 (x + y) cos 21 (x − y)
cos x − cos y = −2 sin 12 (x + y) sin 12 (x − y)
sin x cos y = 21 [sin(x + y) + sin(x − y)]
cos x sin y = 21 [sin(x + y) − sin(x − y)]
cos x cos y = 21 [cos(x + y) + cos(x − y)]
sin x sin y = 12 [cos(x − y) − cos(x + y)]
tan x ± tan y =
1+tan x
1−tan x =
sin x±sin y
cos x+cos y
sin x+sin y
sin x−sin y
2
sin(x±y)
cos x cos y
tan( π4 + x)
= tan 12 (x ± y)
=
tan
tan
1
2 (x+y)
1
2 (x−y)
± sin(x±y)
sin x sin y
cot x+1
π
=
cot(
cot x−1
4 − x)
sin x±sin y
1
cos x−cos y = − cot 2 (x ∓
cot x ± cot y =
y)
sin x − sin2 y = sin(x + y) sin(x − y)
cos2 x − cos2 y = − sin(x + y) sin(x − y)
cos2 x − sin2 y = cos(x + y) cos(x − y)
INVERSE TRIGONOMETRIC FUNCTIONS
The following table gives the domains of the inverse trigonometric functions.
Interval containing principal value
Function
x positive or zero
y = arcsin x and y = arctan x
0≤y≤
y = arccos x and y = arccotx
0≤y≤
y = arcsecx and y = arccscx
0≤y≤
π
2
π
2
π
2
x negative
− π2 ≤ y ≤ 0
π
2
≤y≤π
−π ≤ y ≤ − π2
Two notation systems are in common use. For example, the inverse sine is
often denoted by arcsin or sin−1 . In many references, however, “principal
value”is used for the function and is denoted by Arcsin or Sin−1 . Thus, in one
book,
π
π
1
and arcsin x = + 2nπ,
Arc sin = Sin−1 x =
2
6
6
while in another, only lower case is used:
1
π
= .
2
6
Special care is advisable in the use of reference materials.
arcsin
21
SIDE-ANGLE RELATIONS IN PLANE TRIANGLES
Letters A, B, and C denote the angles of a plane triangle with opposite sides a,
b, and c, respectively. The letter s denotes the semi-perimeter of the triangle,
that is,
1
s = (a + b + c).
2
b
c
a
=
=
= diameter of the circumscribed circle
sin A
sin B
sin C
a2 = b2 + c2 − 2 b c cos A
a = b cos C + c cos B
A−B
a−b
C
tan
=
cot
2
a+b
2
2p
sin A =
s(s − a)(s − b)(s − c)
bc
p
area = s(s − a)(s − b)(s − c)
r
(s − b)(s − c)
A
sin =
2
bc
r
A
s(s − a)
cos =
2
bc
s
A
(s − b)(s − c)
tan =
2
s(s − a)
cot 21 C
tan 12 (A + B)
a+b
sin A + sin B
=
=
=
1
a−b
sin A − sin B
tan 2 (A − B)
tan 21 (A − B)
h=d
sin α sin β
d
=
sin(α + β)
cot α + cot β
h=d
sin α sin β ′
d
=
sin(β ′ − α)
cot α − cot β ′
h
α
β
d
h
α
d
β′
22
HYPERBOLIC FUNCTIONS
1
1 x
(e − e−x ),
cosh x = (ex + e−x )
2
2
sinh x
1
ex − e−x
=±
=
tanh x = x
−x
e +e
cosh x
coth x
p
tanh x
1
2
= ±p
sinh x = ± cosh x − 1 = p
2
1 − tanh x
coth2 x − 1
p
1
coth x
cosh x = 1 + sinh2 x = p
= ±p
2
1 − tanh x
coth2 x − 1
p
1
sinh x
cosh2 x − 1
=±
=
tanh x = p
2
cosh
x
coth
x
1 + sinh x
p
1
1 + sinh2 x
cosh x
=
coth x =
= ±p
2
sinh x
tanh x
cosh x − 1
sinh x =
sinh(x ± y) = sinh x cosh y ± cosh x sinh y
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
coth x coth y ± 1
tanh x ± tanh y
coth(x ± y) =
tanh(x ± y) =
1 ± tanh x tanh y
coth y ± coth x
sinh 2x = 2 cosh x sinh x
cosh 2x = cosh2 x + sinh2 x
p
p
cosh−1 x = ln(x + x2 − 1)
sinh−1 x = ln(x + x2 + 1)
1 x + 1 1 1 + x −1
coth
x
=
ln
tanh−1 x = ln
2
1 − x
2 x − 1
Complex hyperbolic functions involve the Euler relations
eiz = cos z + i sin z;
cos z =
eiz + e−iz
,
2
cos z = cosh iz
sin z = −i sinh iz
sin z =
eiz − e−iz
.
