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

$\backslash left\backslash \{\backslash mathcal\{M\}f\backslash right\backslash \}(s)\; =\; \backslash varphi(s)=\backslash int\_0^\{\backslash infty\}\; x^s\; f(x)\backslash frac\{dx\}\{x\}.$

The inverse transform is

$\backslash left\backslash \{\backslash mathcal\{M\}^\{-1\}\backslash varphi\backslash right\backslash \}(x)\; =\; f(x)=\backslash frac\{1\}\{2\; \backslash pi\; i\}\; \backslash int\_\{c-i\; \backslash infty\}^\{c+i\; \backslash infty\}\; x^\{-s\}\; \backslash varphi(s)\; ds.$

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).

Relationship to other transforms

---

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

$\backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{M\}\; f(-\backslash ln\; x)\; \backslash right\backslash \}(s)$

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

$\backslash left\backslash \{\backslash mathcal\{M\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{B\}\; f(e^\{-x\})\backslash right\backslash \}(s)$

The Mellin transform may be thought of as integrating using a kernel x^s with respect to the multiplicative Haar measure, $\backslash frac\{dx\}\{x\}$, which is invariant under dilation $x\; \backslash mapsto\; ax$, so that $\backslash frac\{d(ax)\}\{ax\}\; =\; \backslash frac\{dx\}\{x\}$; 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

$\backslash left\backslash \{\backslash mathcal\{F\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{B\}\; f\backslash right\backslash \}(is)$

= \left\{\mathcal{M} f(-\ln x)\right\}(is)

We may also reverse the process and obtain

$\backslash left\backslash \{\backslash mathcal\{M\}\; f\backslash right\backslash \}(s)\; =\; \backslash left\backslash \{\backslash mathcal\{B\}$

f(e^{-x})\right\}(s) = \left\{\mathcal{F} f(e^{-x})\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 $c>0$, $\backslash Re(y)>0$ and $y^\{-s\}$ on the principal branch, on has

$e^\{-y\}=\; \backslash frac\{1\}\{2\backslash pi\; i\}$

\int_{c-i\infty}^{c+i\infty} \Gamma(s) y^{-s}\;ds

where $\backslash Gamma(s)$ is the gamma function. This integral is known as the Cahen-Mellin integral.

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.

Complex analysis | Integral transforms

Mellin-Transformation