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# Mellin Transform

The Mellin transform is the integral transform defined by

 (1) (2)

It is implemented in the Wolfram Language as MellinTransform[expr, x, s].

The transform exists if the integral

 (3)

is bounded for some , in which case the inverse exists with . The functions and are called a Mellin transform pair, and either can be computed if the other is known.

The following table gives Mellin transforms of common functions (Bracewell 1999, p. 255). Here, is the delta function, is the Heaviside step function, is the gamma function, is the incomplete beta function, is the complementary error function erfc, and is the sine integral.

 convergence

Another example of a Mellin transform is the relationship between the Riemann function and the Riemann zeta function ,

 (4) (5)

A related pair is used in one proof of the prime number theorem (Titchmarsh 1987, pp. 51-54 and equation 3.7.2).

Fourier Transform, Integral Transform, Strassen Formulas

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## References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985.Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 254-257, 1999.Gradshteyn, I. S. and Ryzhik, I. M. "Mellin Transform." §17.41 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 1193-1197, 2000.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 469-471, 1953.Oberhettinger, F. Tables of Mellin Transforms. New York: Springer-Verlag, 1974.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Evaluation of Integrals and the Mellin Transform." Itogi Nauki i Tekhniki, Seriya Matemat. Analiz 27, 3-146, 1989.Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 567, 1995.

Mellin Transform

## Cite this as:

Weisstein, Eric W. "Mellin Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MellinTransform.html