2i
cosh z = cos iz
sinh z = −i sin iz
tan z = −i tanh iz
tanh z = −i tan iz
cot z = i coth iz
coth z = i cot iz
sinh(x ± iy) = sinh x cos y ± i cosh x sin y
cosh(x ± iy) = cosh x cos y ± i sinh x sin y
1
ln ix = ln |x| + (2n + )πi
2
and ln(−x) = ln |x| + (2n + 1)πi (n = 0, ±1, ±2, . . .)
ln z = ln |z| + i arg z;
so
23
FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS
I. Separable equations : N (y) dy = M (x) dx. To solve, integrate both sides:
Z
Z
N (y) dy = M (x) dx + c.
II. Exact equations : M (x, y) dx + N (x, y) dy = 0, where
solve,
integrate
f (x, y) =
Z
∂f
= M (x, y)
∂x
∂f
= N (x, y) :
∂y
and
M (x, y) dx + φ(y) =
Z
∂M
∂N
=
. To
∂y
∂x
N (x, y) dy + ψ(x) = c.
III. Linear equations :
dy
+ p(x) y = q(x).
dx
The solution is given by
Z
1
c
,
y=
P (x)q(x) dx +
P (x)
P (x)
where
P (x) = e
R
p(x) dx
IV. Homogeneous equations :
y
dy
or
=f
dx
x
dy
x
.
=f
dx
y
dy
du
dx
dx
=x
+ u or x = vy,
=y
+ v,
dx
dx
dy
dy
integrate the resulting expression, then resubstitute for u or v.
Substitute y = ux,
V. Equations in the rational form
ax + by + c
dy
=
,
dx
dx + ey + f
where a, b, c, d, e, f are constants such that ae 6= bd. Substitute
x = X − h, y = Y − k,
where h =
to obtain the homogeneous equation
bf − ce
cd − af
, k=
,
ae − bd
ae − bd
aX + bY
dY
=
.
dX
dX + eY
This equation may be solved as in item IV above.
.
24
LINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS
I. Homogeneous equation ay ′′ + by ′ + cy = 0. Substitute y = erx to find the
characteristic equation, and write its roots r1 and r2 :
√
−b ± b2 − 4ac
2
.
ar + br + c = 0;
r1 , r2 =
2a
The solution, yc , is given by the following table.
Case 1. b2 − 4ac > 0,
real and distinct roots.
y c = c1 e r 1 x + c2 e r 2 x
Case 2. b2 − 4ac = 0,
two real equal roots.
yc = (c1 + c2 x)erx
Case 3. b2 − 4ac < 0,
two complex roots.
yc = eαx (c1 cos βx + c2 sin βx),
where
√
−b
4ac − b2
α=
, β=
2a
2a
II. Undetermined Coefficients : ay ′′ + by ′ + cy = F (x).
1. Solve the homogeneous equation ay ′′ + by ′ + cy = 0 for yc .
2. Assume a particular solution yp for any term in F (x) that is one of three
special types, as follows.
A. For a polynomial of degree N, yp is a polynomial of degree N.
B. For an exponential kep(x) , yp = Aep(x) .
C. For j cos ωx + k sin ωx, yp = A cos ωx + B sin ωx.
3. If any term in yp found above appears in yc , multiply the corresponding
trial term by the power of x just high enough that the resulting product
contains no term of yc .
4. For any term in F (x) that is a product of terms of the three types listed
above, let yp be the product of the corresponding trial solutions.
5. Substitute yp into the given O.D.E. and evaluate the unknown coefficients
by equating like terms.
6. The general solution of the given equation is
y = yc + yp .
Evaluate the two arbitrary constants c1 and c2 in the general solution y
by applying boundary/initial conditions, as appropriate.
25
III. Variation of Parameters : ay ′′ + by ′ + cy = F (x)∗ .
1. Write the solution of the homogeneous equation ay ′′ + by ′ + cy = 0 in the
form
y c = c1 y 1 + c2 y 2 .
