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1: 30.11 Radial Spheroidal Wave Functions
Connection Formulas
§30.11(v) Connection with the Ps and Qs Functions
30.11.9 S n m ( 2 ) ( z , γ ) = ( n - m ) ! ( n + m ) ! ( - 1 ) m + 1 Qs n m ( z , γ 2 ) γ K n m ( γ ) A n m ( γ 2 ) A n - m ( γ 2 ) ,
30.11.10 K n m ( γ ) = π 2 ( γ 2 ) m ( - 1 ) m a n , 1 2 ( m - n ) - m ( γ 2 ) Γ ( 3 2 + m ) A n - m ( γ 2 ) Ps n m ( 0 , γ 2 ) , n - m even,
2: 30.16 Methods of Computation
§30.16(ii) Spheroidal Wave Functions of the First Kind
If λ n m ( γ 2 ) is known, then Ps n m ( x , γ 2 ) can be found by summing (30.8.1). …
§30.16(iii) Radial Spheroidal Wave Functions
The coefficients a n , k m ( γ 2 ) calculated in §30.16(ii) can be used to compute S n m ( j ) ( z , γ ) , j = 1 , 2 , 3 , 4 from (30.11.3) as well as the connection coefficients K n m ( γ ) 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 2830.
4: 30.15 Signal Analysis
§30.15 Signal Analysis
§30.15(i) Scaled Spheroidal Wave Functions
§30.15(ii) Integral Equation
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 = . Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions30.12) 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
  • 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.
  • J. Lehner (1941) A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.
  • 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.
  • 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.
  • Lord Kelvin (1905) Deep water ship-waves. Phil. Mag. 9, pp. 733–757.
  • 7: Bibliography K
  • A. Khare and U. Sukhatme (2004) Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.
  • 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..
  • 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.
  • 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.
  • I. V. Komarov, L. I. Ponomarev, and S. Yu. Slavyanov (1976) Sferoidalnye i kulonovskie sferoidalnye funktsii. Izdat. “Nauka”, Moscow (Russian).
  • 8: Bibliography J
  • D. L. Jagerman (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.
  • 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.
  • D. S. Jones (1986) Acoustic and Electromagnetic Waves. Oxford Science Publications, The Clarendon Press Oxford University Press, New York.
  • S. Jorna and C. Springer (1971) Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions p s ¯ n r ( η , h ) and q s ¯ n r ( η , h ) for large h . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.
  • 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
  • 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.
  • 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.
  • 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.
  • E. T. Copson (1933) An approximation connected with e - x . Proc. Edinburgh Math. Soc. (2) 3, pp. 201–206.
  • A. R. Curtis (1964a) Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.
  • 10: Bibliography M
  • J. W. Miles (1975) Asymptotic approximations for prolate spheroidal wave functions. Studies in Appl. Math. 54 (4), pp. 315–349.
  • T. Morita (2013) A connection formula for the q -confluent hypergeometric function. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 050, 13.
  • H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.
  • H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.
  • H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.