Pade' approximations to the exponential function

Abstract

This dissertation contains a Pade' approximation for the exponential function. The Pade' approximation happens to be a rational approximation that is unique and is in fact a generalisation of the truncated Taylor services expansion. It converges much faster than the Taylor series and is very useful among other things in the numerical inversion of Laplace transforms by use of Bromwich's integral. A general definition of the Pade' approximation is given then a Pade' approximation to the exponential function is derived. An analysis of the Pade' approximant and the polynomials forming the approximant is given and finally some analysis of a hypergeometic function containing indeterminate terms is given. This confluent hypergeometic function is part of thePade' approximation for the exponential function. The variables and parameters used in this paper are to be considered complex unless its specifically stipulated that they are real, but the parameters m and n of the Pade' approximant for the exponential function will be assumed to be positive integers even if it is not stated.