Page 746 - 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 f(x)=(2n – 1) if 2a(n – 1) < x <2an;
12 coth(ap), a >0
p n =1, 2, ...
1
13 Arcoth(p/a) sinh(ax)
x
5.7. Expressions With Logarithmic Functions

1 c+i∞ px ˜
˜
No Laplace transform, f(p) Inverse transform, f(x)= e f(p) dp
2πi
c–i∞
1 – ln x – C,
1 ln p
p C = 0.5772 ... is the Euler constant
x n
1 1 1
2 p –n–1 ln p 1+ 2 + 3 + ··· + n – ln x – C n! ,
C = 0.5772 ... is the Euler constant
2 2 2 n–1/2
k n 2+ + + ··· + – ln(4x) – C x ,
3 5 2n–1
3 p –n–1/2 ln p 2 n
k n = √ , C = 0.5772 ...
1 ⋅ 3 ⋅ 5 ... (2n – 1) π
1 ν–1
4 p –ν ln p, ν >0 Γ(ν) x ψ(ν) – ln x , ψ(ν) is the logarithmic
derivative of the gamma function
1 2 2 2
1
5 (ln p) (ln x + C) – π , C = 0.5772 ...
6
p
1 2 2 2
1
6 2 (ln p) x (ln x + C – 1) +1 – π
6
p
ln(p + b) –ax
7 e ln(b – a) – Ei (a – b)x }
p + a
ln p 1 1
8 2 2 cos(ax) Si(ax)+ sin(ax) ln a – Ci(ax)
p + a a a
p ln p
9 2 2 cos(ax) ln a – Ci(ax) – sin(ax) Si(ax)
p + a
p + b 1 –ax –bx
10 ln e – e
p + a x
2
p + b 2 2
11 ln cos(ax) – cos(bx)
2
p + a 2 x
2
p + b 2 2
12 p ln cos(bx)+ bx sin(bx) – cos(ax) – ax sin(ax)
2
p + a 2 x
2
(p + a) + k 2 2 –bx –ax
13 ln 2 2 cos(kx)(e – e
(p + b) + k x
1 1 a

2
14 p ln p + a 2 cos(ax) – 1 + sin(ax)
p x 2 x
1 1 a

2
15 p ln p – a 2 cosh(ax) – 1 – sinh(ax)
p x 2 x
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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