Asymptotic Expansions Of The Modified Bessel Function Of Pure Order And Mexiner Polynomials

Asymptotic is the branch of analysis that deals with problems concerning the determination of the behavior of a function as one of its parameters tends to a specific value, or a sequence as its index tends to infinity. In general, it refers to just two main areas:(1) asymptotic evaluation of integrals, and (2) asymptotic solution to differential equation. In this thesis, we just consider the first area. There are many classical methods for asymptotic evaluation of integrals, such as Laplace’s approximation, Kelvin’s principle of stationary phase, Debye’s method of steepest descent and so on. But these classical methods can not derive a uniform asymptotic expansion. Due to the needs in physical applications, many author developed various methods for derive a uniform asymptotic expansion. So research in this area is very interesting and important.This thesis is divide into three parts:(1) Using the method of Paris to derive a hyperasymptotic expansion involved Hadamard expansion for the case there are infinite many saddle points on the steepest descent path. (2) Modifing the method of Olde Daalhuis and Temme to prove Airy-type expansion is indeed uniform in the unbounded domain, and give a computable error bound for the expansion of modified Bessel function of purely imaginary order. (3) Modifing the rational functions method to representing the remainder and coefficients in parabolic cylinder-type expansion, and extend the domain of the saddle-point parameter to unbounded domain.The hyperasymptotic expansion involved Hadamard expansion is intro-duced by Paris. Hadamard expansions are absolutely convergent series in-volving the incomplete gamma function, by modifing their tails, they decay rapidly like an asymptotic expansion. So these expansions are very good tool for deriving hyperasymptitic expansion. Paris have used this method to deal with various integrals, such as Laplace-type integrals, integrals with clusters of saddles, or with poles, stokes’ phenomena. Now we modify his method to handle the problem when there are infinity many saddle points on the steepest descent path. We will use the modified Bessel function of purely imaginary order to illustrate our theory, and give some numerical results to show the level of accuracy that we can achieved.In the second part of this thesis, we use a new method based on a class of rational functions introduced by Olde Daalhuis and Temme, to prove that the Airy-type expansion of the modified Bessel function of purely imaginary order is indeed uniform even when saddle point parameter go to infinity. By modi-fying this method, we try to give a computable error bound for the expansion of this function. Since the conform mapping is complex, it is very difficult to get a error bound by using the integral method to derive the uniform asymp-totic expansion. Now we can obtain a error bound under this special case, and we also give the numerical result of the error bound of the modified Bessel function of purely imaginary order.In the last part of this thesis, we extend the method of Olde Daalhuis and Temme to deal with the parabolic cylinder-type integral expansions. We note that by using their method one can extend the validity of uniform asymptotic expansion to unbounded domain, but it only can deal with the Airy-type in-tegral. So we introduced a class of similar rational functions, and give a new representations of the coefficients and remainders which are involved Cauchy-integral. And we also obtain uniformly estimates when saddle points param-eter runs through an unbounded interval. We choose Meixner polynomials Mn(nα;β,c) as an example to illustrate our theory, it is worked out in detail.