connection with spheroidal wave functions

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1: 30.11 Radial Spheroidal Wave Functions

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Connection Formulas

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§30.11(v) Connection with the $\mathit{Ps}$ and $\mathit{Qs}$ Functions

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30.11.8

S^{m(1)}_{n}\left(z,\gamma\right)=K_{n}^{m}(\gamma)\mathit{Ps}^{m}_{n}\left(z,% \gamma^{2}\right),

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Symbols:: $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $K_{n}^{m}(\gamma)$ : connection coefficient and $\gamma^{2}$ : real parameter
A&S Ref:: 21.10.1 (in different form)
Permalink:: http://dlmf.nist.gov/30.11.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §30.11(v), §30.11 and Ch.30

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30.11.9

S^{m(2)}_{n}\left(z,\gamma\right)=\frac{(n-m)!}{(n+m)!}\frac{(-1)^{m+1}\mathit% {Qs}^{m}_{n}\left(z,\gamma^{2}\right)}{\gamma K_{n}^{m}(\gamma)A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})},

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Symbols:: $!$ : factorial (as in $n!$ ), $S^{\NVar{m}(\NVar{j})}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma}\right)$ : radial spheroidal wave function, $\mathit{Qs}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument, $z$ : complex variable, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient and $\gamma^{2}$ : real parameter
A&S Ref:: 21.10.2 (in different form)
Permalink:: http://dlmf.nist.gov/30.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §30.11(v), §30.11 and Ch.30

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30.11.10

K_{n}^{m}(\gamma)=\frac{\sqrt{\pi}}{2}\left(\frac{\gamma}{2}\right)^{m}\frac{(% -1)^{m}a_{n,\frac{1}{2}(m-n)}^{-m}(\gamma^{2})}{\Gamma\left(\frac{3}{2}+m% \right)A_{n}^{-m}(\gamma^{2})\mathsf{Ps}^{m}_{n}\left(0,\gamma^{2}\right)},

n-m

even,

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind, $m$ : nonnegative integer, $n\geq m$ : integer degree, $A_{n}^{m}(\gamma^{2})$ , $K_{n}^{m}(\gamma)$ : connection coefficient, $\gamma^{2}$ : real parameter and $a^{m}_{n,k}(\gamma^{2})$ : coefficients
Referenced by:: §30.16(iii)
Permalink:: http://dlmf.nist.gov/30.11.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §30.11(v), §30.11 and Ch.30

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2: 30.16 Methods of Computation

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§30.16(ii) Spheroidal Wave Functions of the First Kind

… ►If

\lambda^{m}_{n}\left(\gamma^{2}\right)

is known, then

\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)

can be found by summing (30.8.1). … ► ►

§30.16(iii) Radial Spheroidal Wave Functions

►The coefficients

a_{n,k}^{m}(\gamma^{2})

calculated in §30.16(ii) can be used to compute

S^{m(j)}_{n}\left(z,\gamma\right)

j=1,2,3,4

from (30.11.3) as well as the connection coefficients

K_{n}^{m}(\gamma)

from (30.11.10) and (30.11.11). …

3: 31.18 Methods of Computation

§31.18 Methods of Computation

… ►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of

z

; see Laĭ (1994) and Lay et al. (1998). …The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

4: 30.15 Signal Analysis

§30.15 Signal Analysis

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§30.15(i) Scaled Spheroidal Wave Functions

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§30.15(ii) Integral Equation

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5: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at

z=0

and

1

, and an irregular singularity of rank 1 at

z=\infty

. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. … ►For properties of the solutions of (31.12.1)–(31.12.4), including connection formulas, see Bühring (1994), Ronveaux (1995, Parts B,C,D,E), Wolf (1998), Lay and Slavyanov (1998), and Slavyanov and Lay (2000). …

6: Bibliography L

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E. W. Leaver (1986) Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:: ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt
Referenced by:: §30.12, §31.17(ii).
Encodings:: BibTeX

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J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:: Document, MathReview (R. D. James), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

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L.-W. Li, M. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan (1998a) Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E 58 (5), pp. 6792–6806.

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Notes:: Mathematica package for eigenvalues and solutions of the spheroidal wave equations
External Links:: Code(SpheroidF), Document
Referenced by:: 2nd item, 3rd item.
Encodings:: BibTeX

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L.-W. Li, T. S. Yeo, P. S. Kooi, and M. S. Leong (1998b) Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions. J. Electromagn. Waves Appl. 12 (6), pp. 709–711.

