Page 769 - 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∞

1 a
2 1 2 1
11 cos s /a , a >0 cos 4 a|ln x| – π
4
2 π
 b
 x
a sin a|ln x| if 0 < x <1,

12 arctan , Re s > –b |ln x|
s + b 
0 if 1 < x
9.4. Expressions With Special Functions
1 σ+i∞ –s
ˆ
ˆ
No Direct transform, f(s) Inverse transform, f(x)= f(s)x ds
2πi σ–i∞
1 Γ(s), Re s >0 e –x
2 Γ(s), –1<Re s <0 e –x – 1

1
3 sin 2 πs Γ(s), –1<Re s <1 sin x
sin(as)Γ(s),
4 Re s > –1, |a| < π exp(–x cos a) sin(x sin a)
2
1
5 cos πs Γ(s), 0 < Re s <1 cos x
2
1 2
6 cos 2 πs Γ(s), –2<Re s <0 –2 sin (x/2)
π
7 cos(as)Γ(s), Re s >0, |a| < exp(–x cos a) cos(x sin a)
2
Γ(s) 1 x √
8 , 0<Re s < e erfc x
cos(πs) 2
Γ(a + s)Γ(b – s), a –a–b
9 Γ(a + b)x (x +1)
–a <Re s < b, a + b >0
Γ(a + s)Γ(b + s), (a+b)/2 √
10 2x K a–b 2 x
Re s > –a, –b
ν–1
Γ(s) (1 – x)
11 , Re s >0, ν >0 Γ(ν) if 0 < x <1,
Γ(s + ν)
0 if 1 < x
Γ(1 – ν – s) 0 if 0 < x <1,
, ν–1
12 Γ(1 – s) (x – 1) if 1 < x
Re s <1 – ν, ν >0 Γ(ν)
Γ(s)
,
Γ(ν – s +1) –ν/2 √
13 ν 3 x J ν 2 x
0<Re s < +
2 4
Γ(s + ν)Γ(s – ν)
14 , Re s > |ν| π –1/2 –x/2 K ν (x/2)
e
Γ(s +1/2)

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 754