Ramanujan's approximation to the exponential function and generalizations

TL;DR

This paper reexamines Ramanujan's approximation to the exponential function using Perron's saddle-point method, enabling generalizations with closed-form asymptotic coefficients, and extends the analysis to the exponential integral.

Contribution

It provides a new analytical framework for Ramanujan's approximations, generalizing previous results and deriving closed-form asymptotic coefficients.

Findings

Generalized Ramanujan's approximation with closed-form coefficients

Extended analysis to exponential integral

Unified approach using Perron's saddle-point method

Abstract

Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion coefficients may be given in closed form. Ramanujan's approximation to the exponential integral is treated similarly.