28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.27 Addition Theorems 28.29 Definitions and Basic Properties

§28.28 Integrals, Integral Representations, and Integral Equations

Referenced by:: §28.10(iii), §28.18
Permalink:: http://dlmf.nist.gov/28.28

§28.28(i) Equations with Elementary Kernels
§28.28(ii) Integrals of Products with Bessel Functions
§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order
§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
§28.28(v) Compendia

§28.28(i) Equations with Elementary Kernels

Notes:: See Arscott (1964b, pp. 80-86) and Meixner and Schäfke (1954, §2.68).
Keywords:: Mathieu functions, integral equations, integrals, modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.28.i

Let

28.28.1 $w=\mathop{\cosh\/}\nolimits z\mathop{\cos\/}\nolimits t\mathop{\cos\/}% \nolimits\alpha+\mathop{\sinh\/}\nolimits z\mathop{\sin\/}\nolimits t\mathop{% \sin\/}\nolimits\alpha.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E1
Encodings:: TeX, pMML, png

Then

28.28.2 $\dfrac{1}{2\pi}\int_{{0}}^{{2\pi}}e^{{2ihw}}\mathop{\mathrm{ce}_{{n}}\/}% \nolimits\!\left(t,h^{2}\right)dt=i^{n}\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!% \left(\alpha,h^{2}\right)\mathop{{\mathrm{Mc}^{{(1)}}_{{n}}}\/}\nolimits\!% \left(z,h\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : parameter, : integer, : complex variable, : Mathieu’s equation solution and $\alpha$ : parameter
A&S Ref:: 20.7.34 20.7.35 20.7.36 20.7.37
Permalink:: http://dlmf.nist.gov/28.28.E2
Encodings:: TeX, pMML, png

28.28.3 $\dfrac{1}{2\pi}\int_{{0}}^{{2\pi}}e^{{2ihw}}\mathop{\mathrm{se}_{{n}}\/}% \nolimits\!\left(t,h^{2}\right)dt=i^{n}\mathop{\mathrm{se}_{{n}}\/}\nolimits\!% \left(\alpha,h^{2}\right)\mathop{{\mathrm{Ms}^{{(1)}}_{{n}}}\/}\nolimits\!% \left(z,h\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : parameter, : integer, : complex variable, : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E3
Encodings:: TeX, pMML, png

28.28.4 $\dfrac{ih}{\pi}\int_{{0}}^{{2\pi}}\frac{\partial w}{\partial\alpha}e^{{2ihw}}% \mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)dt=i^{n}{\mathop{% \mathrm{ce}_{{n}}\/}\nolimits^{{\prime}}}\!\left(\alpha,h^{2}\right)\mathop{{% \mathrm{Mc}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : parameter, : integer, : complex variable, : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E4
Encodings:: TeX, pMML, png

28.28.5 $\dfrac{ih}{\pi}\int_{{0}}^{{2\pi}}\frac{\partial w}{\partial\alpha}e^{{2ihw}}% \mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)dt=i^{n}{\mathop{% \mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(\alpha,h^{2}\right)\mathop{{% \mathrm{Ms}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z,h\right).$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : parameter, : integer, : complex variable, : Mathieu’s equation solution and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/28.28.E5
Encodings:: TeX, pMML, png

In (28.28.7)–(28.28.9) the paths of integration $\mathcal{L}_{j}$ are given by

28.28.6

$\mathcal{L}_{1}\mbox{ : from }-\eta_{1}+i\infty\mbox{ to }2\pi-\eta_{1}+i\infty,$

$\mathcal{L}_{3}\mbox{ : from }-\eta_{1}+i\infty\mbox{ to }\eta_{2}-i\infty,$

$\mathcal{L}_{4}\mbox{ : from }\eta_{2}-i\infty\mbox{ to }2\pi-\eta_{1}+i\infty,$

Symbols:: : integer, $\mathcal{L}_{j}$ : integration paths and $\eta_{1}$ , $\eta_{2}$ : real constants
A&S Ref:: 20.7.16 (in slightly different notation) 20.7.17 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.28.E6
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

where $\eta_{1}$ and $\eta_{2}$ are real constants.

