Page 757 - Handbook Of Integral Equations

∞
ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
1 1 π
10 , a >0 –
e ax +1 2u 2a sinh(πu/a)
1 π πu 1
11 , a >0 coth –
e ax – 1 2a a 2u
e x/2 1
12 – tanh(πu)
2
x
e – 1
√ 2

13 x exp –ax 2 π u exp – u
4a 3/2 4a
1 2 π u
14 exp –ax erf √
x 2 2 a
1 a π √ √ √
15 √ exp – e – 2au cos 2au + sin 2au
x x 2u
1 a π √ √
16 √ exp – e – 2au sin 2au
x x x a

7.4. Expressions With Hyperbolic Functions

∞

ˇ
No Original function, f(x) Sine transform, f s (u)= f(x) sin(ux) dx
0
1 π 1 –1
1 , a >0 tanh 2 πa u
sinh(ax) 2a
2
–1
x π sinh 1 2 πa u
2 , a >0
sinh(ax) 4a cosh 2 1 πa u
2
–1
2
1 –bx 1 2au
3 e sinh(ax), b > |a| arctan
2
2
x 2 u + b – a 2
1 –1
4 , a >0 arctan sinh 1 2 πa u
x cosh(ax)
1 π
1
5 1 – tanh 2 ax , a >0 u – a sinh πa u
–1

π 1
1 –1
6 coth 2 ax – 1, a >0 coth πa u –
a u
–1
cosh(ax) π sinh πb u
7 , |a| < b –1 –1
sinh(bx) 2b cos πab + cosh πb u
1 –1 1 –1
sinh(ax) π sin 2 πab sinh 2 πb u
8 , |a| < b
cosh(bx) b cos πab –1 + cosh πb u
–1
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 741