Extending the domain of the integral form of the digamma function through analytic continuation

Abstract

Special functions, such as the family of hypergeometric functions, play a crucial role in physics, engineering, and applied mathematics. One of these functions is the regularized hypergeometric function, but evaluating its derivatives, which are functions of the digamma function can be challenging to compute numerically with good accuracy due to rounding errors in finite differencing. To avoid this issue, the integral representation of the digamma function is considered to convert the problem from using numerical differentiation to using numerical integration which is considered as a more stable and reliable technique. However, the integral representation of the digamma function is analytically defined only for values of the complex argument with a positive real part. This work extends the domain of the integral representation of the digamma function to include negative values of the real part of the complex argument through analytic continuation.