28.28.7 $\dfrac{1}{\pi}\int_{{\mathcal{L}_{j}}}e^{{2ihw}}\mathop{\mathrm{me}_{{\nu}}\/}% \nolimits\!\left(t,h^{2}\right)dt=e^{{i\nu\pi/{2}}}\mathop{\mathrm{me}_{{\nu}}% \/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}% \nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : parameter, : integer, : complex variable, $\nu$ : complex parameter, : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : integration paths
A&S Ref:: 20.7.18 (in slightly different notation)
Referenced by:: §28.28(i)
Permalink:: http://dlmf.nist.gov/28.28.E7
Encodings:: TeX, pMML, png

28.28.8 $\dfrac{1}{\pi}\int_{{\mathcal{L}_{j}}}2ih\frac{\partial w}{\partial\alpha}e^{{% 2ihw}}\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right)dt=e^{{i\nu% \pi/{2}}}{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\alpha,h^% {2}\right)\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , : parameter, : integer, : complex variable, $\nu$ : complex parameter, : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : integration paths
A&S Ref:: 20.7.19 (in slightly different notation)
Permalink:: http://dlmf.nist.gov/28.28.E8
Encodings:: TeX, pMML, png

28.28.9 $\dfrac{1}{2\pi}\int_{{\mathcal{L}_{1}}}e^{{2ihw}}\mathop{\mathrm{me}_{{\nu}}\/% }\nolimits\!\left(t,h^{2}\right)dt=e^{{i\nu\pi/{2}}}\mathop{\mathrm{me}_{{\nu}% }\/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{M}^{{(1)}}_{{\nu}}}\/% }\nolimits\!\left(z,h\right).$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : parameter, : complex variable, $\nu$ : complex parameter, : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : integration paths
Referenced by:: §28.28(i)
Permalink:: http://dlmf.nist.gov/28.28.E9
Encodings:: TeX, pMML, png

In (28.28.11)–(28.28.14)

28.28.10 $0<\mathop{\mathrm{ph}\/}\nolimits\!\left(h(\mathop{\cosh\/}\nolimits z\pm 1)% \right)<\pi.$

Symbols:: $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E10
Encodings:: TeX, pMML, png

28.28.11 $\int_{0}^{{\infty}}e^{{2ih\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}\nolimits t% }}\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right)dt=\tfrac{1}{2}% \pi ie^{{i\nu\pi}}\mathop{\mathrm{ce}_{{\nu}}\/}\nolimits\!\left(0,h^{2}\right% )\mathop{{\mathrm{M}^{{(3)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Referenced by:: §28.28(i)
Permalink:: http://dlmf.nist.gov/28.28.E11
Encodings:: TeX, pMML, png

28.28.12 $\int_{0}^{{\infty}}e^{{2ih\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}\nolimits t% }}\mathop{\sinh\/}\nolimits z\mathop{\sinh\/}\nolimits t\mathop{\mathrm{Se}_{{% \nu}}\/}\nolimits\!\left(t,h^{2}\right)dt=-\dfrac{\pi}{4h}e^{{i\nu\pi/{2}}}{% \mathop{\mathrm{se}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)% \mathop{{\mathrm{M}^{{(3)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, $\mathop{\mathrm{Se}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, : parameter, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.28.E12
Encodings:: TeX, pMML, png

28.28.13 $\int_{0}^{{\infty}}e^{{2ih\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}\nolimits t% }}\mathop{\sinh\/}\nolimits z\mathop{\sinh\/}\nolimits t\mathop{\mathrm{Fe}_{{% m}}\/}\nolimits\!\left(t,h^{2}\right)dt=-\dfrac{\pi}{4h}i^{{m}}{\mathop{% \mathrm{fe}_{{m}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)\mathop{{% \mathrm{Mc}^{{(3)}}_{{m}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, : differential of , : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\mathrm{Fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E13
Encodings:: TeX, pMML, png

28.28.14 $\int_{0}^{{\infty}}e^{{2ih\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}\nolimits t% }}\mathop{\mathrm{Ge}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)dt=\tfrac{1}{2}% \pi i^{{m+1}}\mathop{\mathrm{ge}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)% \mathop{{\mathrm{Ms}^{{(3)}}_{{m}}}\/}\nolimits\!\left(z,h\right).$

Symbols:: $\mathop{\mathrm{ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, : differential of , : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{Ge}_{{n}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter and : complex variable
Referenced by:: §28.28(i)
Permalink:: http://dlmf.nist.gov/28.28.E14
Encodings:: TeX, pMML, png

In particular, when the integrals (28.28.11), (28.28.14) converge absolutely and uniformly in the half strip $\realpart{z}\geq 0$ , $0\leq\imagpart{z}\leq\pi$ .

