A modification of the Poincare-type asymptotic expansion for functions defined by Laplace transforms is analyzed. This modification is based on an alternative power series expansion of the integrand, and the convergen...
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A modification of the Poincare-type asymptotic expansion for functions defined by Laplace transforms is analyzed. This modification is based on an alternative power series expansion of the integrand, and the convergence properties are seen to be superior to those of the original asymptotic series. The resulting modified asymptotic expansion involves a series of confluent hypergeometric functions U(a, c, z), which can be computed by means of continued fractions in a backward recursion scheme. Numerical examples are included, such as the incomplete gamma function Gamma(a, z) and the modified Bessel function K-nu(z) for large values of z. It is observed that the same procedure can be applied to uniform asymptotic expansions when extra parameters become large as well.
The possibility of extending to generalized hypergeometricfunctions a sum rule for confluent hypergeometric functions found by Temme is considered. (C) 2003 Elsevier B.V. All rights reserved.
The possibility of extending to generalized hypergeometricfunctions a sum rule for confluent hypergeometric functions found by Temme is considered. (C) 2003 Elsevier B.V. All rights reserved.
In this paper, the product of parabolic cylinder functions D-nu(+/- z) D nu+mu-1(z), with different parameters mu and nu, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeome...
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In this paper, the product of parabolic cylinder functions D-nu(+/- z) D nu+mu-1(z), with different parameters mu and nu, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder functions.
We consider the confluenthypergeometric function M(a, b;z) for z is an element of C and Rb > Ra > 0, and the confluenthypergeometric function U(a, b;z) for b is an element of C, Ra > 0, and Rz > 0. We de...
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We consider the confluenthypergeometric function M(a, b;z) for z is an element of C and Rb > Ra > 0, and the confluenthypergeometric function U(a, b;z) for b is an element of C, Ra > 0, and Rz > 0. We derive two convergent expansions of M(a, b;z);one of them in terms of incomplete gamma functions gamma(a,z) and another one in terms of rational functions of e(z) and z. We also derive a convergent expansion of U(a, b;z) in terms of incomplete gamma functions gamma(a,z) and Gamma(a,z). The expansions of M(a, b;z) hold uniformly in either Rz >= 0 or Rz <= 0;the expansion of U(a,b;z) holds uniformly in Rz > 0. The accuracy of the approximations is illustrated by means of some numerical experiments.
In 1946, Magnus presented an addition theorem for the confluenthypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with...
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In 1946, Magnus presented an addition theorem for the confluenthypergeometric function of the second kind U with argument x+y expressed as an integral of a product of two U's, one with argument x and another with argument y. We take advantage of recently obtained asymptotics for U with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of U, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.
We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent und...
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We present a method of high-precision computation of the confluent hypergeometric functions using an effective computational approach of what we termed Franklin-Friedman expansions. These expansions are convergent under mild conditions of the involved amplitude function and for some interesting cases the coefficients can be rapidly computed, thus providing a viable alternative to the conventional dichotomy between series expansion and asymptotic expansion. The present method has been extensively tested in different regimes of the parameters and compared with recently investigated convergent and uniform asymptotic expansions.
Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials , as well as complementary confluent hypergeometric functions. The expansions are valid for n large and smal...
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Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials , as well as complementary confluent hypergeometric functions. The expansions are valid for n large and small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of compared to previous expansions, in particular with respect to higher terms and large values of . Numerical evidence of their accuracy for real and complex values of x is provided.
A general second-order linear differential equation having an irregular singularity of rank one in ∞∞\infty is considered. It is shown that the solutions of this equation can be represented by series in terms of co...
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A general second-order linear differential equation having an irregular singularity of rank one in ∞ is considered. It is shown that the solutions of this equation can be represented by series in terms of confluent hypergeometric functions which describe the full analytic behavior at the singular point
34A20
34E05
33A30
global representations
singular ODEs
series
confluent hypergeometric functions
We obtain new and complete asymptotic expansions of the confluent hypergeometric functions M(a, b;z) and U(a, b;z) for large b and z. The expansions are different in the three different regions: z + a + 1 - b > 0, ...
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We obtain new and complete asymptotic expansions of the confluent hypergeometric functions M(a, b;z) and U(a, b;z) for large b and z. The expansions are different in the three different regions: z + a + 1 - b > 0, z + a + 1 - b < 0 and z + a + 1 - b = 0. The expansions are not of Poincare type and we give explicit expressions for the terms of the expansions. In some cases, the expansions are valid for complex values of the variables too. We give numerical examples which show the accuracy of the expansions. (C) 2009 Elsevier B.V. All rights reserved.
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