Handbook of mathematics for engineers and scienteists part 172 potx
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T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1165
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
14
f(p)
(p + a)
2
+ b
2
1
b
x
0
e
–a(x–t)
sin
b(x – t)
f(t) dt
15
1
p
n
f(p), n = 1, 2,
1
(n – 1)!
x
0
(x – t)
n–1
f(t) dt
16
f
1
(p)
f
2
(p)
x
0
f
1
(t)f
2
(x – t) dt
17
1
√
p
f
1
p
∞
0
cos
2
√
xt
√
πx
f(t) dt
18
1
p
√
p
f
1
p
∞
0
sin
2
√
xt
√
πt
f(t) dt
19
1
p
2ν+1
f
1
p
∞
0
(x/t)
ν
J
2ν
2
√
xt
f(t) dt
20
1
p
f
1
p
∞
0
J
0
2
√
xt
f(t) dt
21
1
p
f
p +
1
p
x
0
J
0
2
√
xt – t
2
f(t) dt
22
1
p
2ν+1
f
p +
a
p
,–
1
2
< ν ≤ 0
x
0
x – t
at
ν
J
2ν
2
√
axt – at
2
f(t) dt
23
f
√
p
∞
0
t
2
√
πx
3
exp
–
t
2
4x
f(t) dt
24
1
√
p
f
√
p
1
√
πx
∞
0
exp
–
t
2
4x
f(t) dt
25
f
p +
√
p
1
2
√
π
x
0
t
(x – t)
3/2
exp
–
t
2
4(x – t)
f(t) dt
26
f
p
2
+ a
2
f(x)–a
x
0
f
√
x
2
– t
2
J
1
(at) dt
27
f
p
2
– a
2
f(x)+a
x
0
f
√
x
2
– t
2
I
1
(at) dt
28
f
p
2
+ a
2
p
2
+ a
2
x
0
J
0
a
√
x
2
– t
2
f(t) dt
29
f
p
2
– a
2
p
2
– a
2
x
0
I
0
a
√
x
2
– t
2
f(t) dt
30
f
(p + a)
2
– b
2
e
–ax
f(x)+be
–ax
x
0
f
√
x
2
– t
2
I
1
(bt) dt
31
f(ln p)
∞
0
x
t–1
Γ(t)
f(t) dt
32
1
p
f
(ln p)
∞
0
x
t
Γ(t + 1)
f(t) dt
33
f
(p – ia)+
f
(p + ia), i
2
=–1
2f(x)cos(ax)
34
i
f(p – ia)–
f(p + ia)
, i
2
=–1
2f(x)sin(ax)
1166 INTEGRAL TRANSFORMS
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
35
d
f(p)
dp
– xf(x)
36
d
n
f(p)
dp
n
(–x)
n
f(x)
37
p
n
d
m
f(p)
dp
m
, m ≥ n
(–1)
m
d
n
dx
n
x
m
f(x)
38
∞
p
f(q) dq
1
x
f(x)
39
1
p
p
0
f(q) dq
∞
x
f(t)
t
dt
40
1
p
∞
p
f(q) dq
x
0
f(t)
t
dt
T3.2.2. Expressions with Rational Functions
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
1
1
p
1
2
1
p + a
e
–ax
3
1
p
2
x
4
1
p(p + a)
1
a
1 – e
–ax
5
1
(p + a)
2
xe
–ax
6
p
(p + a)
2
(1 – ax)e
–ax
7
1
p
2
– a
2
1
a
sinh(ax)
8
p
p
2
– a
2
cosh(ax)
9
1
(p + a)(p + b)
1
a – b
e
–bx
– e
–ax
10
p
(p + a)(p + b)
1
a – b
ae
–ax
– be
–bx
11
1
p
2
+ a
2
1
a
sin(ax)
12
p
p
2
+ a
2
cos(ax)
13
1
(p + b)
2
+ a
2
1
a
e
–bx
sin(ax)
14
p
(p + b)
2
+ a
2
e
–bx
cos(ax)–
b
a
sin(ax)
T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1167
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
15
1
p
3
1
2
x
2
16
1
p
2
(p + a)
1
a
2
e
–ax
+ ax – 1
17
1
