Inverse Laplace Transform - The World of Mathematical Equations

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Auxiliary Sections > Integral Transforms > Tables of Inverse Laplace Transforms > Inverse Laplace
Transforms: Expressions with Exponential Functions
Inverse Laplace Transforms: Expressions with Exponential
Functions
1
No Laplace transform, ˜f(p) Inverse transform, f(x) =
2πi
{
1 p −1 e −ap 0 if 0 < x < a,
, a > 0
1 if a < x.
2 p −1( 1 − e −ap) { 1 if 0 < x < a,
, a > 0
0 if a < x.
{ 0 if 0 < x < a,
3 p −1( e −ap − e −bp) , 0 ≤ a < b 1 if a < x < b,
0 if b < x.
{ 0 if 0 < x < a,
4 p −2( e −ap − e −bp) , 0 ≤ a < b x − a if a < x < b,
b − a if b < x.
{
5 (p + b) −1 e −ap 0 if 0 < x < a,
, a > 0
e −b(x−a) if a < x.
{ 0 if 0 < x < a,
6 p −ν e −ap , ν > 0
(x − a) ν−1
if a < x.
Γ(ν)
∫ c+i∞
c−i∞
e px ˜f(p) dp
7 p −1( e ap − 1 ) −1
, a > 0 f(x) = n if na < x < (n + 1)a; n = 0, 1, . . .
√ a
8 e a/p − 1
x I ( √ )
1 2 ax
9 p −1/2 e a/p 1
√ πx
cosh ( 2 √ ax )
10 p −3/2 e a/p 1
√ πa
sinh ( 2 √ ax )
11 p −ν−1 e a/p , ν > −1 (x/a) ν/2 I ν (2 √ ax )
√ a
12 1 − e −a/p x J ( √ )
1 2 ax
13 p −1/2 e −a/p 1
√ πx
cos ( 2 √ ax )
14 p −3/2 e −a/p 1
√ πa
sin ( 2 √ ax )
15 p −ν−1 e −a/p , ν > −1 (x/a) ν/2 J ν (2 √ ax )
16 exp ( − √ ap ) , a > 0
√ a
(
2 √ π x−3/2 exp − a )
4x

No Laplace transform, ˜f(p) Inverse transform, f(x) =
1
2πi
17
18
19
20
21
1
p exp( − √ ap ) ( √ a
)
, a ≥ 0 erfc
2 √ x
1
√ exp ( − √ ap ) 1
(
, a ≥ 0
√ exp − a )
p πx 4x
1
p √ p exp( − √ ap ) 2 √ x
(
, a ≥ 0
√ exp − a
π 4x
exp ( −k √ p 2 + a 2 )
√
p2 + a 2 , k > 0
exp ( −k √ p 2 − a 2 )
√
p2 − a 2 , k > 0
∫ c+i∞
c−i∞
)
− √ ( √ a
)
a erfc
2 √ x
{ 0 if 0 < x < k,
J 0
(
a
√
x2 − k 2 ) if k < x.
{ 0 if 0 < x < k,
I 0
(
a
√
x2 − k 2 ) if k < x.
e px ˜f(p) dp
Notation: erfc z is the complementary error function, J ν (z) is the Bessel function of the first kind,
I ν (z) is the modified Bessel function of the first kind, Γ(z) is the gamma function.
References
Bateman, H. and Erdélyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954.
Doetsch, G., Einführung in Theorie und Anwendung der Laplace-Transformation, Birkhäuser Verlag, Basel–Stuttgart, 1958.
Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.
Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations , CRC Press, Boca Raton, 1998.
Inverse Laplace Transforms: Expressions with Exponential Functions
Copyright c○ 2005 Andrei D. Polyanin
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