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relation to Heun equation

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1: 31.14 General Fuchsian Equation
Normal Form
2: 29.2 Differential Equations
For the Weierstrass function see §23.2(ii). …
3: 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. …
4: Bibliography T
  • O. I. Tolstikhin and M. Matsuzawa (2001) Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.
  • 5: Bibliography S
  • 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.
  • 6: 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 = . …
    7: 31.3 Basic Solutions
    §31.3(i) Fuchs–Frobenius Solutions at z = 0
    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
    For example, H ( a , q ; α , β , γ , δ ; z ) is equal to
    8: 31.17 Physical Applications
    §31.17 Physical Applications
    §31.17(i) Addition of Three Quantum Spins
    §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. …
    9: Bibliography M
  • R. S. Maier (2005) On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.
  • R. S. Maier (2007) The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.
  • T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.
  • M. Mazzocco (2001b) Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.
  • S. C. Milne (1985c) A new symmetry related to SU ( n ) for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.
  • 10: 31.7 Relations to Other Functions
    §31.7 Relations to Other Functions
    §31.7(i) Reductions to the Gauss Hypergeometric Function
    Other reductions of H to a F 1 2 , with at least one free parameter, exist iff the pair ( a , p ) takes one of a finite number of values, where q = α β p . …
    §31.7(ii) Relations to Lamé Functions
    equation (31.2.1) becomes Lamé’s equation with independent variable ζ ; compare (29.2.1) and (31.2.8). …