Jacobi–Anger expansions
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11—20 of 366 matching pages
11: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►Corresponding expansions for , , can be found by differentiating (20.2.1)–(20.2.4) with respect to . … ►For fixed , each is an entire function of with period ; is odd in and the others are even. For fixed , each of , , , and is an analytic function of for , with a natural boundary , and correspondingly, an analytic function of for with a natural boundary . … ►For , the -zeros of , , are , , , respectively.12: 22.8 Addition Theorems
13: 22.10 Maclaurin Series
14: 14.31 Other Applications
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►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
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§14.31(ii) Conical Functions
… ►The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …15: 18.18 Sums
§18.18 Sums
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… ►See Andrews et al. (1999, Lemma 7.1.1) for the more general expansion of in terms of . … ►Jacobi
…16: 11.13 Methods of Computation
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§11.13(i) Introduction
… ►The treatment of Lommel and Anger–Weber functions is similar. … ►§11.13(ii) Series Expansions
►Although the power-series expansions (11.2.1) and (11.2.2), and the Bessel-function expansions of §11.4(iv) converge for all finite values of , they are cumbersome to use when is large owing to slowness of convergence and cancellation. For large and/or the asymptotic expansions given in §11.6 should be used instead. …17: 22.6 Elementary Identities
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22.6.2
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22.6.5
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22.6.8
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§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)
► …18: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
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22.11.1
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22.11.2
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►Similar expansions for and follow immediately from (22.6.1).
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►Again, similar expansions for and may be derived via (22.6.1).
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19: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.2
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22.12.8
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22.12.11
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22.12.13
20: Bibliography B
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The generating function of Jacobi polynomials.
J. London Math. Soc. 13, pp. 8–12.
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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials.
In Orthogonal Polynomials and Their Applications (Segovia, 1986),
Lecture Notes in Math., Vol. 1329, pp. 203–221.
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Tables of the Anger and Lommel-Weber Functions.
Technical report
Technical Report 53 and AFCRL 796, University Washington Press, Seattle.
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Uniform asymptotic expansion of Charlier polynomials.
Methods Appl. Anal. 1 (3), pp. 294–313.
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A cubic counterpart of Jacobi’s identity and the AGM.
Trans. Amer. Math. Soc. 323 (2), pp. 691–701.
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