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Zbl 1071.65024
Paris, R.B.
Exactification of the method of steepest descents: the Bessel functions of large order and argument. (English)
[J] Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460, No. 2049, 2737-2759 (2004). ISSN 1364-5021; ISSN 1471-2946/e

Summary: The Hadamard expansion procedure applied to Laplace-type integrals taken along contours in the complex plane enables an exact description of the method of steepest descents. This mode of expansion is illustrated by the evaluation of the Bessel functions $J_\nu(\nu x)$ and $Y_\nu(\nu x)$ of large order and argument when $x$ is bounded away from unity. The limit $x\to 1$, corresponding to the coalescence of the active saddles in the integral representations of the Bessel functions, translates into a progressive loss of exponential separation between the different levels of the Hadamard expansion, which renders computation in this limit more difficult. It is shown how a simple modification to this procedure can be employed to deal with the coalescence of the active saddles when $x\to 1$.

MSC 2000:: *65D20 Computation of special functions
33C10 Cylinder functions, etc.
33C15 Confluent hypergeometric functions
33F05 Numerical approximation of special functions

Keywords: asymptotics; hyperasymptotics; Hadamard expansions; Laplace-type integrals; method of steepest descents; Bessel functions; confluent hypergeometric functions; numerical examples

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Zentralblatt MATH Berlin [Germany]