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integral equation for Lamé functions

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1: 29.8 Integral Equations
§29.8 Integral Equations
29.8.2 μ w ( z 1 ) w ( z 2 ) w ( z 3 ) = - 2 K 2 K P ν ( x ) w ( z ) d z ,
2: Bibliography
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • 3: Bibliography S
  • B. D. Sleeman (1968a) Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.
  • 4: 29.2 Differential Equations
    §29.2(i) Lamé’s Equation
    §29.2(ii) Other Forms
    we have …For the Weierstrass function see §23.2(ii). …
    5: 29.3 Definitions and Basic Properties
    §29.3(i) Eigenvalues
    satisfies the continued-fraction equation
    §29.3(iv) Lamé Functions
    §29.3(v) Normalization
    §29.3(vi) Orthogonality
    6: Bibliography E
  • A. Erdélyi (1941b) On Lamé functions. Philos. Mag. (7) 31, pp. 123–130.
  • A. Erdélyi (1941c) On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.
  • A. Erdélyi (1942a) Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.
  • A. Erdélyi (1942b) The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.
  • A. Erdélyi (1944) Certain expansions of solutions of the Heun equation. Quart. J. Math., Oxford Ser. 15, pp. 62–69.
  • 7: Bibliography L
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • C. G. Lambe (1952) Lamé-Wangerin functions. Quart. J. Math., Oxford Ser. (2) 3, pp. 107–114.
  • J. Letessier, G. Valent, and J. Wimp (1994) Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.
  • S. Lewanowicz (1991) Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.
  • Y. L. Luke (1962) Integrals of Bessel Functions. McGraw-Hill Book Co., Inc., New York.
  • 8: 29.1 Special Notation
    ( s ν m ( k 2 ) ) 2 = 4 π 0 K ( Es ν m ( x , k 2 ) ) 2 d x .
    9: 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.
  • 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.
  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
  • H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
  • H. Volkmer (2004b) Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.
  • 10: 29.17 Other Solutions
    §29.17(i) Second Solution
    §29.17(ii) Algebraic Lamé Functions
    Algebraic Lamé functions are solutions of (29.2.1) when ν is half an odd integer. …
    §29.17(iii) Lamé–Wangerin Functions
    Lamé–Wangerin functions are solutions of (29.2.1) with the property that ( sn ( z , k ) ) 1 / 2 w ( z ) is bounded on the line segment from i K to 2 K + i K . …