p(p + a)(p + b)
1
ab(a – b)
a – b + be
–ax
– ae
–bx
18
1
p(p + a)
2
1
a
2
1 – e
–ax
– axe
–ax
19
1
(p + a)(p + b)(p + c)
(c – b)e
–ax
+(a – c)e
–bx
+(b – a)e
–cx
(a – b)(b – c)(c – a)
20
p
(p + a)(p + b)(p + c)
a(b – c)e
–ax
+ b(c – a)e
–bx
+ c(a – b)e
–cx
(a – b)(b – c)(c – a)
21
p
2
(p + a)(p + b)(p + c)
a
2
(c – b)e
–ax
+ b
2
(a – c)e
–bx
+ c
2
(b – a)e
–cx
(a – b)(b – c)(c – a)
22
1
(p + a)(p + b)
2
1
(a – b)
2
e
–ax
– e
–bx
+(a – b)xe
–bx
23
p
(p + a)(p + b)
2
1
(a – b)
2
–ae
–ax
+[a + b(b – a)x
e
–bx
24
p
2
(p + a)(p + b)
2
1
(a – b)
2
a
2
e
–ax
+ b(b – 2a – b
2
x + abx)e
–bx
25
1
(p + a)
3
1
2
x
2
e
–ax
26
p
(p + a)
3
x
1 –
1
2
ax
e
–ax
27
p
2
(p + a)
3
1 – 2ax +
1
2
a
2
x
2
e
–ax
28
1
p(p
2
+ a
2
)
1
a
2
1 –cos(ax)
29
1
p
(p + b)
2
+ a
2
1
a
2
+ b
2
1 – e
–bx
cos(ax)+
b
a
sin(ax)
30
1
(p + a)(p
2
+ b
2
)
1
a
2
+ b
2
e
–ax
+
a
b
sin(bx)–cos(bx)
31
p
(p + a)(p
2
+ b
2
)
1
a
2
+ b
2
–ae
–ax
+ a cos(bx)+b sin(bx)
32
p
2
(p + a)(p
2
+ b
2
)
1
a
2
+ b
2
a
2
e
–ax
– ab sin(bx)+b
2
cos(bx)
33
1
p
3
+ a
3
1
3a
2
e
–ax
–
1
3a
2
e
ax/2
cos(kx)–
√
3 sin(kx)
,
k =
1
2
a
√
3
34
p
p
3
+ a
3
–
1
3a
e
–ax
+
1
3a
e
ax/2
cos(kx)+
√
3 sin(kx)
,
k =
1
2
a
√
3
1168 INTEGRAL TRANSFORMS
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
35
p
2
p
3
+ a
3
1
3
e
–ax
+
2
3
e
ax/2
cos(kx), k =
1
2
a
√
3
36
1
p + a)
(p + b)
2
+ c
2
]
e
–ax
– e
–bx
cos(cx)+ke
–bx
sin(cx)
(a – b)
2
+ c
2
, k =
a – b
c
37
p
p + a)
(p + b)
2
+ c
2
]
–ae
–ax
+ ae
–bx
cos(cx)+ke
–bx
sin(cx)
(a – b)
2
+ c
2
,
k =
b
2
+ c
2
– ab
c
38
p
2
p + a)
(p + b)
2
+ c
2
]
a
2
e
–ax
+(b
2
+c
2
–2ab)e
–bx
cos(cx)+ke
–bx
sin(cx)
(a –b)
2
+c
2
,
k =–ac – bc +
ab
2
– b
3
c
39
1
p
4
1
6
x
3
40
1
p
3
(p + a)
1
a
3
–
1
a
2
x +
1
2a
x
2
–
1
a
3
e
–ax
41
1
p
2
(p + a)
2
1
a
2
x
1 + e
–ax
+
2
a
3
e
–ax
– 1
42
1
p
2
(p + a)(p + b)
–
a + b
a
2
b
2
+
1
ab
x +
1
a
2
(b – a)
e
–ax
+
1
b
2
(a – b)
e
–bx
43
1
(p + a)
2
(p + b)
2
1
(a – b)
2
e
–ax
x +
2
a – b
+ e
–bx
x –
2
a – b
44
1
(p + a)
4
1
6
x
3
e
–ax
45
p
(p + a)
4
1
2
x
2
e
–ax
–
1
6
ax
3
e
–ax
46
1
p
2
(p
2
+ a
2
)
1
a
3
ax –sin(ax)
47
1
p
4
– a
4
1
2a
3
sinh(ax)–sin(ax)
48
p
p
4
– a
4
1
2a
2
cosh(ax)–cos(ax)
49
p
2
p
4
– a
4
1
2a
sinh(ax)+sin(ax)
50
p
3
p
4
– a
4
1
2
cosh(ax)+cos(ax)
51
1
p
4
+ a
4
1
a
3
√
2
cosh ξ sin ξ –sinhξ cos ξ
