scholarly journals Uniform Asymptotic Expansions of a Class of Integrals in Terms of Modified Bessel Functions, with Application to Confluent Hypergeometric Functions

1990 ◽  
Vol 21 (1) ◽  
pp. 241-261 ◽  
Author(s):  
N. M. Temme
Keyword(s):  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

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.

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 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.

UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE WHITTAKER FUNCTIONS Mκ,iμ(z) AND Wκ,iμ(z)

2003 ◽  
Vol 01 (02) ◽  
pp. 199-212 ◽  
Author(s):  
T. M. DUNSTER
Keyword(s):  
Hypergeometric Functions ◽  
Turning Point ◽  
Bessel Functions ◽  
Asymptotic Approximations ◽  
Whittaker Functions ◽  
Modified Bessel Functions ◽  
Airy Functions ◽  
Double Pole ◽  
Confluent Hypergeometric Functions ◽  
Uniform Reduction

Uniform asymptotic approximations are obtained for the Whittaker's confluent hypergeometric functions Mκ, iμ(z) and Wκ, iμ(z), where κ, μ and z are real. Three cases are considered, and when taken together, result in approximations which are valid for κ → ∞ uniformly for 0 ≤ μ < ∞, 0 < z < ∞, and also for μ → ∞ uniformly for 0 ≤ κ < ∞, 0 < z < ∞. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.

scholarly journals INEQUALITIES FOR SOME SPECIAL FUNCTIONS-II

1995 ◽  
Vol 26 (3) ◽  
pp. 235-242
Author(s):  
S. K. BISSU ◽  
C. M. JOSHI
Keyword(s):  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Upper Bounds ◽  
Modified Bessel Functions ◽  
Lower And Upper Bounds ◽  
Confluent Hypergeometric Functions

Some inequalities for Bessel functions, modified Bessel functions of the first kind and of their ratios involving both lower and upper bounds are given. The inequalities improve the results of earlier authours. Also incorporated in the discussion are some inequalities for the ratios of confluent hypergeometric functions of one variable.

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.

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