Abstract
As a basis for the investigations of this chapter we take the series
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.
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© 1969 Springer-Verlag Berlin Heidelberg
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Buchholz, H. (1969). Integrals Depending on Parameters in the Relations for the Various Types of Physical Waves Expressed in Parabolic Coordinates. In: The Confluent Hypergeometric Function. Springer Tracts in Natural Philosophy, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-88396-5_6
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DOI: https://doi.org/10.1007/978-3-642-88396-5_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-88398-9
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