Trigonometry
Trigonometric Functions
T1.
sin2 x + cos2 x = 1
T2.
tan2 x + 1 = sec2 x
T3.
cot2 x + 1 = csc2 x
T4.
sin(x ± y) = sin x cos y ± cos x sin y
T5.
cos(x ± y) = cos x cos y ∓ sin x sin y
T6.
tan(x ± y) =
tan x ± tan y
1 ∓ tan x tan y
T7.
x
sin x
tan( ) =
2
1 + cos x
T8.
sin(2x) = 2 sin x cos x
T9.
cos(2x) = cos2 x − sin2 x
T10.
sin2 x = 1/2(1 − cos(2x))
T11.
cos2 x = 1/2(1 + cos(2x))
T12.
¡
sin x sin y = 1/2 cos(x − y) − cos(x + y))
T13.
¡
cos x cos y = 1/2 cos(x − y) + cos(x + y))
T14.
¡
sin x cos y = 1/2 sin(x − y) + sin(x + y))
1
T15.
c1 cos(ωt) + c2 sin(ωt) = A sin(ωt + φ),
q
where A =
c21 + c22 , φ = 2 arctan
c1
c2 + A
Hyperbolic Functions
T16.
cosh x =
ex + e−x
2
T17.
sinh x =
ex − e−x
2
T18.
cosh2 x − sinh2 x = 1
T19.
tanh2 x + sech2 x = 1
T20.
coth2 x − csch2 x = 1
T21.
sinh(x ± y) = sinh x cosh y ± cosh x sinh y
T22.
cosh(x ± y) = cosh x cosh y ± sinh x sinh y
T23.
tanh(x ± y) =
tanh x ± tanh y
1 ± tanh x tanh y
T24.
sinh(2x) = 2 sinh x cosh x
T25.
cosh(2x) = cosh2 x + sinh2 x
T26.
¡
sinh x sinh y = 1/2 cosh(x + y) − cosh(x − y))
T27.
¡
cosh x cosh y = 1/2 cosh(x + y) + cosh(x − y))
2
¡
sinh x cosh y = 1/2 sinh(x + y) + sinh(x − y))
T28.
Power Series
P1.
P2.
P3.
ex =
sin x =
∞
X
xn
x2
x3
=1+x+
+
+ ··· ,
n!
2!
3!
n=0
∞
X
(−1)n
n=0
∞
X
x3
x5
x2n+1
=x−
+
− ··· ,
(2n + 1)!
3!
5!
(−1)n
cos x =
n=0
x3
2
17 7
+ x5 +
x + ··· ,
3
15
315
tan x = x +
P5.
∞
X
1
=
xn = 1 + x + x2 + x3 + · · · ,
1 − x n=0
∞
X
x2n
x2
x4
=1+
+
+ ··· ,
(2n)!
2!
4!
n=0
cosh x =
P8.
tanh x = x −
P10.
−∞ < x < ∞
−
π
π
<x<
2
2
−1 < x < 1
∞
X
x2n+1
x3
x5
sinh x =
=x+
+
+ ··· ,
(2n + 1)!
3!
5!
n=0
P7.
P9.
−∞ < x < ∞
x2
x4
x2n
=1−
+
− ··· ,
(2n)!
2!
4!
P4.
P6.
−∞ < x < ∞
2
17 7
x3
+ x5 −
x + ··· ,
3
15
315
f (x) = f (a) + f 0 (a)(x − a) +
−∞ < x < ∞
−∞ < x < ∞
−
π
π
<x<
2
2
f 00 (a)
f 000 (a)
(x − a)2 +
(x − a)3 + · · ·
2!
3!
Taylor Series with remainder:
f (x)
RN +1 (x) =
=
f
PN
(N +1)
n=0
(ξ)
(N +1)!
f (n) (a)
n!
(x − a)n + RN +1 (x),
(x − a)N +1
where
for some ξ between a and x.
Derivative formulas
3
1. Product rule: (f (x)g(x))0 = f 0 (x)g(x) + f (x)g 0 (x)
2. Chain rule:
f (g(x))0 = f 0 (g(x))g 0 (x)
du
du dx
=
dt
dx dt
3. Quotient rule: (
4.
5.
6.
f (x) 0
f 0 (x)g(x) − f (x)g 0 (x)
) =
g(x)
g(x)2
d
cx
cx
dx (e ) = c · e
d
dx sin(cx) = c · cos(cx)
d
dx cos(cx) = −c · sin(cx)
Algebra formulas
1.
