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We present a method using contour integration to derive definite integralsand their associated infinite sums which can be expressed as a specialfunction. We give a proof of the basic equation and some examples of themethod. The advantage of using special functions is their analytic continuationwhich widens the range of the parameters of the definite integral over whichthe formula is valid. We give as examples definite integrals of logarithmicfunctions times a trigonometric function. In various cases thesegeneralizations evaluate to known mathematical constants such as Catalan'sconstant and $\pi$.

A Method for Evaluating Definite Integrals in terms of Special Functions with Examples