# 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 Zbl 32.0348.02 [3] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948) MR0942661 Zbl 0017.40404 Zbl 63.0367.05 [4] V.A. Ditkin, A.P. Prudnikov, "Transformations intégrales et calcul opérationnel" , MIR (1978) (Translated from Russian) MR0622209 MR0622210 Zbl 0375.44001