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integral equation for Lamé functions

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11: 29.18 Mathematical Applications
§29.18(i) Sphero-Conal Coordinates
(29.18.5) is the differential equation of spherical Bessel functions10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). … where u 1 , u 2 , u 3 each satisfy the Lamé wave equation (29.11.1). …
§29.18(iv) Other Applications
Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. …
12: 29.4 Graphics
§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs
§29.4(ii) Eigenvalues of Lamé’s Equation: Surfaces
See accompanying text
Figure 29.4.12: b ν 2 ( k 2 ) as a function of ν and k 2 . Magnify 3D Help
§29.4(iii) Lamé Functions: Line Graphs
§29.4(iv) Lamé Functions: Surfaces
13: 31.7 Relations to Other Functions
§31.7 Relations to Other Functions
§31.7(i) Reductions to the Gauss Hypergeometric Function
§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). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …
14: 29.12 Definitions
§29.12 Definitions
§29.12(i) Elliptic-Function Form
The Lamé functions 𝐸𝑐 ν m ( z , k 2 ) , m = 0 , 1 , , ν , and 𝐸𝑠 ν m ( z , k 2 ) , m = 1 , 2 , , ν , are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of z . …
15: Bibliography F
  • B. R. Fabijonas, D. W. Lozier, and F. W. J. Olver (2004) Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.
  • M. V. Fedoryuk (1989) The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).
  • M. V. Fedoryuk (1991) Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).
  • P. J. Forrester and N. S. Witte (2001) Application of the τ -function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Comm. Math. Phys. 219 (2), pp. 357–398.
  • 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.
  • 16: 29.13 Graphics
    §29.13(i) Eigenvalues for Lamé Polynomials
    See accompanying text
    Figure 29.13.4: a 4 m ( k 2 ) , b 4 m ( k 2 ) as functions of k 2 for m = 0 , 1 , 2 , 3 , 4 ( a ’s), m = 1 , 2 , 3 , 4 ( b ’s). Magnify
    §29.13(ii) Lamé Polynomials: Real Variable
    §29.13(iii) Lamé Polynomials: Complex Variable
    See accompanying text
    Figure 29.13.23: | 𝑢𝐸 4 1 ( x + i y , 0.9 ) | for 3 K x 3 K , 0 y 2 K . … Magnify 3D Help
    17: Bibliography I
  • M. Ikonomou, P. Köhler, and A. F. Jacob (1995) Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.
  • E. L. Ince (1926) Ordinary Differential Equations. Longmans, Green and Co., London.
  • E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.
  • E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • 18: 31.8 Solutions via Quadratures
    §31.8 Solutions via Quadratures
    Lastly, λ j , j = 1 , 2 , , 2 g + 1 , are the zeros of the Wronskian of w + ( 𝐦 ; λ ; z ) and w ( 𝐦 ; λ ; z ) . … 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. The curve Γ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for m j . …For more details see Smirnov (2002). …
    19: 29.6 Fourier Series
    §29.6 Fourier Series
    §29.6(i) Function 𝐸𝑐 ν 2 m ( z , k 2 )
    In addition, if H satisfies (29.6.2), then (29.6.3) applies. … Consequently, 𝐸𝑐 ν 2 m ( z , k 2 ) reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). …
    §29.6(ii) Function 𝐸𝑐 ν 2 m + 1 ( z , k 2 )
    20: Bibliography H
  • P. I. Hadži (1968) Computation of certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven 1968 (2), pp. 81–104. (errata insert) (Russian).
  • P. I. Hadži (1969) Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).
  • P. I. Hadži (1970) Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).
  • P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
  • B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.