28.28.15 $\int_{0}^{{\infty}}\mathop{\cos\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits y% \mathop{\cosh\/}\nolimits t\right)\mathop{\mathrm{Ce}_{{2n}}\/}\nolimits\!% \left(t,h^{2}\right)dt=(-1)^{{n+1}}\tfrac{1}{2}\pi\mathop{{\mathrm{Mc}^{{(2)}}% _{{2n}}}\/}\nolimits\!\left(0,h\right)\mathop{\mathrm{ce}_{{2n}}\/}\nolimits\!% \left(y,h^{2}\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : parameter, : integer and : real variable
Permalink:: http://dlmf.nist.gov/28.28.E15
Encodings:: TeX, pMML, png

28.28.16 $\int_{0}^{{\infty}}\mathop{\sin\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits y% \mathop{\cosh\/}\nolimits t\right)\mathop{\mathrm{Ce}_{{2n}}\/}\nolimits\!% \left(t,h^{2}\right)dt=-\dfrac{\pi A_{0}^{{2n}}(h^{2})}{2\mathop{\mathrm{ce}_{% {2n}}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}\*\left(\mathop{\mathrm{% ce}_{{2n}}\/}\nolimits\!\left(y,h^{2}\right)\mp\dfrac{2}{\pi C_{{2n}}(h^{2})}% \mathop{\mathrm{fe}_{{2n}}\/}\nolimits\!\left(y,h^{2}\right)\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{fe}_{{n}}\/}\nolimits\!\left(z,q\right)$ : second solution, Mathieu’s equation, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{Ce}_{{\nu}}\/}\nolimits\!\left(z,q\right)$ : modified Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, : parameter, : integer, : real variable, $A_{{m}}(q)$ : Fourier coefficient and $C_{n}(q)$ : factor
Permalink:: http://dlmf.nist.gov/28.28.E16
Encodings:: TeX, pMML, png

where the upper or lower sign is taken according as $0\leq y\leq\pi$ or $\pi\leq y\leq 2\pi$ . For $A_{0}^{{2n}}(q)$ and $C_{{2n}}(q)$ see §§28.4 and 28.5(i).

For details and further equations see Meixner et al. (1980, §2.1.1) and Sips (1970).

§28.28(ii) Integrals of Products with Bessel Functions

Notes:: See Arscott (1964b, §4.6).
Keywords:: Mathieu functions, integrals, modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.28.ii

With the notations of §28.4 for $A_{m}^{n}(q)$ and $B_{m}^{n}(q)$ , §28.14 for $c_{n}^{{\nu}}(q)$ , and (28.23.1) for $\mathcal{C}_{{\mu}}^{{(j)}}$ , ,

28.28.17 $\dfrac{1}{\pi}\int_{0}^{\pi}\mathcal{C}^{{(j)}}_{{\nu+2s}}(2hR)e^{{-i(\nu+2s)% \phi}}\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{s}% c^{{\nu}}_{{2s}}(h^{2})\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left% (z,h\right),$ $s\in\Integer$ ,

where and $\phi=\phi(z,t)$ are analytic functions for $\realpart{z}>0$ and real with

28.28.18

$R(z,t)=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(2z\right)+\mathop{% \cos\/}\nolimits\!\left(2t\right))\right)^{{\ifrac{1}{2}}},$

$R(z,0)=\mathop{\cosh\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, : complex variable and : function
Permalink:: http://dlmf.nist.gov/28.28.E18
Encodings:: TeX, TeX, pMML, pMML, png, png

and

28.28.19

$e^{{2i\phi}}=\dfrac{\mathop{\cosh\/}\nolimits\!\left(z+it\right)}{\mathop{% \cosh\/}\nolimits\!\left(z-it\right)},$

$\phi(z,0)=0.$

Symbols:: : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, : complex variable and $\phi(z,t)$ : function
Permalink:: http://dlmf.nist.gov/28.28.E19
Encodings:: TeX, TeX, pMML, pMML, png, png