2. Assume a particular solution yp for any term in the form
yp = u 1 y1 + u 2 y2 ,
where u1 and u2 are functions determined by the conditions
u′1 y1 + u′2 y2 = 0
u′1 y1′ + u′2 y2′ = G(x) where G(x) =
F (x)
.
a
3. The solution of this system of equations is given by
y2 G
y1 G
and u′2 =
,
W
W
where W denotes the Wronskian determinant:
y1 y2 6= 0.
W = W (x) = ′
y1 y2′ u′1 = −
4. Integrate to find the unknown functions:
Z
Z
y1 (x)G(x)
y2 (x)G(x)
dx and u2 (x) =
dx.
u1 (x) = −
W (x)
W (x)
5. Form the particular solution yp = u1 y1 + u2 y2 .
6. The general solution of the given equation is
y = yc + yp .
Evaluate the two arbitrary constants c1 and c2 in the general solution y
by applying boundary/initial conditions, as appropriate.
IV. Laplace Transforms : ay ′′ + by ′ + cy = F (x), y(0) = y0 and y ′ (0) = y0′ .
1. Take the Laplace transform (see pages 9-10) of both sides of the given
equation, using the given initial conditions. The result is an algebraic
equation for F (s), the Laplace transform of the solution.
2. Solve the algebraic equation obtained in Step 1 to find F (s) explicitly.
3. The inverse Laplace transform of F (s) is the solution of the given initial
value problem.
*The coefficients may also be functions, i.e., b = b(x) and c = c(x).
26
VECTOR ALGEBRA
1. Scalar and vector products. Given vectors
a = α1 e1 + α2 e2 + α3 e3 = ax i + ay j + az k
b = β1 e1 + β2 e2 + β3 e3 = bx i + by j + bz k,
c = γ1 e1 + γ2 e2 + γ3 e3 = cx i + cy j + cz k,
let φ be the angle from a to b. The scalar (dot) product of a and b is the
scalar
a · b = |a||b| cos φ = ax bx + ay by + az bz .
The vector (cross) product of a and b is the vector of magnitude
|a × b| = |a||b| sin φ,
perpendicular to a and b and in the direction of the axial motion of a
right-handed screw turning a into b. In coordinate form,
e2 × e3 α1 β1 a × b = e3 × e1 α2 β2 e1 × e2 α3 β3 i ax b x ax ay az ax ay az k
j+
i+
= j ay b y = bx by bz bx by bz k az b z = (ay bz − az by )i + (az bx − ax bz )j + (ax by − ay bx )k.
The box product of a, b, and c is the scalar quantity given by
a · b × c ≡ [abc] = [bca] = [cab] = −[bac] = −[cba] = −[acb]
α1 β1 γ1 a x b x cx = α2 β2 γ2 [e1 e2 e3 ] = ay by cy .
α3 β3 γ3 a z b z cz The product of two box products is given by
a·d a·e
[abc][def ] = b · d b · e
c·d c·e
A special case gives Gram′ s determinant :
a·a a·b a·c
[abc]2 = b · a b · b b · c c·a c·b c·c
= [(a × b)(b × c)(c × a)]
a · f b · f .
c·f = (a · a)(b · b)(c · c) + 2(a · b)(b · c)(a · c) − (a · a)(b · c)2 −
(b · b)(a · c)2 − (c · c)(a · b)2 .
27
VECTOR ALGEBRA (Continued)
The vector triple product of vectors a, b, and c is given by
b
c a × (b × c) = (a · c)b − (a · b)c = .
a · b a · c
Three additional formulas for scalar and vector products:
a·c b·c
.
(a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c) = a ·d b ·d
(a × b)2 = (a · a)(b · b) − (a · b)2
(a × b) × (c × d) = [acd]b − [bcd]a = [abd]c − [abc]d.
