Introduction

Abstract

The integral

$$ {\text{M}}[\phi {\text{(x),}}\,{\text{z}}] = \Phi {\text{(z)}} = \int\limits_0^{\infty } {{{\text{x}}^{{z - 1}}}\phi {\text{(x)dx}}} $$

((1))

is called the Mellin transform of the function Φ (x) with respect to the complex parameter

$$ z = \sigma + {\text{i}}\tau $$

((2))

The substitution x = e^-t transforms (1) into a two-sided Laplace integral

$$ \Phi (z) = \int\limits_{{ - \infty }}^{\infty } {\phi ({{\text{e}}^{{{\text{ - t}}}}}){{\text{e}}^{{{\text{ - tz}}}}}{\text{dt}}} $$

((3))

or into the sum of two one-sided Laplace integrals of parameter z and -z (3′)

$$ \Phi (z) = \int\limits_0^{\infty } {\phi ({{\text{e}}^{{{\text{ - t}}}}}){{\text{e}}^{{{\text{ - tz}}}}}{\text{dt}}} + \int\limits_0^{\infty } {\phi ({{\text{e}}^{\text{t}}}){{\text{e}}^{{{\text{ - t( - z)}}}}}{\text{dt}}} $$

((3′))

Denote the abscissas of absolute and ordinary convergence by β and α respectively for the first integral in (3) and by β′ and α′ for the second integral. Then it is evident that the domains of absolute and ordinary convergence of the integral (1) consist of the respective strips.

$$ \beta < {\rm Re} \,z < - \beta ';\quad \alpha < {\rm Re} \,z < - \alpha ' $$

for the inversion of the integral (1)

$$ \phi (x) = {M^{{ - 1}}}[\phi (z);x] $$

((4))

exists the following theorem.