2.
3.
4.
√
2
b −4ac
Quadratic formula: Roots of ax2 + bx + c = 0 are: x = −b± 2a
2
2
a − b = (a − b)(a + b)
a3 − b3 = (a − b)(a2 + ab + b2 ) and a3 + b3 = (a + b)(a2 − ab + b2 )
an − bn = (a − b)(an−1 + ban−2 + ... + bn−2 a + bn−1 )
Geometry formulas
1. area of a triangle of base b, height h: A = bh/2
2. volume of a sphere of radius r: V = 4πr3 /3
2. surface area of a sphere of radius r: V = 4πr2
4
Table of Integrals
A constant of integration should be added to each formula. The letters a, b,
m, and n denote constants; u and v denote functions of an independent variable
such as x.
Standard Integrals
Z
un du =
I1.
Z
I2.
un+1
,
n+1
n 6= −1
du
= ln |u|
u
Z
eu du = eu
I3.
Z
au du =
I4.
au
,
ln a
a>0
Z
I5.
cos u du = sin u
Z
I6.
sin u du = − cos u
Z
sec2 u du = tan u
I7.
Z
csc2 u du = − cot u
I8.
Z
I9.
sec u tan u du = sec u
Z
I10.
csc u cot u du = − csc u
Z
I11.
tan u du = − ln | cos u|
Z
I12.
cot u du = ln | sin u|
5
Z
I13.
sec u du = ln | sec u + tan u|
Z
I14.
csc u du = ln | csc u − cot u|
Z
³u´
du
1
arctan
=
a2 + u2
a
a
Z
³u´
du
= arcsin
a
a2 − u2
Z
Z
u dv = uv − v du
I15.
√
I16.
I17.
Integrals involving au + b
Z
(au + b)n du =
I18.
Z
I19.
Z
I20.
Z
I21.
I22.
Z
I23.
I24.
Z
I25.
I26.
n 6= −1
1
du
= ln |au + b|
au + b
a
u du
u
b
= − 2 ln |au + b|
au + b
a a
u du
b
1
= 2
+
ln |au + b|
(au + b)2
a (au + b) a2
¯
¯
Z
1 ¯¯ u ¯¯
du
= ln¯
u(au + b)
b
au + b ¯
√
2(3au − 2b)
u au + b du =
(au + b)3/2
15a2
Z
u du
2(au − 2b) √
√
=
au + b
3a2
au + b
√
u2 au + b du =
Z
(au + b)n+1
,
(n + 1)a
¢
2 ¡ 2
8b − 12abu + 15a2 u2 (au + b)3/2
3
105a
¢√
u2 du
2 ¡ 2
2 2
√
=
8b
−
4abu
+
3a
u
au + b
15a3
au + b
6
Integrals involving u2 ± a2
Z
I27.
¯
¯
du
1 ¯¯ u − a ¯¯
=
ln
u2 − a2
2a ¯ u + a ¯
Z
¯
¯
u du
= 1/2 ln ¯u2 ± a2 ¯
u2 ± a2
¯
¯
Z
u2 du
a ¯¯ u − a ¯¯
ln
=
u
+
u2 − a2
2 ¯u + a¯
Z
³u´
u2 du
=
u
−
a
arctan
u2 + a2
a
¯
¯
Z
¯ u2 ¯
du
1
¯
¯
=
±
ln
u(u2 ± a2 )
2a2 ¯ u2 ± a2 ¯
I28.
I29.
I30.
I31.
Integrals involving
Z
√
I32.
Z
I33.
√
u2 ± a2
p
u du
= u2 ± a2
2
2
u ±a
p
¢3/2
1¡ 2
u u2 ± a2 du =
u ± a2
3
Z
I34.
I35.
I36.
I37.
I38.
I39.