, ξ =
ax
√
2
52
p
p
4
+ a
4
1
a
2
sin
ax
√
2
sinh
ax
√
2
53
p
2
p
4
+ a
4
1
a
√
2
cos ξ sinh ξ +sinξ coshξ
, ξ =
ax
√
2
T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1169
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
54
1
(p
2
+ a
2
)
2
1
2a
3
sin(ax)–ax cos(ax)
55
p
(p
2
+ a
2
)
2
1
2a
x sin(ax)
56
p
2
(p
2
+ a
2
)
2
1
2a
sin(ax)+ax cos(ax)
57
p
3
(p
2
+ a
2
)
2
cos(ax)–
1
2
ax sin(ax)
58
1
(p + b)
2
+ a
2
2
1
2a
3
e
–bx
sin(ax)–ax cos(ax)
59
1
(p
2
– a
2
)(p
2
– b
2
)
1
a
2
– b
2
1
a
sinh(ax)–
1
b
sinh(bx)
60
p
(p
2
– a
2
)(p
2
– b
2
)
cosh(ax)–cosh(bx)
a
2
– b
2
61
p
2
(p
2
– a
2
)(p
2
– b
2
)
a sinh(ax)–b sinh(bx)
a
2
– b
2
62
p
3
(p
2
– a
2
)(p
2
– b
2
)
a
2
cosh(ax)–b
2
cosh(bx)
a
2
– b
2
63
1
(p
2
+ a
2
)(p
2
+ b
2
)
1
b
2
– a
2
1
a
sin(ax)–
1
b
sin(bx)
64
p
(p
2
+ a
2
)(p
2
+ b
2
)
cos(ax)–cos(bx)
b
2
– a
2
65
p
2
(p
2
+ a
2
)(p
2
+ b
2
)
–a sin(ax)+b sin(bx)
b
2
– a
2
66
p
3
(p
2
+ a
2
)(p
2
+ b
2
)
–a
2
cos(ax)+b
2
cos(bx)
b
2
– a
2
67
1
p
n
, n = 1, 2,
1
(n – 1)!
x
n–1
68
1
(p + a)
n
, n = 1, 2,
1
(n – 1)!
x
n–1
e
–ax
69
1
p(p + a)
n
, n = 1, 2,
a
–n
1 – e
–ax
e
n
(ax)
, e
n
(z)=1 +
z
1!
+ ···+
z
n
n!
70
1
p
2n
+ a
2n
, n = 1, 2,
–
1
na
2n
n
k=1
exp(a
k
x)
a
k
cos(b
k
x)–b
k
sin(b
k
x)
,
a
k
= a cos ϕ
k
, b
k
= a sinϕ
k
, ϕ
k
=
π(2k – 1)
2n
71
1
p
2n
– a
2n
, n = 1, 2,
1
na
2n–1
sinh(ax)+
1
na
2n
n
k=2
exp(a
k
x)
×
a
k
cos(b
k
x)–b
k
sin(b
k
x)
,
a
k
= a cos ϕ
k
, b
k
= a sinϕ
k
, ϕ
k
=
π(k – 1)
n
1170 INTEGRAL TRANSFORMS
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
72
1
p
2n+1
+ a
2n+1
, n = 0, 1,
e
–ax
(2n + 1)a
2n
–
2
(2n + 1)a
2n+1
n
k=1
exp(a
k
x)
×
a
k
cos(b
k
x)–b
k
sin(b
k
x)
,
a
k
= a cos ϕ
k
, b
k
= a sinϕ
k
, ϕ
k
=
π(2k – 1)
2n + 1
73
1
p
2n+1
– a
2n+1
, n = 0, 1,
e
ax
(2n + 1)a
2n
+
2
(2n + 1)a
2n+1
n
k=1
exp(a
k
x)
×
a
k
cos(b
k
x)–b
k
sin(b
k
x)
,
a
k
= a cos ϕ
k
, b
k
= a sinϕ
k
, ϕ
k
=
2πk
2n + 1
74
Q(p)
P (p)
,
P (p)=(p – a
1
) (p – a
n
);
Q(p) is a polynomial of degree
≤ n – 1; a
i
≠ a
j
if i ≠ j
n
k=1
Q(a
k
)
P
(a
k
)
exp
a
k
x
,
(the prime stand for the differentiation)
75
Q(p)
P (p)
,
P (p)=(p – a
1
)
m
1
(p – a
n
)
m
n
;
Q(p) is a polynomial of degree
< m
1
+ m
2
+ ···+ m
n
– 1;
a
i
≠ a
j
if i ≠ j
n
k=1
m
k
l=1
Φ
kl
(a
k
)
(m
k
– l)! (l – 1)!
