A novel solution of a lateral connection problem

Abstract

Weber's parabolic cylinder equation, ( d 2y dz 2 ) + [v + 1 2 − ( z 2 4 )] y = 0 , (∗) has as solutions the parabolic cylinder functions, D v(z) ~ z v exp(− z 2 4 ) , z → + ∞. (∗∗) The expansion (∗∗) is generally not valid for z → − ∞. This situation leads to the so-called “lateral connection problem” for (∗). A novel method of solution of this problem based on the Hadamard factorization theorem applied to the “lateral connection coefficient” is given. Unlike previous methods, explicit contour integrals for D v ( z) are not required.