  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 $\pi$.