x
m
k
–l
exp
a
k
x
,
Φ
kl
(p)=
d
l–1
dp
l–1
Q(p)
P
k
(p)
, P
k
(p)=
P (p)
(p – a
k
)
m
k
76
Q(p)+pR(p)
P (p)
,
P (p)=(p
2
+ a
2
1
) (p
2
+ a
2
n
);
Q(p)andR(p) are polynomials
of degree ≤ 2n – 2; a
l
≠ a
j
, l ≠ j
n
k=1
Q(ia
k
)sin(a
k
x)+a
k
R(ia
k
)cos(a
k
x)
a
k
P
k
(ia
k
)
,
P
m
(p)=
P (p)
p
2
+ a
2
m
, i
2
=–1
T3.2.3. Expressions with Square Roots
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
1
1
√
p
1
√
πx
2
√
p – a
–
p – b
e
bx
– e
ax
2
√
πx
3
3
1
√
p + a
1
√
πx
e
–ax
4
p + a
p
– 1
1
2
ae
–ax/2
I
1
1
2
ax
+ I
0
1
2
ax
5
√
p + a
p + b
e
–ax
√
πx
+(a – b)
1/2
e
–bx
erf
(a – b)
1/2
x
1/2
6
1
p
√
p
2
x
π
7
1
(p + a)
√
p + b
(b – a)
–1/2
e
–ax
erf
(b – a)
1/2
x
1/2
T3.2. TABLES OF INVERSE LAPLACE TRANSFORMS 1171
No. Laplace transform,
f(p) Inverse transform, f(x)=
1
2πi
c+i∞
c–i∞
e
px
f(p) dp
8
1
√
p (p – a)
1
√
a
e
ax
erf
√
ax
9
1
p
3/2
(p – a)
a
–3/2
e
ax
erf
√
ax
– 2a
–1
π
–1/2
x
1/2
10
1
√
p + a
π
–1/2
x
–1/2
– ae
a
2
x
erfc
a
√
x
11
a
p
√
p + a
1 – e
a
2
x
erfc
a
√
x
12
1
p + a
√
p
e
a
2
x
erfc
a
√
x
13
1
√
p +
√
a
2
1 –
2
√
π
(ax)
1/2
+(1 – 2ax)e
ax
erf
√
ax
– 1
14
1
p
√
p +
√
a
2
1
a
+
2x –
1
a
e
ax
erfc
√
ax
–
2
√
πa
√
x
15
1
√
p
√
p + a
2
2π
–1/2
x
1/2
– 2axe
a
2
x
erfc
a
√
x
16
1
√
p + a
3
2
√
π
(a
2
x + 1)
√
x
– ax(2a
2
x + 3)e
a
2
x
erfc
a
√
x
17
p
–n–1/2
, n = 1, 2,
2
n
1×3× × (2n – 1)
√
π
x
n–1/2
18
(p + a)
–n–1/2
2
n
1×3× × (2n – 1)
√
π
x
n–1/2
e
–ax
19
1
p
2
+ a
2
J
0
(ax)
20
1
p
2
– a
2
I
0
(ax)
21
1
p
2
+ ap + b
exp
–
1
2
ax
J
0
(b –
1
4
a
2
1/2
x
22
p
2
+ a
2
– p
1/2
1
√
2πx
3
sin(ax)
23
1
p
2
+ a
2
p
2
+ a
2
+ p
1/2
√
2
√
πx
cos(ax)
24
1
p
2
– a
2
p
2
– a
2
+ p
1/2
√
2
√
πx
cosh(ax)
25
p
2
+ a
2
+ p
–n
na
–n
x
–1
J
n
(ax)
26
p
2
– a
2
+ p
–n
na
–n
x
–1
I
n
(ax)
27
p
2
+ a
2
–n–1/2
(x/a)
n
J
n
(ax)
1×3×5× × (2n – 1)
28
p
2
– a
2
–n–1/2
(x/a)
n
I
n
(ax)
1×3×5× × (2n – 1)
