Computing the Kummer function U(a, b, z) for small values of the arguments

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• 作者：Amparo Gil ; ^a ; ^{amparo.gil@unican.es" class="auth_mail" title="E-mail the corresponding author} ; Javier Segura^b ; ^{javier.segura@unican.es" class="auth_mail" title="E-mail the corresponding author} ; Nico M. Temme¹ ; ^c ; ^{nico.temme@cwi.nl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Kummer function ; Numerical computation ; Asymptotic approximation ; Power series
• 刊名：Applied Mathematics and Computation
• 出版年：2015
• 出版时间：15 November 2015
• 年：2015
• 卷：271
• 期：Complete
• 页码：532-539
• 全文大小：376 K

文摘

We describe methods for computing the Kummer function U(a, b, z) for small values of z, with special attention to small values of b. For these values of b the connection formula that represents U(a, b, z) as a linear combination of two ₁F₁-functions needs a limiting procedure. We use the power series of the ₁F₁-functions and consider the terms for which this limiting procedure is needed. We give recursion relations for higher terms in the expansion, and we consider the derivative U′(a, b, z) as well. We also discuss the performance for small |z| of an asymptotic approximation of the Kummer function in terms of modified Bessel functions.