Mellin Transform - EqWorldeqworld.ipmnet.ru/en/auxiliary/inttrans/mellin.pdf · Title: Mellin...
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EqWorld http://eqworld.ipmnet.ru Auxiliary Sections > Integral Transforms > Mellin Transforms: General Formulas Mellin Transforms: General Formulas No Original function, f (x) Mellin transform, ˆ f (s ) = Z ∞ 0 f (x)x s-1 dx 1 af 1 (x) + bf 2 (x) a ˆ f 1 (s ) + b ˆ f 2 (s ) 2 f (ax), a > 0 a -s ˆ f (s ) 3 x a f (x) ˆ f (s + a) 4 f (1/x) ˆ f (-s ) 5 f ( x β ) , β > 0 1 β ˆ f ‡ s β · 6 f ( x -β ) , β > 0 1 β ˆ f ‡ - s β · 7 x λ f ( ax β ) , a, β > 0 1 β a - s+ λ β ˆ f ‡ s + λ β · 8 x λ f ( ax -β ) , a, β > 0 1 β a s+ λ β ˆ f ‡ - s + λ β · 9 f 0 x (x) -(s -1) ˆ f (s -1) 10 xf 0 x (x) -s ˆ f (s ) 11 f (n) x (x) (-1) n Γ(s ) Γ(s - n) ˆ f (s - n) 12 ‡ x d dx · n f (x) (-1) n s n ˆ f (s ) 13 ‡ d dx x · n f (x) (-1) n (s -1) n ˆ f (s ) 14 x α Z ∞ 0 t β f 1 (xt)f 2 (t) dt ˆ f 1 (s + α) ˆ f 2 (1- s - α + β) 15 x α Z ∞ 0 t β f 1 ‡ x t · f 2 (t) dt ˆ f 1 (s + α) ˆ f 2 (s + α + β + 1 ) Notation: Γ(z) is the gamma function. References Bateman, H. and Erd´ elyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954. Doetsch, G., Einf¨ uhrung in Theorie und Anwendung der Laplace-Transformation, Birkh¨ auser Verlag, Basel–Stuttgart, 1958. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations , CRC Press, Boca Raton, 1998. Mellin Transforms: General Formulas Copyright c 2005 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/auxiliary/inttrans/mellin.pdf
Transcript of Mellin Transform - EqWorldeqworld.ipmnet.ru/en/auxiliary/inttrans/mellin.pdf · Title: Mellin...
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Auxiliary Sections > Integral Transforms > Mellin Transforms: General Formulas
Mellin Transforms: General Formulas
No Original function, f (x) Mellin transform, f (s) =∫ ∞
0f (x)xs−1 dx
1 af1(x) + bf2(x) af1(s) + bf2(s)
2 f (ax), a > 0 a−s f (s)
3 xaf (x) f (s + a)
4 f (1/x) f (−s)
5 f(xβ
), β > 0
1β
f( s
β
)
6 f(x−β
), β > 0
1β
f(
−s
β
)
7 xλf(axβ
), a,β > 0
1β
a− s+λ
β f( s + λ
β
)
8 xλf(ax−β
), a,β > 0
1β
as+λβ f
(−
s + λ
β
)
9 f ′x(x) −(s − 1)f (s − 1)
10 xf ′x(x) −sf (s)
11 f (n)x (x) (−1)n
Γ(s)Γ(s − n)
f (s − n)
12(x
d
dx
)n
f (x) (−1)nsnf (s)
13( d
dxx)n
f (x) (−1)n(s − 1)nf (s)
14 xα
∫ ∞
0tβf1(xt)f2(t) dt f1(s + α)f2(1 − s − α + β)
15 xα
∫ ∞
0tβf1
( x
t
)f2(t) dt f1(s + α)f2(s + α + β + 1)
Notation: Γ(z) is the gamma function.
ReferencesBateman, H. and Erdelyi, A., Tables of Integral Transforms. Vols. 1 and 2, McGraw-Hill Book Co., New York, 1954.Doetsch, G.,Einfuhrung in Theorie und Anwendung der Laplace-Transformation, Birkhauser Verlag, Basel–Stuttgart, 1958.Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965.Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
Mellin Transforms: General Formulas
Copyright c© 2005 Andrei D. Polyanin http://eqworld.ipmnet.ru/en/auxiliary/inttrans/mellin.pdf
