Page 767 - 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
√ – I 1 (–a ln x)if 0 < x <1,
2
2
12 s – a – s, Re s > |a| ln x
0 if 1 < x

s + a aI 0 (–a ln x)+ aI 1 (–a ln x)if 0 < x <1,
13 – 1, Re s > |a|
s – a 0 if 1 < x

1 a ν–1
–ν x (– ln x) if 0 < x <1,
14 (s + a) , Re s > –a, ν >0 Γ(ν)
0 if 1 < x
–ν
–1
s (s + a) , a –ν Γ(ν) –1 γ(ν, –a ln x)if 0 < x <1,
15
Re s >0, Re s > –a, ν >0 0 if 1 < x
–ν
–1
s (s + a) , –a –ν Γ(ν) –1 Γ(ν, –a ln x)if 0 < x <1,
16
–a <Re s <0, ν >0 –a –ν if 1 < x
√ ν–1/2
π (– ln x) I ν–1/2 (–a ln x)
2 –ν
2
17 (s – a ) , Re s > |a|, ν >0 ν–1/2 if 0 < x <1,
Γ(ν)(2a)
0 if 1 < x
 ν–1/2
(– ln x) K ν–1/2 (–a ln x)
√ if 0 < x <1,


 π Γ(ν)(2a) ν–1/2
2 –ν
2
18 (a – s ) , Re s < |a|, ν >0 ν–1/2
 (ln x) K ν–1/2 (a ln x)
√ if 1 < x

 ν–1/2
π Γ(ν)(2a)
9.2. Expressions With Exponential and Logarithmic
Functions
1 σ+i∞ –s
ˆ
ˆ
No Direct transform, f(s) Inverse transform, f(x)= f(s)x ds
2πi σ–i∞
1 ln x
2
2
1 exp(as ), a >0 √ exp –
2 πa 4a
1–ν


 a 2 a|ln x|
–ν –a/s
2 s e , Re s >0; a, ν >0 ln x J ν–1 2 if 0 < x <1,

0 if 1 < x
 1/2
 (a/π) a
√ exp – if 0 < x <1,
3 exp – as , Re s >0, a >0 2|ln x| 3/2 4|ln x|

0 if 1 < x

a
1 √ erfc √ if 0 < x <1,
4 exp –a s , Re s >0 2 |ln x|
s
0 if 1 < x

a
– erf √ if 0 < x <1,
1 √
5 exp –a s – 1 , Re s >0 2 |ln x|
s
0 if 1 < x
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 752