VECTOR ANALYSIS
Differential operators
∂Φ
∂Φ
∂Φ
i+
j+
k
∂x
∂y
∂z
∂Fx
∂Fy
∂Fz
∇ · F(x, y, z) = div F(x, y, z) ≡
+
+
∂x
∂y
∂z
∇ × F = curl F(x, y, z)
∂Fx
∂Fy
∂Fy ∂Fz ∂Fx ∂Fz
i+
j+
k
−
−
−
≡
∂y
∂z
∂z
∂x
∂x
∂y
i
j
k ∂
∂
∂ = ∂x
∂y
∂z F
Fy Fz x
∂F
∂F
∂F
+ Gy
+ Gz
(G · ∇)F = Gx
∂x
∂y
∂z
= (G · ∇Fx )i + (G · ∇Fy )j + (G · ∇Fz )k
∂2
∂2
∂2 ∇2 = (∇ · ∇) ≡
+
+
(Laplace operator)
∂x2
∂y 2
∂z 2
div grad Φ = ∇ · (∇Φ) = ∇2 Φ
∇Φ(x, y, z) = grad Φ(x, y, z) ≡
grad div F = ∇(∇ · F) = ∇2 F + ∇ × (∇ × F)
curl curl F = ∇ × (∇ × F) = ∇(∇ · F) − ∇2 F
curl grad Φ = ∇ × (∇Φ) = 0
div curl F = ∇ · (∇ × F) = 0
∇(ΦΨ) = Ψ∇Φ + Φ∇Ψ
∇2 (ΦΨ) = Ψ∇2 Φ + 2(∇Φ) · (∇Ψ) + Φ∇2 Ψ
28
VECTOR ANALYSIS (Continued)
∇(F · G) = (F · ∇)G + (G · ∇)F + F × (∇ × G) + G × (∇ × F)
∇ · (ΦF) = Φ∇ · F + (∇Φ) · F
∇ · (F × G) = G · ∇ × F − F · ∇ × G
(G · ∇)(ΦF) = F(G · ∇Φ) + Φ(G · ∇)F
∇ × (ΦF) = Φ∇ × F + (∇Φ) × F
∇ × (F × G) = (G · ∇)F − (F · ∇)G + F(∇ · G) − G(∇ · F)
Cylindrical : x = r cos θ, y = r sin θ, z = z
∂Φ
1 ∂Φ
∂Φ
θ̂ +
k̂
r̂ +
∂r
r ∂θ
∂z
1 ∂(rFr ) 1 ∂Fθ
∂Fz
div F = ∇ · F =
+
+
r ∂r
r ∂θ
∂z
1 1r r̂
θ̂
k̂
r ∂
∂
∂ curl F = ∇ × F = ∂r
∂θ
∂z Fr rFθ Fz ∂Φ
1 ∂2Φ ∂2Φ
1 ∂
2
r
+ 2 2 +
Laplacian Φ = ∇ Φ =
r ∂r
∂r
r ∂θ
∂z 2
grad Φ = ∇Φ =
Spherical : x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ
1 ∂Φ
1 ∂Φ
∂Φ
ρ̂ +
φ̂ +
θ̂
∂ρ
ρ ∂φ
ρ sin φ ∂θ
1 ∂(ρ2 Fρ )
1 ∂(sin φ Fφ )
1 ∂Fθ
div F = ∇ · F = 2
+
+
ρ
∂ρ
ρ sin φ
∂φ
ρ sin φ ∂θ
1
1
1
ρ2 sin φ ρ̂ ρ sin
φ φ̂
ρ θ̂
∂
∂
∂
curl F = ∇ × F = ∂ρ
∂φ
∂θ
Fρ
ρFφ
ρ sin φ Fθ 1
∂Φ
1
∂
∂2Φ
1 ∂
2 ∂Φ
2
ρ
+ 2
sin φ
+ 2 2
∇ Φ= 2
ρ ∂ρ
∂ρ
ρ sin φ ∂φ
∂φ
ρ sin φ ∂θ2
grad Φ = ∇Φ =
Integral Theorems
R
R
∇ · F(r) dV = S F(r) · dA (Divergence theorem)
V
R
R
R
∇Φ · ∇Ψ dV + V Ψ∇2 Φ dV = S (Ψ∇Φ) · dA (Green’s theorem)
V
R
R
2
2
V (Ψ∇ Φ − Φ∇ Ψ) dV = S (Ψ∇Φ − Φ∇Ψ) · dA (symmetric form)
R
R
2
(Gauss’ theorem)
V ∇ Φ dV = S ∇Φ · dA
H
R
S [∇ × F(r)] · dA = C F(r) · dr (Stokes’ theorem)
29
FUNDAMENTALS OF SIGNALS AND NOISE
Complex numbers (j 2 = −1)
a + bj = Rejθ ,
where
p
R = a2 + b 2 ,

arctan ab ,



arctan ab ± π,
θ=
π

,

 π2
−2,
Aejθ · Bejφ = ABej(θ+φ)
ejθ = ej(θ+2nπ) ,
a > 0;
a < 0;
a = 0, b > 0;
a = 0, b < 0.
n = ±1, ±2, . . .
