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integral equation for Lamé functions

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1: 29.8 Integral Equations
§29.8 Integral Equations
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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
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  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • 3: Bibliography S
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  • 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
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    §29.2(i) Lamé’s Equation
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    §29.2(ii) Other Forms
    ►we have …For the Weierstrass function see §23.2(ii). … ►
    5: 29.3 Definitions and Basic Properties
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    §29.3(i) Eigenvalues
    ►satisfies the continued-fraction equation►
    §29.3(iv) Lamé Functions
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    §29.3(v) Normalization
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    §29.3(vi) Orthogonality
    6: Bibliography E
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  • A. Erdélyi (1941b) On Lamé functions. Philos. Mag. (7) 31, pp. 123–130.
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  • A. Erdélyi (1941c) On algebraic Lamé functions. Philos. Mag. (7) 32, pp. 348–350.
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  • A. Erdélyi (1942a) Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.
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  • A. Erdélyi (1942b) The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.
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  • W. N. Everitt (2005a) A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.
  • 7: Bibliography L
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  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
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  • C. G. Lambe (1952) Lamé-Wangerin functions. Quart. J. Math., Oxford Ser. (2) 3, pp. 107–114.
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  • 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.
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  • S. Lewanowicz (1991) Evaluation of Bessel function integrals with algebraic singularities. J. Comput. Appl. Math. 37 (1-3), pp. 101–112.
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  • Y. L. Luke (1962) Integrals of Bessel Functions. McGraw-Hill Book Co., Inc., New York.
  • 8: 29.1 Special Notation
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    ( s ν m ⁡ ( k 2 ) ) 2 = 4 π ⁢ 0 K ⁡ ( 𝐸𝑠 ν m ⁡ ( x , k 2 ) ) 2 ⁢ d x .
    9: Bibliography V
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  • G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.
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  • 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.
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  • H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.
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  • H. Volkmer (1984) Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.
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  • H. Volkmer (2004b) Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.
  • 10: 29.17 Other Solutions
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    §29.17(i) Second Solution
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    §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 ⁡ . …