scholarly journals Voros Coefficients at the Origin and at the Infinity of the Generalized Hypergeometric Differential Equations with a Large Parameter

2022 ◽  
Author(s):  
Takashi Aoki ◽  
◽  
Shofu Uchida ◽  
Keyword(s):  
Differential Equations ◽  
Large Parameter

Voros coefficients of the generalized hypergeometric differential equations with a large parameter are defined and their explicit forms are given for the origin and for the infinity. It is shown that they are Borel summable in some specified regions in the space of parameters and their Borel sums in the regions are given.

Convergent Liouville–Green expansions for second-order linear differential equations, with an application to Bessel functions

1993 ◽  
Vol 440 (1908) ◽  
pp. 37-54 ◽  
Keyword(s):  
Differential Equations ◽  
Bessel Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Large Parameter ◽  
Order Linear ◽  
Factorial Series ◽  
Independent Variable ◽  
Linear Differential ◽  
Asymptotic Validity

A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville–Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re ( u ) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.

Uniform approximate solutions of certain nth-order differential equations. I

1963 ◽  
Vol 59 (1) ◽  
pp. 95-110 ◽  
Author(s):  
J. Heading
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Second Order ◽  
Large Parameter ◽  
Integral Methods ◽  
History Of ◽  
Transcendental Functions ◽  
Linear Differential ◽  
Image Position ◽  
Complete History

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)are given in terms of solutions either of the equationor of the equationwhere h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).

Sign in / Sign up

Export Citation Format

Share Document