§28.10 Integral Equations

Permalink:: http://dlmf.nist.gov/28.10

§28.10(i) Equations with Elementary Kernels
§28.10(ii) Equations with Bessel-Function Kernels
§28.10(iii) Further Equations

§28.10(i) Equations with Elementary Kernels

Notes:: See Arscott (1964b, pp. 79-86) and Meixner and Schäfke (1954, p. 186).
Keywords:: Mathieu functions, integral equations
Permalink:: http://dlmf.nist.gov/28.10.i

With the notation of §28.4 for Fourier coefficients,

28.10.1 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\cos\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\right)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{A_{0}^{{2n}}(h^{2})}{\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

28.10.2 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\cosh\/}\nolimits\!\left(2h\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\right)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{A_{0}^{{2n}}(h^{2})}{\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(0,h^{2}\right)}\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.3 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\sin\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\right)\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(t,h^{2}\right)dt=-\frac{hA_{1}^{{2n+1}}(h^{2})}{{\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits^{{\prime}}}\!\left(\frac{1}{2}\pi,h^{2}\right)}\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.4 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\mathop{\cosh\/}\nolimits\!\left(2h\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\right)\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{A_{1}^{{2n+1}}(h^{2})}{2\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(0,h^{2}\right)}\mathop{\mathrm{ce}_{{2n+1}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.5 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\sinh\/}\nolimits\!\left(2h\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\right)\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{hB_{1}^{{2n+1}}(h^{2})}{{\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)}\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.6 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\mathop{\cos\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\right)\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{B_{1}^{{2n+1}}(h^{2})}{2\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}\mathop{\mathrm{se}_{{2n+1}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.7 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\mathop{\sin\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\right)\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(t,h^{2}\right)dt=-\frac{hB_{2}^{{2n+2}}(h^{2})}{2{\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits^{{\prime}}}\!\left(\frac{1}{2}\pi,h^{2}\right)}\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(z,h^{2}\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.8 $\frac{2}{\pi}\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{\cos\/}\nolimits z\mathop{\cos\/}\nolimits t\mathop{\sinh\/}\nolimits\!\left(2h\mathop{\sin\/}\nolimits z\mathop{\sin\/}\nolimits t\right)\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(t,h^{2}\right)dt=\frac{hB_{2}^{{2n+2}}(h^{2})}{2{\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)}\mathop{\mathrm{se}_{{2n+2}}\/}\nolimits\!\left(z,h^{2}\right).$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

§28.10(ii) Equations with Bessel-Function Kernels

Notes:: See Meixner et al. (1980, §2.1.2).
Keywords:: Mathieu functions, integral equations
Permalink:: http://dlmf.nist.gov/28.10.ii

28.10.9 $\int _{{0}}^{{\ifrac{\pi}{2}}}\mathop{J_{{0}}\/}\nolimits\!\left(2\sqrt{q({\mathop{\cos\/}\nolimits^{{2}}}\tau-{\mathop{\sin\/}\nolimits^{{2}}}\zeta)}\right)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(\tau,q\right)d\tau=w_{{\mbox{\tiny II}}}(\tfrac{1}{2}\pi;\mathop{a_{{2n}}\/}\nolimits\!\left(q\right),q)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!\left(\zeta,q\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

28.10.10 $\int _{0}^{\pi}\mathop{J_{{0}}\/}\nolimits\!\left(2\sqrt{q}(\mathop{\cos\/}\nolimits\tau+\mathop{\cos\/}\nolimits\zeta)\right)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(\tau,q\right)d\tau=w_{{\mbox{\tiny II}}}(\pi;\mathop{a_{{n}}\/}\nolimits\!\left(q\right),q)\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(\zeta,q\right).$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{a_{{n}}\/}\nolimits\!\left(q\right)$ : eigenvalues of Mathieu equation, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : 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

§28.10(iii) Further Equations

Keywords:: Mathieu functions, integral equations
Permalink:: http://dlmf.nist.gov/28.10.iii

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).

§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