Unit phasor
π
j = ej 2
ejπ = −1
1 = ej·0 = ej2π
π
−j = e−j 2 = ej
3π
2
Euler Identity
e
±jx
= cos x ± j sin x;
(
cos x =
sin x =
ejx +e−jx
;
2
ejx −e−jx
.
2j
Fourier series (Periodic signals. For the real number form, see p. 13.)
x(t) =
∞
X
cn ej2πnf1 t ,
where
n=−∞
cn =
1
T1
Z
T1
2
T
− 21
x(t)e−j2πnf1 t dt,
T1 = period,
1
= fundamental frequency,
f1 =
T1
(For real signals, c−n = c∗n (complex conjugate).)
30
Fourier transform (aperiodic signals)
X(f ) =
Z
∞
x(t)e−j2πf t dt;
x(t) =
−∞
Z
∞
X(f )ej2πtf df.
−∞
The correspondence between x(t) and X(f ) is denoted by
x(t) ↔ X(f ).
Transform of Fourier Series
∞
X
If x(t) =
cn ej2πnf1 t ,
then X(f ) =
n=−∞
∞
X
n=−∞
cn δ(f − nf1 ).
Convolution (∗) and Correlation (⋆)
x ∗ y=
Z
∞
x(τ )y(t − τ ) dτ ≡
−∞
x ⋆ y=
Z
∞
Z
∞
−∞
y(τ )x(t − τ ) dτ = y ∗ x
x(τ )y(t + τ ) dτ
−∞
Properties of the delta function
Z
∞
−∞
Z
∞
δ(t) dt = 1
−∞
δ(t − t0 )x(t) dt = x(t0 )
x(t) ∗ δ(t − t0 ) = x(t − t0 )
x(t)δ(t − t0 ) = x(t0 )δ(t − t0 )
Parseval theorem (Average Power in a periodic signal)
1
P =
T
Z
T
2
− T2
|x(t)|2 dt =
∞
X
n=−∞
|cn |2
Rayleigh theorem (Total Energy in an aperiodic signal)
E=
Z
∞
−∞
|x(t)|2 dt =
Z
∞
−∞
|X(f )|2 df
31
Fourier theorems
If x(t) ↔ X(f ) is a transform pair , then
LINEARITY
ax(t) + by(t) ↔ aX(f ) + bY (f )
SHIFT/MODULATION
x(t − t0 ) ↔ X(f )e−j2πt0 f
x(t)ej2πf0 t ↔ X(f − f0 )
PRODUCT/CONVOLUTION
x(t)y(t) ↔ X ∗ Y
x ∗ y ↔ X(f )Y (f )
1
f
x(at) ↔
X
|a|
a
d
x(t) ↔ j2πf X(f )
dt
Z t
X(f )
x(τ ) dτ ↔
.
j2πf
−∞
SCALE
DIFFERENTIATION
INTEGRATION
Fourier pairs
Time Domain
Frequency Domain
h(t) = 1 − u T0 (|t|)
↔
2
H(f ) =
h(t)
1
-T0/2
h(t) =
T0
T0/2
sin(2πf0 t)
2πf0 t
1
sin(πT0 f )
πf
t
↔
1/T0 2/T0
H(f ) =
f
1
(1 − uf0 (|f |))
2f0
h(t)
1/2f0 1/f0
H(f)
H(f)
1/2f0
t
-f0
f0
f
32
Fourier pairs Continued
Time Domain
h(t) = K
h(t)
Frequency Domain
H(f ) = Kδ(f )
H(f)
K
↔
K
t
K
h(t) = Kδ(t)
h(t)
f
↔
H(f ) = K
H(f)
K
f
t
h(t) = A cos(2πf0 t)
A
A
[δ(f + f0 ) + δ(f − f0 )]
2
H(f)
A/2
↔
h(t)
H(f ) =
-f0
1/f0 t
h(t) = A sin(2πf0 t)
A
H(f ) =
1/f0
f0
-f0
t
|t|
h(t) = 1 −
1 − uT0 (|t|)
T0
h(t)
1
-T0
T0
f
A
j[δ(f + f0 ) − δ(f − f0 )]
2
Im[H(f)]
A/2
↔
h(t)
f0
t
↔
f
H(f ) =
T0
sin2 (πT0 f )
T0 π 2 f 2
H(f)
1/T0 2/T0 f
33
Fourier pairs Continued
Time Domain
h(t) = e−α|t|
Frequency Domain
2α
H(f ) = 2
α + 4π 2 f 2
↔
h(t)
1
2/α H(f)
t
h(t) = exp(−αt2 )
↔
f
H(f ) =
h(t)
1
r
−π 2 f 2
π
exp
α
α
√π/α

H(f)
t
h(t) =
e−αt ,
0,
t>0
t≤0
f
