# connection with spheroidal wave functions

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##### 1: 30.11 Radial Spheroidal Wave Functions
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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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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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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,
##### 2: 30.16 Methods of Computation
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###### §30.16(ii) SpheroidalWaveFunctions of the First Kind
βΊIf $\lambda^{m}_{n}\left(\gamma^{2}\right)$ is known, then $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ can be found by summing (30.8.1). … βΊ βΊ
βΊThe 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
βΊSubsequently, 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 LaΔ­ (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 2830.
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##### 5: 31.12 Confluent Forms of Heun’s Equation
βΊThis has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\infty$. βΊMathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions30.12) are special cases of solutions of the confluent Heun equation. … βΊFor 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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• J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.
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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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• 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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• Lord Kelvin (1905) Deep water ship-waves. Phil. Mag. 9, pp. 733–757.
• ##### 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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• 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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• 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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• 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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• I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov (1976) Sferoidalnye i kulonovskie sferoidalnye funktsii. Izdat. “Nauka”, Moscow (Russian).
• ##### 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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• 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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• D. S. Jones (1986) Acoustic and Electromagnetic Waves. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.
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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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• N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.
• ##### 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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• 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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• 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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• E. T. Copson (1933) An approximation connected with $e^{-x}$ . Proc. Edinburgh Math. Soc. (2) 3, pp. 201–206.
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• A. R. Curtis (1964a) Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.
• ##### 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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• 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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• 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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• 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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• H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.