A special kind of integral by residue theorem

Heguang Chen

doi:10.1117/12.2639528

17 May 2022 A special kind of integral by residue theorem

Heguang Chen

Proceedings Volume 12259, 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2022); 122591W (2022) https://doi.org/10.1117/12.2639528
Event: 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing, 2022, Kunming, China

Abstract

This paper presents the recurrence formula of a specific kind of improper integral with the bounds of 0 and positive infinity, involving a certain power of the natural logarithm of z, divided by the sum of one and a certain power of z, with respect to z. We use two integer variables as the power of the natural logarithm and the power of z in the denominator of the integrand to parameterize this integral. Moreover, we present some discoveries for particular integral among this integral type as well. To compute the integral, we get inspiration from the residue theorem to construct a contour as the new integration path. Besides, we cope with the redundant paths via integral transformations as well as mathematical induction. After computing it, we conclude a recurrence formula and try different values of the parameters for some observations. To prove the observations, we apply some properties of the infinity, the series, and the Euler numbers. This recurrence formula makes this integral easier for us to calculate, and these observations reveal more beautiful properties of this integral category.

Citation Download Citation

Heguang Chen "A special kind of integral by residue theorem", Proc. SPIE 12259, 2nd International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2022), 122591W (17 May 2022); https://doi.org/10.1117/12.2639528

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