Mellin transform

Encyclopedia : M : ME : MEL : Mellin transform

In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.

The Mellin transform of a function f is

[\left\f\right\}(s) = \varphi(s)=\int_0^ x^s f(x)\frac.]

The inverse transform is

[\left\^\varphi\right\}(x) = f(x)=\frac \int_^ x^ \varphi(s) ds.]

The notation implies this is a line integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem.

The transform is named after the Finnish mathematician Robert Hjalmar Mellin (1854 - 1933).

Contents

1 Relationship to other transforms
2 Cahen-Mellin integral
3 Examples
4 References

Relationship to other transforms

The two-sided Laplace transform may be defined in terms of the Mellin transform by

[ \left\ f\right\}(s) = \left\ f(-\ln x) \right\}(s)]

and conversely we can get the Mellin transform from the two-sided Laplace transform by

[\left\ f\right\}(s) = \left\ f(e^)\right\}(s)]

The Mellin transform may be thought of as integrating using a kernel x^s with respect to the multiplicative Haar measure, [\frac], which is invariant under dilation [x \mapsto ax], so that [\frac = \frac]; the two-sided Laplace transform integrates with respect to the additive Haar measure [dx], which is translation invariant, so that [d(x+a) = dx].

We also may define the Fourier transform in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then

[\left\ f\right\}(s) = \left\ f\right\}(is) = \left\ f(-\ln x)\right\}(is)]

We may also reverse the process and obtain

[\left\ f\right\}(s) = \left\ f(e^)\right\}(s) = \left\ f(e^)\right\}(-is)]

The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson-Mellin-Newton cycle.

Cahen-Mellin integral

For [c>0], [\Re(y)>0] and [y^] on the principal branch, on has

[e^= \frac\int_^ \Gamma(s) y^\;ds]

where [\Gamma(s)] is the gamma function. This integral is known as the Cahen-Mellin integral^{[#endnote_Hard16]}.

Examples

Perron's formula describes the inverse Mellin transform applied to a Dirichlet series.
The Mellin transform is used in some proofs of the prime counting function and occurs in discussions of the Riemann zeta function.
Inverse Mellin transforms commonly occur in Riesz means.

References

↑ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp.119-196. (See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)
Paris, R. B., and Kaminsky, D., Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001.
A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4
[Tables of Integral Transforms] at EqWorld: The World of Mathematical Equations.
, [Mellin Transform] at MathWorld.

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.