Integrals Depending on Parameters in the Relations for the Various Types of Physical Waves Expressed in Parabolic Coordinates

Abstract

As a basis for the investigations of this chapter we take the series

$$\begin{array}{*{20}c} {K\left( {s,t;h} \right)} \hfill & { = \sum\limits_{\lambda = 0}^\infty {\frac{{\Gamma \left( {\lambda + \alpha + 1} \right)}} {{\lambda !}}h^\lambda .M_{x_1 + \lambda ,\mu _1 /2} \left( s \right) \cdot M_{x_2 + \lambda ,\mu _2 /2} \left( t \right)} } \hfill \\ {} \hfill & {\left( {\left| h \right| < 1;s,t\,arbitrary;\,\alpha \ne - 1, - 1,...} \right),} \hfill \\ \end{array}$$

(1)

whose sum was first evaluated by Erdelyi [17]. It converges absolutely for any real or complex values of the parameters x ₁, x ₂, μ₁, μ₂ α s and t inside the unit circle |h| < 1 as long as α ≠ − 1, −2,.... For such a fixed value of h the convergence with regard to s and t is in fact uniform, provided s and t remain confined to any arbitrary closed region in their plane. For the case when s and t are real, we show in Chapter III that convergence persists even when h = 1. The convergence is conditional when −1/2 < Re(α − (<₁ + <₂)/2) < +1/2 and absolute when Re(α − (μ₁ + μ₂)/2) < − 1/2.