¯
¯
p
du
¯
¯
= ln¯u + u2 ± a2 ¯
u2 ± a2
Z
¯
p
up 2
a2 ¯¯
u2 du
¯
√
=
u ± a2 ∓
ln¯u + u2 ± a2 ¯
2
2
2
2
u ±a
¯
¯
Z
¯
du
1 ¯¯
u
¯
√
√
= ln¯
2
2
2
2
a
u u +a
a+ u +a ¯
Z
³u´
du
1
√
= arcsec
a
a
u u2 − a2
√
Z
u2 ± a2
du
√
=∓
2
2
2
a2 u
u u ±a
√
Z p
u2 ± a2 du =
¯
p
up 2
a2 ¯¯
¯
u ± a2 ±
ln¯u + u2 ± a2 ¯
2
2
7
Z
I40.
u2
¯
p
¢3/2 a2 u p
u¡ 2
a4 ¯¯
¯
u ± a2
u2 ± a2 −
ln¯u + u2 ± a2 ¯
∓
4
8
8
¯
¯
Z √ 2
¯ a + √u2 + a2 ¯
p
u + a2
¯
¯
2
2
du = u + a − a ln¯
¯
¯
¯
u
u
Z √ 2
³u´
p
u − a2
du = u2 − a2 − a arcsec
u
a
√
√
Z
¯
¯
p
u2 ± a2
u2 ± a2
¯
2 ± a2 ¯¯
+
ln
u
+
du
=
−
u
¯
u2
u
p
u2 ± a2 du =
I41.
I42.
I43.
Integrals involving
Z p
I44.
a2 − u2 du =
Z
√
I45.
√
a2 − u2
³u´
up 2
a2
a − u2 +
arcsin
2
2
a
p
u du
= − a2 − u2
a2 − u2
Z
p
¢3/2
1¡ 2
u a2 − u2 du = −
a − u2
3
¯
Z √ 2
¯ a + √a2 − u2
p
a − u2
¯
du = a2 − u2 − a ln ¯
¯
u
u
¯
¯
√
Z
1 ¯¯ a + a2 − u2 ¯¯
du
√
= − ln ¯
¯
¯
a ¯
u
u a2 − u2
I46.
I47.
I48.
Z
I49.
I50.
I51.
I52.
u2
p
¯
¯
¯
¯
¯
³u´
¢3/2 a2 u p
u¡ 2
a4
a − u2
+
a2 − u2 +
arcsin
4
8
8
a
√
Z √ 2
³u´
a − u2
a2 − u2
du
=
−
−
arcsin
u2
u
a
Z
³u´
up 2
a2
u2 du
√
=−
a − u2 +
arcsin
2
2
a
a2 − u2
√
Z
du
a2 − u2
√
=−
a2 u
u2 a2 − u2
a2 − u2 du = −
Integrals involving trigonometric functions
8
Z
I53.
sin2 (au) du =
u sin(2au)
−
2
4a
cos2 (au) du =
u sin(2au)
+
2
4a
Z
I54.
Z
1
sin (au) du =
a
µ
cos3 (au)
− cos(au)
3
3
I55.
Z
1
cos (au) du =
a
3
I56.
µ
sin3 (au)
sin(au) −
3
Z
sin2 (au) cos2 (au) du =
I57.
Z
tan2 (au) du =
I58.
Z
sec3 (au) du =
I60.
csc3 (au) du = −
u sin(au) du =
1
(sin(au) − au cos(au))
a2
u cos(au) du =
1
(cos(au) + au sin(au))
a2
Z
I63.
Z
I64.
u2 sin(au) du =
¢
1 ¡
2au sin(au) − ( a2 u2 − 2 ) cos(au)
a3
u2 cos(au) du =
¢
1 ¡
2au cos(au) + ( a2 u2 − 2 ) sin(au)
3
a
Z
I65.
Z
I66.
sin(au) sin(bu) du =
sin(a − b)u sin(a + b)u
−
,
2(a − b)
2(a + b)
a2 6= b2
cos(au) cos(bu) du =
sin(a − b)u sin(a + b)u
+
,
2(a − b)
2(a + b)
a2 6= b2
Z
I67.
Z
I68.
1
tan(au) − u
a
1
1
csc(au) cot(au) +
ln | csc(au) − cot(au) |
2a
2a
Z
I62.
1
u
−
sin(4au)
8 32a
1
1
sec(au) tan(au) +
ln | sec(au) + tan(au) |
2a
2a
Z
I61.
¶
1
cot2 (au) du = − cot(au) − u
a
I59.
Z
¶
sin(au) cos(bu) du = −
cos(a − b)u cos(a + b)u
−
,
2(a − b)
2(a + b)
9
a2 6= b2
Z
1
n−1
sin u du = − sinn−1 u cos u +
n
n
Z
n
I69.
sinn−2 u du
Integrals involving hyperbolic functions
Z
I70.
sinh(au) du =
Z
sinh2 (au) du =
I71.
1
u
sinh(2au) −
4a
2
Z
I72.
cosh(au) du =
Z
cosh2 (au) du =
I73.
