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Register a new userIntegrals in Gradshteyn and Ryzhik: Hyperbolic and trigonometric function
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Authors
Mark W. Coffey
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The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.
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It is shown that generalized trigonometric functions and generalized hyperbolic functions can be transformed from each other. As an application of this transformation, a number of properties for one immediately lead to the corresponding properties for the other. In this way, Mitrinovi{c}-Adamovi{c}-type inequalities, multiple-angle formulas, and double-angle formulas for both can be produced.
We aim to introduce the generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $ and to present some formulas of the Riemann-Liouville fractional integration and differentiation operators. Further, we also derive certain integral formulas involving the newly defined generalized multiindex Bessel function $J_{left( beta _{j}right) _{m},kappa ,b}^{left( alpha _{j}right)_{m},gamma ,c}left[ zright] $. We prove that such integrals are expressed in terms of the Fox-Wright function $_{p}Psi_{q}(z)$. The results presented here are of general in nature and easily reducible to new and known results.
In this paper the discussion of the effect of trigonometric series on the theory of integration is continued from an earlier paper by Gluchoff, Trigonometric series and theories of integration, Math.Mag., 67 (1994), 3--20.
Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain special cases of the main results presented here are also pointed out for the extended Gauss hypergeometric and confluent hypergeometric functions.
An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results encompassing various aspects of the special function theory.
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