§28.10 Integral Equations

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Permalink:: http://dlmf.nist.gov/28.10
See also:: Annotations for Ch.28

§28.10(i) Equations with Elementary Kernels

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Keywords:: Mathieu functions, integral equations, with elementary kernels
Notes:: See Arscott (1964b, pp. 79-86) and Meixner and Schäfke (1954, p. 186).
Permalink:: http://dlmf.nist.gov/28.10.i
See also:: Annotations for §28.10 and Ch.28

With the notation of §28.4 for Fourier coefficients,

28.10.1	$\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos\left(2h\cos z\cos t% \right)\operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t$	$\displaystyle=\frac{A_{0}^{2n}(h^{2})}{\operatorname{ce}_{2n}\left(\frac{1}{2}% \pi,h^{2}\right)}\operatorname{ce}_{2n}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.20 (in different form) Permalink: http://dlmf.nist.gov/28.10.E1 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28
28.10.2	$\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cosh\left(2h\sin z\sin t% \right)\operatorname{ce}_{2n}\left(t,h^{2}\right)\,\mathrm{d}t$	$\displaystyle=\frac{A_{0}^{2n}(h^{2})}{\operatorname{ce}_{2n}\left(0,h^{2}% \right)}\operatorname{ce}_{2n}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.21 (in different form) Permalink: http://dlmf.nist.gov/28.10.E2 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28
28.10.3	$\displaystyle\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin\left(2h\cos z\cos t% \right)\operatorname{ce}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t$	$\displaystyle=-\frac{hA_{1}^{2n+1}(h^{2})}{\operatorname{ce}_{2n+1}'\left(% \frac{1}{2}\pi,h^{2}\right)}\operatorname{ce}_{2n+1}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.20 (in different form) Permalink: http://dlmf.nist.gov/28.10.E3 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

28.10.4		$\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t\cosh\left(2h\sin z\sin t% \right)\operatorname{ce}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{A_{1}^{% 2n+1}(h^{2})}{2\operatorname{ce}_{2n+1}\left(0,h^{2}\right)}\operatorname{ce}_% {2n+1}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.21 (in different form) Permalink: http://dlmf.nist.gov/28.10.E4 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

28.10.5		$\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sinh\left(2h\sin z\sin t\right)% \operatorname{se}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{hB_{1}^{2n+1}(% h^{2})}{\operatorname{se}_{2n+1}'\left(0,h^{2}\right)}\operatorname{se}_{2n+1}% \left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.23 (in different form) Permalink: http://dlmf.nist.gov/28.10.E5 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

28.10.6		$\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t\cos\left(2h\cos z\cos t% \right)\operatorname{se}_{2n+1}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{B_{1}^{% 2n+1}(h^{2})}{2\operatorname{se}_{2n+1}\left(\frac{1}{2}\pi,h^{2}\right)}% \operatorname{se}_{2n+1}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.22 (in different form) Permalink: http://dlmf.nist.gov/28.10.E6 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

28.10.7		$\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\sin z\sin t\sin\left(2h\cos z\cos t% \right)\operatorname{se}_{2n+2}\left(t,h^{2}\right)\,\mathrm{d}t=-\frac{hB_{2}% ^{2n+2}(h^{2})}{2\operatorname{se}_{2n+2}'\left(\frac{1}{2}\pi,h^{2}\right)}% \operatorname{se}_{2n+2}\left(z,h^{2}\right),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.22 (in different form) Permalink: http://dlmf.nist.gov/28.10.E7 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

28.10.8		$\frac{2}{\pi}\int_{0}^{\ifrac{\pi}{2}}\cos z\cos t\sinh\left(2h\sin z\sin t% \right)\operatorname{se}_{2n+2}\left(t,h^{2}\right)\,\mathrm{d}t=\frac{hB_{2}^% {2n+2}(h^{2})}{2\operatorname{se}_{2n+2}'\left(0,h^{2}\right)}\operatorname{se% }_{2n+2}\left(z,h^{2}\right).$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\sin\NVar{z}$ : sine function, $h$ : parameter, $n$ : integer, $z$ : complex variable and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.23 (in different form) Permalink: http://dlmf.nist.gov/28.10.E8 Encodings: TeX, pMML, png See also: Annotations for §28.10(i), §28.10 and Ch.28

§28.10(ii) Equations with Bessel-Function Kernels

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Keywords:: Mathieu functions, integral equations, with Bessel-function kernels
Notes:: See Meixner et al. (1980, §2.1.2).
Permalink:: http://dlmf.nist.gov/28.10.ii
See also:: Annotations for §28.10 and Ch.28

28.10.9	$\displaystyle\int_{0}^{\ifrac{\pi}{2}}J_{0}\left(2\sqrt{q({\cos}^{2}\tau-{\sin% }^{2}\zeta)}\right)\operatorname{ce}_{2n}\left(\tau,q\right)\,\mathrm{d}\tau$	$\displaystyle=w_{\mbox{\tiny II}}(\tfrac{1}{2}\pi;a_{2n}\left(q\right),q)% \operatorname{ce}_{2n}\left(\zeta,q\right),$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $q=h^{2}$ : parameter, $n$ : integer and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution Permalink: http://dlmf.nist.gov/28.10.E9 Encodings: TeX, pMML, png See also: Annotations for §28.10(ii), §28.10 and Ch.28
28.10.10	$\displaystyle\int_{0}^{\pi}J_{0}\left(2\sqrt{q}(\cos\tau+\cos\zeta)\right)% \operatorname{ce}_{n}\left(\tau,q\right)\,\mathrm{d}\tau$	$\displaystyle=w_{\mbox{\tiny II}}(\pi;a_{n}\left(q\right),q)\operatorname{ce}_% {n}\left(\zeta,q\right).$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $q=h^{2}$ : parameter, $n$ : integer and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution Permalink: http://dlmf.nist.gov/28.10.E10 Encodings: TeX, pMML, png See also: Annotations for §28.10(ii), §28.10 and Ch.28

§28.10(iii) Further Equations

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Keywords:: Mathieu functions, compendia, integral equations, variable boundaries
Permalink:: http://dlmf.nist.gov/28.10.iii
See also:: Annotations for §28.10 and Ch.28

See §28.28. See also Prudnikov et al. (1990, pp. 359–368), Erdélyi et al. (1955, p. 115), and Gradshteyn and Ryzhik (2015, §§6.91–6.93). For relations with variable boundaries see Volkmer (1983).

§28.10 Integral Equations

Contents

§28.10(i) Equations with Elementary Kernels

§28.10(ii) Equations with Bessel-Function Kernels

§28.10(iii) Further Equations