# Part 2. Infinite series and logarithmic integrals associated to differentiation with respect to parameters of the Whittaker $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ function

## TL;DR

This paper derives formulas for the derivatives of the Whittaker function with respect to its parameters, expressing them as infinite sums and integrals, and provides closed-form solutions for specific parameter values.

## Contribution

It introduces new methods to compute parameter derivatives of the Whittaker function using infinite series and integrals, including closed-form expressions for special cases.

## Key findings

- Derived derivatives of Whittaker function using hypergeometric functions
- Expressed derivatives as infinite sums and integrals involving elementary functions
- Provided closed-form solutions for particular parameter values

## Abstract

First derivatives with respect to the parameters of the Whittaker function $\mathrm{W}_{\kappa ,\mu }\left( x\right) $ are calculated. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Also, it is possible to obtain these parameter derivatives in terms of infinite integrals with integrands containing elementary functions (products of algebraic, exponential and logarithmic functions) from the integral representation of $\mathrm{W}_{\kappa ,\mu }\left( x\right) $. These infinite sums and integrals can be expressed in closed-form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ and its derivative with respect to $\kappa $, as well as some reduction formulas for the integral Whittaker functions $\mathrm{Wi}_{\kappa ,\mu }\left( x\right) $ and $\mathrm{wi}_{\kappa ,\mu }\left( x\right) $ are calculated.