Page 747 - Handbook Of Integral Equations

5.8. Expressions With Trigonometric Functions

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
sin(a/p) 1 √ √
1 √ √ sinh 2ax sin 2ax
p πx
sin(a/p) 1 √ √
2 √ √ cosh 2ax sin 2ax
p p πa
cos(a/p) 1 √ √
3 √ √ cosh 2ax cos 2ax
p πx
cos(a/p) 1 √ √
4 √ √ sinh 2ax cos 2ax
p p πa
1 √ √ 1 a
5 √ exp – ap sin ap √ sin
p πx 2x
1 √ √ 1 a
6 √ exp – ap cos ap √ cos
p πx 2x
a 1
7 arctan sin(ax)
p x
1 a
8 arctan Si(ax)
p p
a 1
9 p arctan – a ax cos(ax) – sin(ax)
p x 2
2ap 2 √
2
10 arctan 2 2 sin(ax) cos x a + b 2
p + b x

5.9. Expressions With Special Functions

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
1 x 2
2 √
1 exp ap erfc p a √ exp –
πa 4a
1 2 √ x
2 exp ap erfc p a erf √
p 2 a
0 if 0 < x < a,

√ √
3 erfc ap , a >0 a
√ if a < x.
πx x – a
√
√ a
ap
4 e erfc ap √
π x (x + a)
1 ap √ 1
5 √ e erfc ap √
p π(x + a)
1 √
6 erf a/p sin 2 ax
πx

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 731