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11: 16.25 Methods of Computation
Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …
12: 36.10 Differential Equations
§36.10 Differential Equations
§36.10(i) Equations for Ψ K ( x )
K = 2 , cusp: … K = 3 , swallowtail: … In terms of the normal forms (36.2.2) and (36.2.3), the Ψ ( U ) ( x ) satisfy the following operator equations
13: Bibliography E
  • A. Erdélyi (1942a) Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.
  • 14: 28.32 Mathematical Applications
    This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting ζ = i ξ , z = η in (28.32.3)). … defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to z uniformly on compact subsets of . …
    15: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • 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.
  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..
  • 16: 19.4 Derivatives and Differential Equations
    §19.4(ii) Differential Equations
    If ϕ = π / 2 , then these two equations become hypergeometric differential equations (15.10.1) for K ( k ) and E ( k ) . An analogous differential equation of third order for Π ( ϕ , α 2 , k ) is given in Byrd and Friedman (1971, 118.03).
    17: Bibliography K
  • A. Ya. Kazakov and S. Yu. Slavyanov (1996) Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.
  • A. V. Kitaev, C. K. Law, and J. B. McLeod (1994) Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.
  • 18: 28.28 Integrals, Integral Representations, and Integral Equations
    §28.28 Integrals, Integral Representations, and Integral Equations
    §28.28(i) Equations with Elementary Kernels
    28.28.2 1 2 π 0 2 π e 2 i h w ce n ( t , h 2 ) d t = i n ce n ( α , h 2 ) Mc n ( 1 ) ( z , h ) ,
    28.28.21 4 π 0 π / 2 𝒞 2 + 1 ( j ) ( 2 h R ) cos ( ( 2 + 1 ) ϕ ) ce 2 m + 1 ( t , h 2 ) d t = ( - 1 ) + m A 2 + 1 2 m + 1 ( h 2 ) Mc 2 m + 1 ( j ) ( z , h ) ,
    28.28.22 4 π 0 π / 2 𝒞 2 + 1 ( j ) ( 2 h R ) sin ( ( 2 + 1 ) ϕ ) se 2 m + 1 ( t , h 2 ) d t = ( - 1 ) + m B 2 + 1 2 m + 1 ( h 2 ) Ms 2 m + 1 ( j ) ( z , h ) ,
    19: 29.3 Definitions and Basic Properties
    For each pair of values of ν and k there are four infinite unbounded sets of real eigenvalues h for which equation (29.2.1) has even or odd solutions with periods 2 K or 4 K . …
    20: Bibliography F
  • Y. Fukui and T. Horiguchi (1992) Characteristic values of the integral equation satisfied by the Mathieu functions and its application to a system with chirality-pair interaction on a one-dimensional lattice. Phys. A 190 (3-4), pp. 346–362.