Digital Library of Mathematical Functions
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28 Mathieu Functions and Hill’s EquationMathieu Functions of Integer Order

§28.10 Integral Equations

Contents

§28.10(i) Equations with Elementary Kernels

With the notation of §28.4 for Fourier coefficients,

28.10.1 2π0π/2cos(2hcoszcost)ce2n(t,h2)t =A02n(h2)ce2n(12π,h2)ce2n(z,h2),
28.10.2 2π0π/2cosh(2hsinzsint)ce2n(t,h2)t =A02n(h2)ce2n(0,h2)ce2n(z,h2),
28.10.3 2π0π/2sin(2hcoszcost)ce2n+1(t,h2)t =-hA12n+1(h2)ce2n+1(12π,h2)ce2n+1(z,h2),
28.10.4 2π0π/2coszcostcosh(2hsinzsint)ce2n+1(t,h2)t=A12n+1(h2)2ce2n+1(0,h2)ce2n+1(z,h2),
28.10.5 2π0π/2sinh(2hsinzsint)se2n+1(t,h2)t=hB12n+1(h2)se2n+1(0,h2)se2n+1(z,h2),
28.10.6 2π0π/2sinzsintcos(2hcoszcost)se2n+1(t,h2)t=B12n+1(h2)2se2n+1(12π,h2)se2n+1(z,h2),
28.10.7 2π0π/2sinzsintsin(2hcoszcost)se2n+2(t,h2)t=-hB22n+2(h2)2se2n+2(12π,h2)se2n+2(z,h2),
28.10.8 2π0π/2coszcostsinh(2hsinzsint)se2n+2(t,h2)t=hB22n+2(h2)2se2n+2(0,h2)se2n+2(z,h2).

§28.10(ii) Equations with Bessel-Function Kernels

§28.10(iii) Further Equations

See §28.28. See also Prudnikov et al. (1990, pp. 359–368), Erdélyi et al. (1955, p. 115), and Gradshteyn and Ryzhik (2000, pp. 755–759). For relations with variable boundaries see Volkmer (1983).