# Mellin transform

The integral transform

The substitution reduces it to the Laplace transform. The Mellin transform is used for solving a specific class of planar problems for harmonic functions in a sectorial domain, of problems in elasticity theory, etc.

The inversion theorem. Suppose that and that the function has bounded variation in a neighbourhood of the point . Then

The representation theorem. Suppose that the function is summable with respect to on and has bounded variation in a neighbourhood of the point . Then

where

#### References

[1] | H. Mellin, "Ueber die fundamentelle Wichtigkeit des Satzes von Cauchy für die Theorie der Gamma- und hypergeometrischen Funktionen" Acta Soc. Sci. Fennica , 21 : 1 (1896) pp. 1–115 |

[2] | H. Mellin, "Ueber den Zusammenhang zwischen linearen Differential- und Differenzengleichungen" Acta Math. , 25 (1902) pp. 139–164 |

[3] | E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) |

[4] | V.A. Ditkin, A.P. Prudnikov, "Transformations intégrales et calcul opérationnel" , MIR (1978) (Translated from Russian) |

#### Comments

If denotes the Mellin transform of , then the Parseval equality takes the form:

if .

The Mellin transform also serves to link Dirichlet series with automorphic functions (cf. Automorphic function); in particular, the inversion formula plays a role in the proof of a functional equation for Dirichlet series similar to that for the Riemann zeta-function. Cf. [a1]–[a5].

#### References

[a1] | E. Hecke, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" Math. Ann. , 112 (1936) pp. 664–699 |

[a2] | A. Weil, "Ueber die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung" Math. Ann. , 168 (1967) pp. 149–156 |

[a3] | A. Weil, "Zeta functions and Mellin transforms" , Algebraic geometry (Bombay Coll., 1968) , Oxford Univ. Press & Tata Inst. (1968) pp. 409–426 |

[a4] | A. Ogg, "Modular forms and Dirichlet series" , Benjamin (1969) |

[a5] | G. Shimura, "Introduction to the arithmetic theory of automorphic functions" , Princeton Univ. Press & Iwanami-Shoten (1971) pp. §3.6, pp 89–94 |

**How to Cite This Entry:**

Mellin transform.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mellin_transform&oldid=15680