The Integral Representation for the Product of Two Parabolic Cylinder Functions $D_{\nu }(x)D_{\nu }(-x)$ at $Re\nu <0$ by Means of the Fundamental Solution of a Landau-Type Operator

Abstract

The fundamental solution (Green's function) of a first order matrix ordinary differential equation arising in a Landau-type problem is calculated by two methods. The coincidence of the two representations results in the integral formula for the product of two parabolic cylinder functions $D_{\nu }(x)D_{\nu }(-x)$ at $Re\nu <0$, $x$ is real.

13 pages, LaTeX, notes added

References in corpus (1)

• Higher Corrections to the Mass Current in the Weakly Inhomogeneous A-phase of Helium-3

Cited by in corpus (3)

• Integral representations for products of two parabolic cylinder functions with different arguments and orders
• Product of parabolic cylinder functions involving Laplace transforms of confluent hypergeometric functions
• Some integral representations and limits for (products of) the parabolic cylinder function