Page 731 - Handbook Of Integral Equations

∞
˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
2
p – 2a 2
2
10 cosh (ax)
3
2
p – 4a p

11 x ν–1 cosh(ax), ν >0 1 Γ(ν) (p – a) –ν +(p + a) –ν
2
√
√ 1 πa a/p
12 cosh 2 ax + √ e erf a/p
p p p
√ √
13 x cosh 2 ax π 1/2 –5/2 1 p + a e a/p
p
2
1 √
e
14 √ cosh 2 ax π 1/2 –1/2 a/p
p
x
1 2 √
15 √ cosh ax 1 2 π 1/2 –1/2 e a/p +1
p
x
4.5. Expressions With Logarithmic Functions
∞

˜
No Original function, f(x) Laplace transform, f(p)= e –px f(x) dx
0
1
– (ln p + C),
1 ln x p
C = 0.5772 ... is the Euler constant
1 p/a
2 ln(1 + ax) – e Ei(–p/a)
p
1 ap
3 ln(x + a) ln a – e Ei(–ap)
p
n! 1 1 1
n 1+ + + ··· + – ln p – C ,
4 x ln x, n =1, 2, ... p n+1 2 3 n
C = 0.5772 ... is the Euler constant
1
5 √ ln x – π/p ln(4p)+ C
x

k n 2 2 2
2+ 3 + 5 + ··· + 2n–1 – ln(4p) – C ,
p n+1/2
n–1/2
6 x ln x, n =1, 2, ... √
π
k n =1 ⋅ 3 ⋅ 5 ... (2n – 1) , C = 0.5772 ...
2 n

Γ(ν)p –ν ψ(ν) – ln p , ψ(ν) is the logarithmic
7 x ν–1 ln x, ν >0
derivative of the gamma function
2
2
1

8 (ln x) 2 1 (ln x + C) + π , C = 0.5772 ...
6
p
ln(p + a)+ C
–ax
9 e ln x – , C = 0.5772 ...
p + a
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 715