Page 768 - Handbook Of Integral Equations

1 σ+i∞ –s
ˆ
ˆ
No Direct transform, f(s) Inverse transform, f(x)= f(s)x ds
2πi σ–i∞
a – 2|ln x| a

√ √  exp – if 0 < x <1,
6 s exp – as , Re s >0 4 π|ln x| 5 4|ln x|

0 if 1 < x

1 a
1 √  √ exp – if 0 < x <1,
7 √ exp – as , Re s >0 π|ln x| 4|ln x|
s 
0 if 1 < x
a b
s + a x – x if 0 < x <1,
8 ln , Re s > –a, –b ln x
s + b
0 if 1 < x
ψ(ν) – ln |ln x|
|ln x| ν–1 if 0 < x <1,
9 s –ν ln s, Re s >0, ν >0 Γ(ν)
0 if 1 < x
9.3. Expressions With Trigonometric Functions

1 σ+i∞ –s
ˆ
ˆ
No Direct transform, f(s) Inverse transform, f(x)= f(s)x ds
2πi σ–i∞
π 1
1 , 0<Re s <1
sin(πs) x +1
π n
, –n <Re s <1 – n, n x
2 sin(πs) (–1)
n = ... , –1, 0, 1, 2, ... x +1
π 2 ln x
3 , 0<Re s <1
2
sin (πs) x – 1
π 2 ln x
4 sin (πs) , n <Re s < n +1, x (x – 1)
2
n
n = ... , –1, 0, 1, 2, ...
2
2π 3 π +ln x
2
5 , 0<Re s <1
3
sin (πs) x +1
2π 3 π +ln x
2
2
6 sin (πs) , n <Re s < n +1, n
3
n = ... , –1, 0, 1, 2, ... (–x) (x +1)

1 a
2 1 2 1
7 sin s /a , a >0 sin a|ln x| – π
2 π 4 4
π 1 1 √ x
8 , – <Re s < 2
2
cos(πs) x +1
π 1 1
, n – <Re s < n + x 1/2–n
9 cos(πs) 2 2 (–1) n
n = ... , –1, 0, 1, 2, ... x +1
cos(βs) 1
2
10 , –1<Re s <0, |β| < π ln(x +2x cos β +1)
s cos(πs) 2π
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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