↔
H(f ) =
h(t)
1
1
α + 2πjf
1/α |H(f)|
t
h(t) = sgn(t)
f
↔
H(f ) =
−j
πf
Im[H(f)]
h(t)
1
-1
t
f
34
Linear time − invariant systems
Basic Properties
If
x(t)
y(t)
g(t)
q(t)
then
ax(t − t0 ) + bg(t − t0 )
ay(t − t0 ) + bq(t − t0 )
Transform properties
Time Domain
Input
Output
δ(t)
h(t)
x(t)
y(t) = x(t) ∗ h(t)
system
Frequency Domain
h(t) ↔ H(f )
X(f )
Y (f ) = X(f ) · H(f )
h(t) ≡ impulse response;
H(f ) ≡ transfer function
Random signals
Z
∞
Z
∞
1
< x, x >= kxk =
T
Z
1
<x>=
T
Z
x=
xp(x) dx
Mean value
−∞
x2 =
x2 p(x) dx
Second moment
−∞
2
T
2
− T2
|x(t)|2 dt
Norm square
T
2
x(t) dt
Time average
− T2
For an ergodic process:
Second moment = Norm square
and
Mean value = Time average.
35
Autocorrelation
1
Rxx (τ ) = lim
T →∞ T
Z
T
2
x(t)x(t + τ ) dt
− T2
Power Spectral Density
The power spectral density Gxx is the Fourier transform of the autocorrelation function:
Z ∞
Z ∞
Rxx (t) =
Gxx (f )ej2πtf df ↔ Gxx (f ) =
Rxx (t)e−j2πf t dt
−∞
−∞
Noise power (Gxx denotes the power spectral density)
P Total =
Z
Gxx df = Rxx (0) = x2 = (x)2
P ac = N = σx2 = noise power
P dc = Rxx (∞) = (x)2
Noise bandwidth (Gxx denotes the power spectral density)
Gyy = Gxx · |H|2
Ny =
Z
∞
−∞
(linear system)
Gxx · |H|2 df = noise power output
N = n0 BN = noise power, where
BN =
Z
∞
−∞
n0
= white noise
2
|H|2 df = noise bandwidth
36
Ideal filter (low − pass)
h(t) =
sin(2πf0 t)
2πf0
1
↔
H(f ) =
1 − uf0 (|f |)
2f0
H(f)
1/2f0
h(t)
-f0
t
1/2f0 1/f0
f0
f
RC filter (low − pass)
VC = h(t)∗E(t)
h(t) =
VC
R
1
1 −t/RC
e
RC
↔
H(f ) =
h(t)
1
1 + 2πjf RC
1
|H(f)|
C
E(t)
t
f
Ideal filter (high − pass)
h(t) = δ(t) −
sin(2πf0 t)
πt
H(f ) = uf0 (|f |)
↔
h(t)
H(f)
1/2f0 1/f0
1
t
-f0
f0
f
RC filter (high − pass)
VR = h(t)∗E(t)
h(t) = δ(t)−
1 −t/RC
e
RC
h(t)
VR
R
↔
H(f ) =
1
2πjf RC
1 + 2πjf RC
|H(f)|
C
E(t)
t
f
37
Ideal filter (band − pass)
h(t) =
sin(2πf2 t) sin(2πf1 t)
−
πt
πt
H(f ) = uf2 (|f |) − uf1 (|f |)
↔
H(f)
h(t)
1
t
-f2
-f1
f1
f2 f
LRC filter (band − pass)
VR = h(t) ∗ E(t)
h(t)
↔
H(f ) =
2πjf RC
1 − (2πf )2 LC + 2πjf RC
h(t)
VR
C
R
1
|H(f)|
L
t
E(t)
−1/(2π√
LC
 ) 1/(2π√
LC) f
Ideal sampling
Sampled signal
xδ (t) = x(t) · sδ (t)
where
sδ =
∞
X
k=−∞
δ(t − kTs )
Spectrum after sampling
Xδ (f ) = X(f ) ∗ Sδ (f )
where
Sδ (f ) = fs
∞
X
n=−∞
δ(f − nfs )
Sampling theorem
If
1
or
fs > 2W
(“Nyquist rate”)
2W
then the signal can be reconstructed with a filter of bandwidth B, given
by:
W < B < fs − W.
Ts <