This note presents and discusses a class of real integrals involving a hyperbolic function by means of complex integration. An integral representation is obtained which appears in several fields of physics, including statistical mechanics, condensed matter and quantum optics and more in the so called dispersion relations.

The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed…