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)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.
Highlights
We evaluate several of these definite integrals of the form 0 eby −1
We will derive integrals as indicated in the abstract in terms of special functions which by means of analytic continuation gives a greater range to the parameters in the integral
Some special cases of these integrals have been reported in Gradshteyn and Ryzhik [2]
Summary
We will derive integrals as indicated in the abstract in terms of special functions which by means of analytic continuation gives a greater range to the parameters in the integral. This method involves using a form of Equation (1) multiplying both sides by a function, take a definite integral of both sides. This yields a definite integral in terms of a contour integral. We multiply both sides of Equation (1) by another function and take the infinite sum of both sides such that the contour integral of both equations are the same This method has been used by us in previous work, [3,4,5,6,7]
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