Z
I74.
sinh(au) cosh(bu) du =
1
cosh(au)
a
1
sinh(au)
a
u
1
+
sinh(2au)
2 4a
cosh ((a + b) u) cosh ((a − b) u)
+
2(a + b)
2(a − b)
Z
I75.
sinh(au) cosh(au) du =
1
cosh(2au)
4a
Z
I76.
tanh u du = ln(cosh u)
Z
sech u du = arctan(sinh u) = 2 arctan (eu )
I77.
Integrals involving exponential functions
Z
ueau du =
I78.
eau
(au − 1)
a2
Z
¢
eau ¡ 2 2
a u − 2au + 2
3
a
Z
Z
1 n au n
n au
un−1 eau du
u e du = u e −
a
a
u2 eau du =
I79.
I80.
Z
I81.
eau sin(bu) du =
eau
(a sin(bu) − b cos(bu))
a2 + b2
10
Z
eau cos(bu) du =
I82.
eau
(a cos(bu) + b sin(bu))
a2 + b2
Integrals involving inverse trigonometric functions
Z
I83.
arcsin
Z
I84.
arccos
Z
I85.
arctan
³u´
a
³u´
a
³u´
a
³u´
du = u arcsin
a
du = u arccos
du = u arctan
+
³u´
a
³u´
a
−
−
p
a2 − u2
p
a2 − u2
¢
a ¡ 2
ln a + u2
2
Integrals involving inverse hyperbolic functions
Z
I86.
arcsinh
³u´
a
du = u arcsinh
³u´
a
−
p
u2 + a2
I87.
Z
arccosh
³u´
a
Z
I88.
arctanh
¡u¢
√
u2 − a2
¡ ¢ √
= u arccosh ua + u2 − a2
du = u arccosh
³u´
a
a
−
du = u arctanh
³u´
a
+
arccosh
¡u¢
arccosh
¡u¢
a
a
¢
a ¡ 2
ln a − u2
2
Integrals involving logarithm functions
Z
I89.
ln u du = u(ln u − 1)
·
Z
I90.
un ln u du = un+1
¸
ln u
1
−
,
n + 1 (n + 1)2
11
n 6= −1
> 0;
< 0.
Wallis’ Formulas
I91.
Z
π/2
sinm x dx
=
0
=
R π/2
0
cosm x dx
(m−1)(m−3)...(2 or 1)
m(m−2)...(3 or 2) k,
where k = 1 if m is odd and k = π/2 if m is even.
Z
I92.
π/2
sinm x cosn x dx =
0
(m − 1)(m − 3) . . . (2 or 1)(n − 1)(n − 3) . . . (2 or 1)
k,
(m + n)(m + n − 2) . . . (2 or 1)
where k = π/2 if both m and n are even and k = 1 otherwise.
Gamma Function
Z
∞
Γ(x) =
tx−1 e−t dt,
x>0
0
Γ(x + 1) = xΓ(x)
Γ ( 1/2) =
Γ(n + 1) = n!,
√
π
if n is a non-negative integer
Fourier Series
f (x) =
∞
nπx
nπx ´
a0 X ³
an cos
+
+ bn sin
2
L
L
n=1
12
where
1
an =
L
1
L
bn =
Z
1
L
a0 =
Z
L
f (x) dx,
−L
L
f (x) cos
nπx
dx,
L
f (x) sin
nπx
dx.
L
−L
Z
L
−L
f (x) =
∞
X
cn eniπx/L
n=−∞
where
1
cn =
2L
Z
L
f (x)e−niπx/L dx.
−L
Bessel Functions
ν = arbitrary real number; n = integer
1. Definition.
x2 y 00 + xy 0 + (λ2 x2 − ν 2 )y = 0 has solutions
y = yν (λx) = c1 Jν (λx) + c2 J−ν (λx),
where
Jν (t) =
(ν 6= n),
∞
X
(−1)m tν+2m
.
2ν+2m m! Γ(ν + m + 1)
m=0
2. General Properties.
J−n (t) = (−1)n Jn (t);
J0 (0) = 1;
Jn (0) = 0,
3. Identities.
d ν
(a)
[t Jν (t)] = tν Jν−1 (t).
dt
(b)
d
J0 (t) = −J1 (t).
dt
d −ν
[t Jν (t)] = −t−ν Jν+1 (t);
dt
13
(n ≥ 1).
