Journal ArticleDOI
Asymptotic solutions to differential-difference equations
T Dosdale,G Duggan,G J Morgan +2 more
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Asymptotic expansions for Bessel functions starting from the differential-difference equations they satisfy are derived in this paper, which are the well known Green-Liouville expansions obtainable from either the pure differential or pure difference equations satisfied by Bessel function.Abstract:
Asymptotic expansions are derived for Bessel functions starting from the differential-difference equations they satisfy. These are just the well known Green-Liouville expansions obtainable from either the pure differential or pure difference equations satisfied by Bessel functions. The Bessel functions are considered because of their well known properties but the method described should be applicable to many more general situations.read more
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Asymptotic analysis of the Hermite polynomials from their differential–difference equation
TL;DR: In this article, the Hermite polynomials H n (x) and their zeros were analyzed asymptotically, as n → ∞ and obtained asymPTotic approximations from the differential-difference equation which they satisfy, using the ray method.
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Asymptotic analysis of the Hermite polynomials from their differential-difference equation
TL;DR: In this paper, the authors analyzed the Hermite polynomials and their zeros asymptotically, as $n\to\infty.$ and obtained asymPTotic approximations from the differential-difference equation which they satisfy, using the ray method.
Journal ArticleDOI
Asymptotic analysis of generalized Hermite polynomials
TL;DR: In this article, the authors analyzed the polynomials considered by Gould and Hopper and derived asymptotic approximations for large values of $n$ from their differential-difference equation using a discrete ray method.
Journal ArticleDOI
Asymptotic analysis of a family of polynomials associated with the inverse error function
Diego Dominici,Charles Knessl +1 more
TL;DR: In this paper, the sequence of polynomials defined by the differential-difference equation Pn+1(x) = P'(n)(x)+x(n+ 1)P-n(x), asymptotically as n -> infinity.
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Asymptotic analysis of the derivatives of the inverse error function
TL;DR: In this paper, a high-order Taylor expansion of the error function was proposed to obtain a very good approximation of the original error function through a high order Taylor expansion around $x = 0.
References
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Journal ArticleDOI
WKB methods for difference equations II
R. B. Dingle,G. J. Morgan +1 more
TL;DR: In this paper, a more general comparison equation theory for difference equations is developed, exploiting the fact that a difference equation can be considered as a differential equation of infinite order, and applying the theory to first order difference equations a useful generalization of the Euler-Maclaurin summation formula is found.
Journal ArticleDOI
Solution of the Raman-Nath equation for light diffracted by ultrasound at normal incidence
TL;DR: In this article, the number of diffracted beams which appear with appreciable intensity when light at normal incidence traverses ultrasonic waves in a liquid is derived for an approximate method for solving the Raman-Nath equation with an analogue computer.