## Uniform asymptotic approximations of integrals

##### Abstract

In this thesis uniform asymptotic approximations of integrals are discussed. In order
to derive these approximations, two well-known methods are used i.e., the saddle point
method and the Bleistein method. To start with this, examples are given to demonstrate
these two methods and a general idea of how to approach these techniques. The asymptotics of the hypergeometric functions with large parameters are discussed
i.e., 2F1 (a + e1λ, b + e2λ
c + e3λ
; z)where ej = 0,±1, j = 1, 2, 3 as |λ|→ ∞, which are valid
in large regions of the complex z-plane, where a, b and c are fixed. The saddle point
method is applied where the saddle point gives a dominant contributions to the integral
representations of the hypergeometric functions and Bleistein’s method is adopted to
obtain the uniform asymptotic approximations of some cases where the coalescence takes
place between the critical points of the integrals.
As a special case, the uniform asymptotic approximation of the hypergeometric function
where the third parameter is large, is obtained. A new method to estimate the remainder
term in the Bleistein method is proposed which is created to deal with new type of
integrals in which the usual methods for the remainder estimates fail.
Finally, using the asymptotic property of the hypergeometric function when the third
parameter is large, the uniform asymptotic approximation of the monic Meixner Sobolev polynomials Sn(x) as n → ∞ , is obtained in terms of Airy functions. The asymptotic
approximations for the location of the zeros of these polynomials are also discussed. As
a limit case, a new asymptotic approximation for the large zeros of the classical Meixner
polynomials is provided.