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relations to Heun functions

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1: 31.7 Relations to Other Functions
§31.7(i) Reductions to the Gauss Hypergeometric Function
§31.7(ii) Relations to Lamé Functions
2: 31.8 Solutions via Quadratures
For m = ( m 0 , 0 , 0 , 0 ) , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …
3: 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 = . …
4: 29.2 Differential Equations
For the Weierstrass function see §23.2(ii). …
5: Bibliography V
  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
  • O. Vallée and M. Soares (2010) Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • 6: 31.17 Physical Applications
    §31.17 Physical Applications
    For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986).
    §31.17(ii) Other Applications
    For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …
    7: 31.3 Basic Solutions
    H ( a , q ; α , β , γ , δ ; z ) denotes the solution of (31.2.1) that corresponds to the exponent 0 at z = 0 and assumes the value 1 there. …
    §31.3(ii) Fuchs–Frobenius Solutions at Other Singularities
    §31.3(iii) Equivalent Expressions
    Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). …For example, H ( a , q ; α , β , γ , δ ; z ) is equal to
    8: Bibliography M
  • A. J. MacLeod (1993) Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • F. Matta and A. Reichel (1971) Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.
  • G. J. Miel (1981) Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.
  • S. C. Milne (1985c) A new symmetry related to SU ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • 9: Bibliography S
  • J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.
  • J. Segura (2001) Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.
  • S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
  • B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.
  • B. D. Sleeman (1969) Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.
  • 10: 31.14 General Fuchsian Equation
    Heun’s equation (31.2.1) corresponds to N = 3 .
    Normal Form
    The algorithm returns a list of solutions if they exist. …