We perform an asymptotic evaluation of the Hankel transform, , for arbitrarily large of an entire exponential type function, , of type by shifting the contour of integration in the complex plane. Under the situation that has an odd parity with respect to and the condition that the asymptotic parameter is greater than the type… Read More »
\begin{theorem} The reciprocal of assumes the following family of integral representations (1) (2) for , and , where is the Bessel function of the first kind and is the Euler gamma function. \end{theorem}
Integrals are a mainstay in physics. Sooner or later we will have to evaluate an integral and when one arises its either we pick up the venerable Table of Integrals, Series, and Products by Gradhsteyn and Ryzhik (GR) or tap our keyboards and let Mathematica or Maple do the integration for us. We are lucky… Read More »
In the publication found here or here, I introduced the concept of analytic principal value (APV) to allow meaningful assignment of values to the class of divergent integrals given by (1) with . The basic assumption in the definition of the APV is that the function has a complex extension , obtained by replacing… Read More »
Beauty is one of those abstractions that is difficult to define or quantify. But we know it when we see one. I arrived at the following integral trying to make sense of some divergent integrals: (1) There is manifest symmetry in the integrand and the value of the integral involves the ratio of two… Read More »