(c)
d
Jν (t)
dt
= 1/2 (Jν−1 (t) − Jν+1 (t))
= Jν−1 (t) − νt Jν (t)
= νt Jν (t) − Jν+1 (t) .
(d) Recurrence Relation.
Jν+1 (t) =
2ν
Jν (t) − Jν−1 (t)
t
4. Orthogonality.
Solutions yν (λ0 x), yν (λ1 x), . . . , yν (λn x), . . .
of the differential system
x2 y 00 + xy 0 + (λ2 x2 − ν 2 )y = 0,
x1 ≤ x ≤ x2
¯
µ
¶¯
d
¯
ak yν (λxk ) − bk
yν (λx) ¯
= 0,
k = 1, 2
¯
dx
x=xk
have the orthogonality property:
Z x2
xyν (λn x)yν (λm x) dx = 0,
(m 6= n)
x1
and
Z x2
x1
xyν2 (λm x) dx =
"
µ
¶2 #x2
¡ 2 2
¢ 2
d
1
2
2
λm x − ν yν (λm x) + x
yν (λm x)
,
2λ2m
dx
x1
5. Integrals.
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)
t J0 (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
14
(m = n) .
α
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Zeros and Associated Values of Bessel Functions
j0,α
J00 (j0,α )
j1,α
J10 (j1,α )
2.40483
-0.519147
3.83170
-0.402760
5.52008
0.340265
7.01559
0.300116
8.65373
-0.271452
10.1735
-0.249705
11.7915
0.232460
13.3237
0.218359
14.9309
-0.206546
16.4706
-0.196465
18.0711
0.187729
19.6159
0.180063
21.2116
-0.173266
22.7601
-0.167185
24.3525
0.161702
25.9037
0.156725
27.4935
-0.152181
29.0468
-0.148011
30.6346
0.144166
32.1897
0.140606
33.7758
-0.137297
35.3323
-0.134211
36.9171
0.131325
38.4748
0.128617
40.0584
-0.126069
41.6171
-0.123668
43.1998
0.121399
44.7593
0.119250
46.3412
-0.117211
47.9015
-0.115274
49.4826
0.113429
51.0435
0.111670
52.6241
-0.109991
54.1856
-0.108385
55.7655
0.106848
57.3275
0.105374
58.9070
-0.103960
60.4695
-0.102601
62.0485
0.101293
63.6114
0.100035
15
Table of Laplace Transforms
f (t) ≡ L−1 {F (s)} (t)
F (s) ≡ L {f (t)} (s)
Z
L1.
f (t)
∞
F (s) =
e−st f (t) dt
0
L2.
1, H(t), U (t)
1
s
L3.
U (t − a)
e−as
s
L4.
L5.
tn
n!
(n = 1, 2, 3, . . .)
ta
sn+1
Γ(a + 1)
sa+1
(a > −1)
1
s−a
L6.
eat
L7.
sin(ωt)
L8.
cos(ωt)
s
s2 + ω 2
L9.
f 0 (t)
sF (s) − f (0)
L10.
f 00 (t)
s2 F (s) − sf (0) − f 0 (0)
L11.
tn f (t)
s2
ω
+ ω2
(−1)n F (n) (s)
(n = 1, 2, 3, . . .)
L12.
eat f (t)
F (s − a)
L13.
eat L−1 {F (s + a)}
F (s)
RP
L14.
0
f (t + P ) = f (t)
16
e−st f (t) dt
1 − e−sP
Table of Laplace Transforms
f (t) ≡ L−1 {F (s)} (t)
F (s) ≡ L {f (t)} (s)
L15.
f (t)U (t − a)
e−as L {f (t + a)}
L16.
f (t − a)U (t − a)
e−as F (s)
Z
t
f (z) dz
F (s)
s
f (z)g(t − z) dz
F (s)G(s)
L17.
0
Z
t
L18.
0
L19.
Z
f (t)
t
∞
F (z) dz
s
L20.
¢
1 ¡ at
e −1
a
1
s (s − a)
L21.
tn eat , n = 1, 2, 3, . . .
n!
