Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations I. General theory, and application to modified Bessel functions of large order

1964 ◽  
Vol 281 (1384) ◽  
pp. 99-110 ◽  
Keyword(s):  
General Theory ◽  
Differential Equations ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Asymptotic Solutions ◽  
Modified Bessel Functions ◽  
Mellin Transforms ◽  
Convenient Procedure ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The problem of deriving Green-type asymptotic solutions from differential equations of general form d 2 y /dz 2 = X(a 2 >, z)y , for large values of a 2 , is reformulated. Combination of this formulation with the method of Mellin transforms leads further to a particularly convenient procedure for finding asymptotic expansions valid in transitional regions, and general uniform expansions. The methods are illustrated by detailed calculations for modified Bessel functions.

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations III. The confluent hypergeometric function U ( a , c , z ) for large | c |

1965 ◽  
Vol 284 (1399) ◽  
pp. 531-539 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Confluent Hypergeometric Function ◽  
Parabolic Cylinder ◽  
Exponential Integral ◽  
Parabolic Cylinder Function ◽  
Uniform Expansions ◽  
Kummer's Equation ◽  
Uniform Expansion ◽  
Special Case ◽  
Uniform Asymptotic Expansions

The methods introduced by Jorna (1964 a , b ) are applied to Kummer’s equation, and Green-type, transitional and uniform expansions derived for solutions of the type denoted in Slater (1960) by U ( a , c , z ) which are valid for large | c |. The subsidiary function in the uniform expansion is essentially a parabolic cylinder function of order ½ a . The general exponential integral is studied as a special case. Here the uniform expansion involves as subsidiary function the extensively tabulated error integral.

Asymptotic expansions and converging factors IV. Confluent hypergeometric, parabolic cylinder, modified Bessel, and ordinary Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 270-283 ◽  
Keyword(s):  
Exponential Type ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Modified Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Parabolic Cylinder Functions

A theory of confluent hypergeometric functions is developed, based upon the methods described in the first three papers (I, II and III) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘ basic converging factors ’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of exponential-type integrals, parabolic cylinder functions, modified Bessel functions, and ordinary Bessel functions.

Asymptotic expansions and converging factors V. Lommel, Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions

1959 ◽  
Vol 249 (1257) ◽  
pp. 284-292 ◽  
Keyword(s):  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Lommel Functions ◽  
Special Cases

A theory of Lommel functions is developed, based upon the methods described in the first four papers (I to IV) of this series for replacing the divergent parts of asymptotic expansions by easily calculable series involving one or other of the four ‘basic converging factors’ which were investigated and tabulated in I. This theory is then illustrated by application to the special cases of Struve, modified Struve, Anger and Weber functions, and integrals of ordinary and modified Bessel functions.

Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

1992 ◽  
Vol 121 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Wave Equation ◽  
Half Plane ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Small Positive Constant ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Uniform Asymptotic Expansions

SynopsisUniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p̅s̅ r n ( η , h ) and q̅s̅ r n ( η , h ) for large h

1971 ◽  
Vol 321 (1547) ◽  
pp. 545-555 ◽  
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Green Method ◽  
Oblate Spheroidal ◽  
Direct Iteration ◽  
Uniform Expansion ◽  
Formal Techniques ◽  
Uniform Asymptotic Expansions

The formal techniques of earlier papers (Jorna 1964 a , b , 1965 a , b ) are applied to the differential equation for oblate spheroidal wavefunctions, y ( z , h ) say, with h 2 large. The integro- differential equation arising in the reformulated Liouville–Green method is solved by: (i) direct iteration, yielding asymptotic expansions valid in the region │ z │≃ ½π; (ii) taking its Mellin transform and solving the resulting difference equation iteratively. This approach leads to new asymptotic expansions valid for z ≃ 0 and π , and also to the more general uniform expansion. Both methods yield, concurrently, expansions for the eigenvalues and the corresponding functions themselves. As a particular application, expansions are derived for the periodic angular oblate spheroidal wavefunctions p̅s̅ ( z , h ).

scholarly journals LINEAR DIFFERENCE EQUATIONS WITH A TRANSITION POINT AT THE ORIGIN

2013 ◽  
Vol 12 (01) ◽  
pp. 75-106 ◽  
Author(s):  
LI-HUA CAO ◽  
YU-TIAN LI
Keyword(s):  
Difference Equation ◽  
Asymptotic Expansions ◽  
Transition Point ◽  
Bessel Functions ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Small Shift ◽  
Linearly Independent ◽  
Uniform Asymptotic Expansion ◽  
Uniform Asymptotic Expansions

A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation [Formula: see text] where An and Bn have asymptotic expansions of the form [Formula: see text] with θ ≠ 0 and α0 ≠ 0 being real numbers, and β0 = ±2. Our result holds uniformly for the scaled variable t in an infinite interval containing the transition point t1 = 0, where t = (n + τ0)-θx and τ0 is a small shift. In particular, it is shown how the Bessel functions Jν and Yν get involved in the uniform asymptotic expansions of the solutions to the above linear difference equation. As an illustration of the main result, we derive a uniform asymptotic expansion for the orthogonal polynomials associated with the Laguerre-type weight xα exp (-qmxm), x > 0, where m is a positive integer, α > -1 and qm > 0.

An extension of the method of steepest descents

1957 ◽  
Vol 53 (3) ◽  
pp. 599-611 ◽  
Author(s):  
C. Chester ◽  
B. Friedman ◽  
F. Ursell
Keyword(s):  
Complex Variable ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Airy Function ◽  
Saddle Points ◽  
Positive Parameter ◽  
Implicit Relation ◽  
Image Position ◽  
Uniform Expansions ◽  
Uniformly Regular

ABSTRACTIn the integralthe functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle points varies with α, and if for some a (say α = 0) two saddle points coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In the present paper we consider this case of two nearly coincident saddle points and construct uniform expansions as follows. A new complex variable u is introduced by the implicit relationwhere the parameters ζ(α), A(α) are determined explicitly from the condition that the (u, z) transformation is uniformly regular near z = 0, α = 0 (see § 2 below). We show that with these values of the parameters there is one branch of the transformation which is uniformly regular. By taking u on this branch as a new variable of integration we obtain for the integral uniformly asymptotic expansions of the formwhere Ai and Ai′ are the Airy function and its derivative respectively, and A(α), ζ(α) are the parameters in the transformation. The application to Bessel functions of large order is briefly described.

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