Page 745 - Handbook Of Integral Equations

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
√ √ 1 –5/2 a
21 p exp – ap , a >0 √ (a – 2x)x exp –
4 π 4x
1 √ 1 a
22 √ exp – ap , a ≥ 0 √ exp –
p πx 4x
√ √
1 √ 2 x a √ a
23 √ exp – ap , a ≥ 0 √ exp – – a erfc √
p p π 4x 2 x

2
exp –k p + a 2 0 if 0 < x < k,
24 , k >0 √ 2 2
2
p + a 2 J 0 a x – k if k < x.

2 2
exp –k p – a 0 if 0 < x < k,
√
25 , k >0 I 0 a x – k 2 if k < x.
2
2
p – a 2
5.6. Expressions With Hyperbolic Functions
1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
1 f(x)=2n if a(2n – 1) < x < a(2n + 1);
1 , a >0
p sinh(ap) n =0, 1, 2, ... (x >0)
1 f(x)=2n(x – an)if a(2n – 1) < x < a(2n + 1);
2 , a >0
p sinh(ap) n =0, 1, 2, ... (x >0)
2
sinh(a/p) 1 √ √
3 √ √ cosh 2 ax – cos 2 ax
p 2 πx
sinh(a/p) 1 √ √
4 √ √ sinh 2 ax – sin 2 ax
p p 2 πa
√ √
–ν–1
1
ν/2
5 p sinh(a/p), ν > –2 (x/a) I ν 2 ax – J ν 2 ax
2

0if a(4n – 1) < x < a(4n + 1),
1 f(x)=
6 , a >0 2if a(4n +1) < x < a(4n + 3),
p cosh(ap)
n =0, 1, 2, ... (x >0)
n
1 x – (–1) (x – 2an)if2n – 1< x/a <2n +1;
7 2 , a >0
p cosh(ap) n =0, 1, 2, ... (x >0)
cosh(a/p) 1 √ √
8 √ √ cosh 2 ax + cos 2 ax
p 2 πx
cosh(a/p) 1 √ √
9 √ √ sinh 2 ax + sin 2 ax
p p 2 πa
√ √
–ν–1
ν/2
1
10 p cosh(a/p), ν > –1 (x/a) I ν 2 ax + J ν 2 ax
2
1 f(x)=(–1) n–1 if 2a(n – 1) < x <2an;
11 tanh(ap), a >0
p n =1, 2, ...
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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