Proof that the Natural Logarithm Can Be Represented by the Gaussian Hypergeometric Function

Abstract

The Gaussian or ordinary hypergeometric function is a special function represented by the hypergeometric series that includes many other special functions as speci fic or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. This paper claims that the natural logarithm can be represented by the Gaussian hypergeometric function.