An Overview of the Methods

This second volume on Innovative Integrals and Their Applications, as the first one, presents a variety of sophisticated techniques to derive integral identities involving many of the known special functions of mathematical physics and the fundamental constants, such the Euler-Mascheroni constant. This first chapter overviews these methods, special functions, and the related areas of applications in the sciences and engineering, from the gamma function to the hypergeometric function to the Riemann-Hurwitz zeta function. The methods include the by now classic power substitution and its many variants, the interchange of the order of integration, and new ones such as permutation symmetry, and the use of pairs of Laplace transforms and Fourier transforms.