A Ramanujan integral and its derivatives: computation and analysis

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

Mathematics of Computation Pub Date : 2023-08-15 DOI:10.1090/mcom/3892

Walter Gautschi, Gradimir Milovanovic

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引用次数: 0

Abstract

The principal tool of computation used in this paper is classical Gaussian quadrature on the interval [0,1], which happens to be particularly effective here. Explicit expressions are found for the derivatives of the Ramanujan integral in question, and it is proved that the latter is completely monotone on ( 0 , ∞ ) (0,\infty ) . As a byproduct, known series expansions for incomplete gamma functions are examined with regard to their convergence properties. The paper also pays attention to another famous integral, the Euler integral — better known as the gamma function — revitalizing a largely neglected part of the function, the part corresponding to negative values of the argument, which plays a prominent role in our work.

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拉马努金积分及其导数:计算与分析

本文使用的主要计算工具是区间[0,1]上的经典高斯正交，它在这里特别有效。得到了所讨论的Ramanujan积分的导数的显式表达式，并证明了Ramanujan积分在(0，∞)(0,\infty)上是完全单调的。作为一个副产品，已知的不完全函数的级数展开式是关于其收敛性的。本文还关注了另一个著名的积分，欧拉积分-更广为人知的是伽马函数-重新激活了函数中很大程度上被忽视的部分，即对应于参数负值的部分，这部分在我们的工作中起着突出的作用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics of Computation 数学-应用数学

CiteScore

3.90

自引率

5.00%

发文量

审稿时长

7.0 months

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.