On asymptotic expansion of a generalization of integral transform with the Tricomi function as the kernel

Author:

Yuri~V.~Vasil'ev

2010 Mathematics Subject Classification:

44A15, 33C15, 41A60.

Keywords:

asymptotic expansions, G-transform, Parseval formula for the Mellin transform, Tricomi function.

Abstract:

The article is devoted to the study of an asymptotic behavior of the integral transform ({\bf\Psi}_{\sigma,\kappa;1,1,h}f)(\lambda) =\lambda^{\sigma}\int_{0}^{\infty}\Psi(a,c,h\lambda t) t^{\kappa}f(t)\d t$, $\lambda,h>0$, $a,c,\sigma,\kappa \in \mathbb{C}, involving the Tricomi function \Psi(a,c,z) in the kernel. It is proved that ({\bf\Psi}_{\sigma,\kappa;1,1,h}f)(\lambda) has power or power-logarithmic asymptotic expansion, as \lambda\to 0^{+}$ and $\lambda\to\infty, provided that f(t) has power asymptotic behavior at infinity and zero, respectively.

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Vol. 10 (18), 2015