Definite Integral of Algebraic, Exponential and Hyperbolic Functions Expressed in Terms of Special Functions

Robert Reynolds and Allan Stauffer
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Robert Reynolds: Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada
Allan Stauffer: Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

Mathematics, 2020, vol. 8, issue 9, 1-10

Abstract: While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫ 0 ∞ ( a + y ) k − ( a − y ) k e b y − 1 d y , ∫ 0 ∞ ( a + y ) k − ( a − y ) k e b y + 1 d y , ∫ 0 ∞ ( a + y ) k − ( a − y ) k sinh ( b y ) d y and ∫ 0 ∞ ( a + y ) k + ( a − y ) k cosh ( b y ) d y in terms of a special function where k , a and b are arbitrary complex numbers.

Keywords: hyperbolic sine; hyperbolic cosine; algebraic function; definite integral; hankel contour; cauchy integral; gradshteyn and ryzhik; bierens de haan (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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