Page 765 - Handbook Of Integral Equations

∞ s–1

ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x dx
0
–s

4 cos(ax), a >0 a Γ(s) cos 1 πs , 0<Re s <1
2
Γ(s) πs –s –s

5 sin(ax) cos(bx), a, b >0 2 sin 2 (a + b) + |a – b| sign(a – b) ,
–1<Re s <1

Γ(s) sin s arctan(b/a)
–ax
6 e sin(bx), a >0 , –1<Re s
2
(a + b )
2 s/2

Γ(s) cos s arctan(b/a)
–ax
7 e cos(bx), a >0 , 0<Re s
(a + b )
2
2 s/2
sin(a ln x)if 0 < x <1,
a
8 – 2 2 , Re s >0
0 if 1 < x s + a
cos(a ln x)if 0 < x <1,
s
9 2 2 , Re s >0
0 if 1 < x s + a
π
10 arctan x – 1 , –1<Re s <0
2s cos πs
2
π
11 arccot x 1 , 0 < Re s <1
2s cos πs
2
8.6. Expressions With Special Functions
∞ s–1

ˆ
No Original function, f(x) Mellin transform, f(s)= f(x)x dx
0
1 1
Γ 2 s + 2
1 erfc x √ , Re s >0
π s
–1
2 Ei(–x) –s Γ(s), Re s >0
–1

3 Si(x) –s sin 1 2 πs Γ(s), –1<Re s <0
–1

4 si(x) –4s sin 1 πs Γ(s), –1<Re s <0
2
–1

5 Ci(x) –s cos 1 πs Γ(s), 0<Re s <1
2
1
2 s–1 Γ 1 2 ν + s 3
2
6 J ν (ax), a >0 1 1 , –ν <Re s < 2
s
a Γ ν – s +1
2 2
2 s–1 s ν s ν π(s – ν)
– Γ + Γ – cos ,
7 Y ν (ax), a >0 πa s 2 2 2 2 2
|ν| <Re s < 3
2
Γ(1/2 – s)Γ(s + ν) 1
–ax
8 e I ν (ax), a >0 √ s , –ν <Re s < 2
π (2a) Γ(1 + ν – s)
2 s–2 s ν s ν
9 K ν (ax), a >0 Γ + Γ – , |ν| <Re s
a s 2 2 2 2
√
π Γ(s – ν)Γ(s + ν)
–ax
10 e K ν (ax), a >0 , |ν| <Re s
s
(2a) Γ(s +1/2)
•
References for Supplement 8: H. Bateman and A. Erd´ elyi (1954), V. A. Ditkin and A. P. Prudnikov (1965).
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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