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External Links:: ISSN 0920-5071, Document, MathReview
Referenced by:: §30.14(v).
Encodings:: BibTeX

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Lord Kelvin (1905) Deep water ship-waves. Phil. Mag. 9, pp. 733–757.

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External Links:: ZentralBlatt
Referenced by:: §36.13.
Encodings:: BibTeX

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7: Bibliography K

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A. Khare and U. Sukhatme (2004) Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.

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External Links:: Pdf, Document
Referenced by:: §22.7(iii).
Encodings:: BibTeX

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B. J. King, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the prolate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 7012 Naval Res. Lab. Washingtion, D.C..

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Referenced by:: §30.18(iii).
Encodings:: BibTeX

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B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:: Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.10.
Encodings:: BibTeX

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G. C. Kokkorakis and J. A. Roumeliotis (1998) Electromagnetic eigenfrequencies in a spheroidal cavity (calculation by spheroidal eigenvectors). J. Electromagn. Waves Appl. 12 (12), pp. 1601–1624.

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External Links:: ISSN 0920-5071, Document, MathReview, ZentralBlatt
Referenced by:: §30.14(v).
Encodings:: BibTeX

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I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov (1976) Sferoidalnye i kulonovskie sferoidalnye funktsii. Izdat. “Nauka”, Moscow (Russian).

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Notes:: Translated title: Spheroidal and Coulomb Spheroidal Functions
External Links:: MathReview
Referenced by:: §30.12, Ch.30, §31.13.
Encodings:: BibTeX

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8: Bibliography J

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D. L. Jagerman (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.

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External Links:: ISSN 0005-8580, MathReview (U. Narayan Bhat), ZentralBlatt
Referenced by:: §8.23.
Encodings:: BibTeX

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H. Jeffreys (1928) The effect on Love waves of heterogeneity in the lower layer. Monthly Notices Roy. Astronom. Soc. Geophysical Supplement 2, pp. 101–111.

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External Links:: ZentralBlatt
Referenced by:: §9.1.
Encodings:: BibTeX

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D. S. Jones (1986) Acoustic and Electromagnetic Waves. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.

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External Links:: ISBN 0-19-853365-9, MathReview (John A. DeSanto)
Referenced by:: §10.73(i).
Encodings:: BibTeX

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S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions

\overline{ps}\,^{r}_{n}(\eta,h)

and

\overline{qs}\,^{r}_{n}(\eta,h)

for large

h

. Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §30.9(ii).
Encodings:: BibTeX

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N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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External Links:: ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt
Referenced by:: §32.11(i), §32.12(ii).
Encodings:: BibTeX

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9: Bibliography C

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B. C. Carlson (1985) The hypergeometric function and the

R

-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:: International conference on special functions: theory and computation (Turin, 1984)
External Links:: ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §19.27(vi).
Encodings:: BibTeX

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P. A. Clarkson and J. B. McLeod (1988) A connection formula for the second Painlevé transcendent. Arch. Rational Mech. Anal. 103 (2), pp. 97–138.

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External Links:: ISSN 0945-8396, Document, MathReview (Reinhard Schäfke), ZentralBlatt
Referenced by:: §32.11(ii).
Encodings:: BibTeX

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W. C. Connett, C. Markett, and A. L. Schwartz (1993) Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

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External Links:: ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt
Referenced by:: §30.10, §30.11(vi).
Encodings:: BibTeX

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E. T. Copson (1933) An approximation connected with

e^{-x}

. Proc. Edinburgh Math. Soc. (2) 3, pp. 201–206.

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External Links:: Document, ZentralBlatt
Referenced by:: §8.11(v).
Encodings:: BibTeX

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A. R. Curtis (1964a) Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-06156-3, MathReview (John Todd), ZentralBlatt
Referenced by:: item Curtis (1964a):, §33.14(i), §33.20(iv), §33.21(ii), §33.23(vi), 2nd item, §33.9(ii), Notations P, Notations Q.
Encodings:: BibTeX

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10: Bibliography M

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J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.

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External Links:: MathReview (M. L. Mehta), ZentralBlatt
Referenced by:: §30.9(i).
Encodings:: BibTeX

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T. Morita (2013) A connection formula for the

q

-confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.

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External Links:: ISSN 1815-0659, Document, MathReview (Jeremy Lovejoy), ZentralBlatt
Referenced by:: §17.6(ii), §17.6(ii).
Encodings:: BibTeX

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H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.9(ii), §30.9(ii).
Encodings:: BibTeX

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H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.9(i), §30.9(i).
Encodings:: BibTeX

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H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

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