integral equation for Lamé functions
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11: 29.18 Mathematical Applications
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§29.18(i) Sphero-Conal Coordinates
… ►(29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1). … ►where , , 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
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§29.4(i) Eigenvalues of Lamé’s Equation: Line Graphs
… ►§29.4(ii) Eigenvalues of Lamé’s Equation: Surfaces
… ► ►§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 ; 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 , , and , , are called the Lamé polynomials. There are eight types of Lamé polynomials, defined as follows: …In consequence they are doubly-periodic meromorphic functions of . …15: Bibliography F
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Computation of complex Airy functions and their zeros using asymptotics and the differential equation.
ACM Trans. Math. Software 30 (4), pp. 471–490.
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The Lamé wave equation.
Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).
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Asymptotics of the spectrum of the Heun equation and of Heun functions.
Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).
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Application of the -function theory of Painlevé equations to random matrices: PIV, PII and the GUE.
Comm. Math. Phys. 219 (2), pp. 357–398.
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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.
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16: 29.13 Graphics
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§29.13(i) Eigenvalues for Lamé Polynomials
… ► ►§29.13(ii) Lamé Polynomials: Real Variable
… ►§29.13(iii) Lamé Polynomials: Complex Variable
… ►17: Bibliography I
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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.
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Ordinary Differential Equations.
Longmans, Green and Co., London.
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The periodic Lamé functions.
Proc. Roy. Soc. Edinburgh 60, pp. 47–63.
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Further investigations into the periodic Lamé functions.
Proc. Roy. Soc. Edinburgh 60, pp. 83–99.
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-Hermite polynomials, biorthogonal rational functions, and -beta integrals.
Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
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18: 31.8 Solutions via Quadratures
§31.8 Solutions via Quadratures
… ►Lastly, , , are the zeros of the Wronskian of and . … ►For , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. The curve reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for . …For more details see Smirnov (2002). …19: 29.6 Fourier Series
§29.6 Fourier Series
►§29.6(i) Function
… ►In addition, if satisfies (29.6.2), then (29.6.3) applies. … ►Consequently, reduces to a Lamé polynomial; compare §§29.12(i) and 29.15(i). … ►§29.6(ii) Function
…20: Bibliography H
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Computation of certain integrals that contain a probability function.
Bul. Akad. Štiince RSS Moldoven 1968 (2), pp. 81–104. (errata insert) (Russian).
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Certain integrals that contain a probability function and degenerate hypergeometric functions.
Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).
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Some integrals that contain a probability function and hypergeometric functions.
Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).
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Certain integrals that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
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Lamé polynomials of large order.
SIAM J. Math. Anal. 8 (5), pp. 800–842.
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