Evaluation of the Integral \int_{0}^{y} \frac{tanh(x)}{x} dx; y\geqslant 1 from BCS Theory in Analytic Form via Numerical Convergence Method.

Abstract

In the weak-coupling limit of superconductivity BCS theory, the energy-gap equation near the transition  temperature involves the integral  where  providing =Boltzmann constant and  is the transition  temperature parameter of this superconductor, and represents the Debye frequency of the normal-metal lattice associated with this superconductor. This integral is not simple to evaluate since its upper limit is not a constant but a variable. So far, there exists a pure analytic method to carry out this integral but complication arises. One has to deal with intuitive integration by parts and infinite series of Dirichlet Eta function. In this article we devise a new method using simple analytic and numerical techniques to carry out the integral. Based on a remarkable mathematical property that the hyperbolic tangent function may be approximated as unity for large , we replace  with the new function of 2 domains separated by a critical parameter named    between which the function takes the value of hyperbolic tangent function and unity.  Suitable yields the correct integral. We call this “numerical convergence method”. Both techniques yield the same result, i.e. . This result was employed further to find the ratio Cooper pair binding energy/.