(s − a)n+1
L22.
eat − ebt
a−b
1
(s − a)(s − b)
L23.
aeat − bebt
a−b
s
(s − a)(s − b)
L24.
sinh(ωt)
L25.
cosh(ωt)
s
s2 − ω 2
L26.
sin(ωt) − ωt cos(ωt)
2ω 3
(s2 + ω 2 )2
L27.
t sin(ωt)
L28.
sin(ωt) + ωt cos(ωt)
s2
(s2
ω
− ω2
2ωs
+ ω 2 )2
2ωs2
+ ω 2 )2
(s2
17
Table of Laplace Transforms
f (t) ≡ L−1 {F (s)} (t)
F (s) ≡ L {f (t)} (s)
L29.
b sin(at) − a sin(bt)
ab(b2 − a2 )
1
(s2 + a2 )(s2 + b2 )
L30.
cos(at) − cos(bt)
b2 − a2
s
(s2 + a2 )(s2 + b2 )
L31.
a sin(at) − b sin(bt)
a2 − b2
s2
(s2 + a2 )(s2 + b2 )
L32.
e−bt sin(ωt)
ω
(s + b)2 + ω 2
L33.
e−bt cos(ωt)
s+b
(s + b)2 + ω 2
L34.
1 − cos(ωt)
ω2
s(s2 + ω 2 )
L35.
ωt − sin(ωt)
L36.
δ(t − a)
e−sa
L37.
δ(t − a)f (t)
f (a)e−sa
ω3
+ ω2 )
s2 (s2
a > 0, s > 0
a > 0, s > 0
Partial fraction decomposition (PFD) of N (x)/D(x): Let N (x) be a
polynomial of lower degree than another polynomial D(x).
1. Factor D(x) into a product of distinct terms of the form (ax + b)r or
(ax2 + bx + c)s , for some integers r > 0, s > 0. For each term (ax + b)r the PFD
A1
Ar
of N (x)/D(x) contains a sum of terms of the form (ax+b)
+ · · · + (ax+b)
r for
2
s
some contstants Ai , and for each term (ax + bx + c) the PFD of N (x)/D(x)
B1 x+C1
Bs x+Cs
contains a sum of terms of the form (ax
2 +bx+c) + · · · + (ax2 +bx+c)s . for some
(x)
constants Bi , Cj . N
D(x) is the sum of all these.
2. The next step is to solve for these A’s, B’s, C’s occurring in the numerators. Cross multiply both sides by D(x) and expand out the resulting polynomial
identity for N (x) in terms of the A’s, B’s, C’s. Equating coefficients of powers
of x on both sides gives rise to a linear system of equations for the A’s, B’s, C’s
which you can solve.
18
Table of Fourier Transforms
Complex Fourier Integral:
f (t) =
1
2π
Z
∞
Cω eiωt dω,
−∞
where Cω = F (ω) = F {f (t)} (ω) = fˆ(ω) =
a > 0 everywhere it appears)
F −1 {F (ω)} =
1
2π
R∞
−∞
F (ω)eiωt dω
R∞
−∞
f (t)e−iωt dt. (In the table
F {f (t)} =
R∞
−∞
F1.
f (t)
F (ω)
F2.
e−at H(t)
1
a+iω
F3.
eat H(−t) + e−at H(t)
2a
a2 +ω 2
F4.
−eat H(−t) + e−at H(t)

if − k < t < 0
 c
b
if 0 < t < m

0 otherwise
−2iω
a2 +ω 2
F5.
1
iω
©
f (t)e−iωt dt
(b − c) + ceiωk − be−iωm
2k
ω
F6.
k(H(t + a) − H(t − a))
F7.
f (t − t0 )
e−iωt0 F (ω)
F8.
eiω0 t f (t)
F (ω − ω0 )
F9.
f (kt)
1
ω
|k| F ( k )
F10.
f (−t)
F (−ω)
F11.
F (t)
2πf (−ω)
19
sin(aω)
ª
Table of Fourier Transforms
F −1 {F (ω)} =
1
2π
R∞
−∞
F (ω)eiωt dω
F {f (t)} =
R∞
−∞
f (t)e−iωt dt
2a
a2 +ω 2
F12.
e−a|t|
F13.
e−at
F14.
1
a2 +t2
π −a|ω|
ae
F15.
f (n) (t) (n = 0, 1, 2, 3, ...)
(iω)n F (ω)
F16.
tn f (t) (n = 0, 1, 2, 3, ...)
in F (n) (ω)
F17.
R∞
−∞
pπ
2
f (u)g(t − u)du
F18.
f (t)g(t)
F19.
δ(t)
2
−ω
4a
ae
F (ω)G(ω)
1
2π
R∞
−∞
F (x)G(ω − x)dx
1
20
Last modified 11-15-2006 by Prof A. M. Gaglione and Prof. W. D. Joyner.
21