Open AccessDefinite Integral of Logarithmic Trigonometric Functions Expressed in terms of the Incomplete Gamma Function
Author(s) -
Robert Reynolds,
A D Stauffer
Publication year - 2021
Publication title -
european journal of pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.245
H-Index - 5
ISSN - 1307-5543
DOI - 10.29020/nybg.ejpam.v14i4.4063
Subject(s) - mathematics , trigonometric functions , logarithm , trigonometric integral , gamma function , inverse trigonometric functions , mathematical analysis , proofs of trigonometric identities , methods of contour integration , integration using euler's formula , range (aeronautics) , function (biology) , trigonometry , constant (computer programming) , positive definite matrix , improper integral , differentiation of trigonometric functions , integral equation , singular integral , polynomial , eigenvalues and eigenvectors , materials science , geometry , physics , composite material , quantum mechanics , evolutionary biology , linear interpolation , computer science , bicubic interpolation , biology , programming language
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π.