Page 770 - 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∞
Γ(s + ν)Γ(1/2 – s)
, 1/2 –x/2
15 Γ(1 + ν – s) π e I ν (x/2)
–ν <Re s < 1
2
b a
ψ(s + a) – ψ(s + b), x – x
16 if 0 < x <1,
Re s > –a, –b 1 – x
0 if 1 < x
17 Γ(s)ψ(s), Re s >0 e –x ln x
0 if 0 < x < a,

18 Γ(s, a), a >0 –x
e if a < x
–1 –a(x+1)
19 Γ(s)Γ(1 – s, a), Re s >0, a >0 (x +1) e
e if 0 < x < a,
–x
20 γ(s, a), Re s >0, a >0
0 if a < x
0 if 0 < x < e ,
 –a
 √
2
√  cos b a – ln x
2
2
a
21 J 0 a b – s 2 , a >0 √ if e –a < x < e ,
π a – ln x
 2 2

a
0 if e < x
1 if 0 < x < e ,
–1
–1
–1
22 s I 0 (s), Re s >0 π arccos(ln x)if e –1 < x < e,
0 if e < x
 ν
2 sin(πν)
–1
 – √ if 0 < x < e ,

 2
 πF(x) ln x – 1


cos ν arccos(ln x)
23 I ν (s), Re s >0  √ if e –1 < x < e,

 2
 π 1 – ln x
0 if e < x,

√ √ 2ν
F(x)= –1 – ln x + 1 – ln x
 ν
2 sin(πν)
–1
 if 0 < x < e ,

 πνF(x)


sin ν arccos(ln x)
–1
24 s I ν (s), Re s >0  if e –1 < x < e,

 πν

0 if e < x,
√ √ 2ν
F(x)= –1 – ln x + 1 – ln x
 –1
 0 if 0 < x < e ,
2
 (1 – ln x) ν–1/2
–ν
25 s I ν (s), Re s > – 1 2 √ if e –1 < x < e,
ν
 π 2 Γ(ν +1/2)

0 if e < x
–1
–1
26 s K 0 (s), Re s >0 Arcosh(– ln x)if 0 < x < e ,
–1
0 if e < x
√ 2 –1
–1
27 s K 1 (s), Re s >0 ln x – 1if 0 < x < e ,
–1
0 if e < x
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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