### Abstract

Original language | English |
---|---|

Pages (from-to) | 2737-2759 |

Number of pages | 23 |

Journal | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 460 |

Issue number | 2049 |

DOIs | |

State | Published - Sep 2004 |

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### Cite this

*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*460*(2049), 2737-2759. DOI: 10.1098/rspa.2004.1307

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*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, vol 460, no. 2049, pp. 2737-2759. DOI: 10.1098/rspa.2004.1307

**Exactification of the method of steepest descents : the Bessel functions of large order and argument.** / Paris, Richard B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Exactification of the method of steepest descents

T2 - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

AU - Paris,Richard B.

PY - 2004/9

Y1 - 2004/9

N2 - 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 Jv(? x) and Yv(v x) of large order and argument when x is bounded away from unity. The limit x → 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 → 1.

AB - 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 Jv(? x) and Yv(v x) of large order and argument when x is bounded away from unity. The limit x → 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 → 1.

U2 - 10.1098/rspa.2004.1307

DO - 10.1098/rspa.2004.1307

M3 - Article

VL - 460

SP - 2737

EP - 2759

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

SN - 1471-2946

IS - 2049

ER -