Large-Parameter Asymptotic Expansions for the Legendre and Allied Functions

  title={Large-Parameter Asymptotic Expansions for the Legendre and Allied Functions},
  author={G Nemes and Adri B. Olde Daalhuis},
  journal={SIAM J. Math. Anal.},
Surprisingly, apart from some special cases, simple asymptotic expansions for the associated Legendre functions $P_\nu ^\mu (z)$ and $Q_\nu ^\mu (z)$ for large degree $\nu$ or large order $\mu$ are not available in the literature. The main purpose of the present paper is to fill this gap by deriving simple (inverse) factorial expansions for these functions and provide sharp and realistic bounds on their error terms. Analogous results for the Ferrers functions and the closely related Gegenbauer… Expand
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We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of theExpand
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We study the perturbative quantization of 2-dimensional massive scalar field theory with polynomial (or power series) potential on manifolds with boundary. We prove that it fits into the functorialExpand
Images of point charges in conducting ellipses and prolate spheroids
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  • Journal of Mathematical Physics
  • 2021
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  • R. P. Thorne
  • Mathematics
  • Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1957
New expansions for the Legendre functions and are obtained; m and n are large positive numbers, is kept fixed as is an unrestricted complex variable. Three groups of expansions are obtained. TheExpand
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For fixed m with $m + \frac{1}{2} > 0$, an asymptotic expansion for large n is obtained for the Legendre function $Q_n^{ - m} (\cosh z)$ that is uniformly valid for z in the unbounded intervalExpand
Uniform asymptotic expansions for associated Legendre functions of large order
  • T. M. Dunster
  • Mathematics
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2003
Uniform asymptotic expansions are obtained for the associated Legendre functions and , and the Ferrers functions and , as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 −Expand
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  • F. Olver
  • Mathematics
  • Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1975
By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for theExpand
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We perform global and local analysis of oscillatory and damped spherically symmetric fundamental solutions for Helmholtz operators $\big({-}\Delta\pm\beta^2\big)$ in $d$-dimensional, $R$-radiusExpand
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  • Mathematics
  • Proceedings of the Edinburgh Mathematical Society
  • 2004
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