Page 744 - Handbook Of Integral Equations

5.5. Expressions With Exponential Functions

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
0if 0 < x < a,
–1 –ap
1 p e , a >0
1if a < x.
1if 0 < x < a,
–ap
–1

2 p 1 – e , a >0
0if a < x.
0if 0 < x < a,

3 p –1 e –ap – e –bp , 0 ≤ a < b 1if a < x < b,
0if 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.
0 if 0 < x < a,
–1 –ap
5 (p + b) e , a >0 –b(x–a)
e if a < x.
0 if 0 < x < a,

–ν –ap
6 p e , ν >0 (x – a) ν–1 if a < x.
Γ(ν)
7 p –1 e ap – 1 –1 , a >0 f(x)= n if na < x <(n +1)a; n = 0,1,2, ...

a √
8 e a/p – 1 I 1 2 ax
x
1 √
e
9 p –1/2 a/p √ cosh 2 ax
πx
1 √
10 p –3/2 a/p √ sinh 2 ax
e
πa

x √ 1 √
11 p –5/2 a/p cosh 2 ax – √ sinh 2 ax
e
πa 2 πa 3
√
ν/2
12 p –ν–1 a/p , ν > –1 (x/a) I ν (2 ax
e

a √
13 1 – e –a/p J 1 2 ax
x
1 √
e
14 p –1/2 –a/p √ cos 2 ax
πx
1 √
e
15 p –3/2 –a/p √ sin 2 ax
πa
1 √ x √
16 p –5/2 –a/p √ sin 2 ax – cos 2 ax
e
2 πa 3 πa
√
ν/2
17 p –ν–1 –a/p , ν > –1 (x/a) J ν (2 ax
e
√
√ a –3/2 a
18 exp – ap , a >0 √ x exp –
2 π 4x
√
√ a –7/2 a
19 p exp – ap , a >0 √ (a – 6x)x exp –
8 π 4x
√
1 √ a
20 exp – ap , a ≥ 0 erfc √
p 2 x
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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