In particular, for integer $\nu$ and $\ell=0,1,2,\dots$ ,

28.28.20 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{{(j)}}_{{2\ell}}(2hR)\mathop{\cos\/}% \nolimits\!\left(2\ell\phi\right)\mathop{\mathrm{ce}_{{2m}}\/}\nolimits\!\left% (t,h^{2}\right)dt=\varepsilon_{\ell}(-1)^{{\ell+m}}A^{{2m}}_{{2\ell}}(h^{2})% \mathop{{\mathrm{Mc}^{{(j)}}_{{2m}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions, $\phi(z,t)$ : function, : function, $\varepsilon_{\ell}$ : constants and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.7.26 (in different form and only for $\ell=0$ )
Permalink:: http://dlmf.nist.gov/28.28.E20
Encodings:: TeX, pMML, png

where again $\varepsilon_{0}=2$ and $\varepsilon_{\ell}=1$ , $\ell=1,2,3,\ldots$ .

28.28.21 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{{(j)}}_{{2\ell+1}}(2hR)\mathop{\cos\/% }\nolimits\!\left((2\ell+1)\phi\right)\mathop{\mathrm{ce}_{{2m+1}}\/}\nolimits% \!\left(t,h^{2}\right)dt=(-1)^{{\ell+m}}A^{{2m+1}}_{{2\ell+1}}(h^{2})\mathop{{% \mathrm{Mc}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions, $\phi(z,t)$ : function, : function and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.7.26 (in different form and only for $\ell=0$ )
Permalink:: http://dlmf.nist.gov/28.28.E21
Encodings:: TeX, pMML, png

28.28.22 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{{(j)}}_{{2\ell+1}}(2hR)\mathop{\sin\/% }\nolimits\!\left((2\ell+1)\phi\right)\mathop{\mathrm{se}_{{2m+1}}\/}\nolimits% \!\left(t,h^{2}\right)dt=(-1)^{{\ell+m}}B^{{2m+1}}_{{2\ell+1}}(h^{2})\mathop{{% \mathrm{Ms}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions, $\phi(z,t)$ : function, : function and $B_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.7.27 (in different form and only for $\ell=0$ )
Permalink:: http://dlmf.nist.gov/28.28.E22
Encodings:: TeX, pMML, png

28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{{(j)}}_{{2\ell+2}}(2hR)\mathop{\sin\/% }\nolimits\!\left((2\ell+2)\phi\right)\mathop{\mathrm{se}_{{2m+2}}\/}\nolimits% \!\left(t,h^{2}\right)dt=(-1)^{{\ell+m}}B^{{2m+2}}_{{2\ell+2}}(h^{2})\mathop{{% \mathrm{Ms}^{{(j)}}_{{2m+2}}}\/}\nolimits\!\left(z,h\right).$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions, $\phi(z,t)$ : function, : function and $B_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.7.27 (in different form and only for $\ell=0$ )
Permalink:: http://dlmf.nist.gov/28.28.E23
Encodings:: TeX, pMML, png

§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

Notes:: See Schäfke (1983).
Permalink:: http://dlmf.nist.gov/28.28.iii

With the parameter suppressed we use the notation

28.28.24

$\mathop{\mathrm{D}_{{0}}\/}\nolimits\!\left(\nu,\mu,z\right)=\mathop{{\mathrm{% M}^{{(3)}}_{{\nu}}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{M}^{{(4)}}_{{% \mu}}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{M}^{{(4)}}_{{\nu}}}\/}% \nolimits\!\left(z\right)\mathop{{\mathrm{M}^{{(3)}}_{{\mu}}}\/}\nolimits\!% \left(z\right),$

$\mathop{\mathrm{D}_{{1}}\/}\nolimits\!\left(\nu,\mu,z\right)={\mathop{{\mathrm% {M}^{{(3)}}_{{\nu}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{M% }^{{(4)}}_{{\mu}}}\/}\nolimits\!\left(z\right)-{\mathop{{\mathrm{M}^{{(4)}}_{{% \nu}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{M}^{{(3)}}_{{% \mu}}}\/}\nolimits\!\left(z\right),$

Defines:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives
Symbols:: $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : parameter, : integer, : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/28.28.E24
Encodings:: TeX, TeX, pMML, pMML, png, png

and assume $\nu\notin\Integer$ and $m\in\Integer$ . Then

28.28.25 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \cos\/}\nolimits t\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right% )\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop% {\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{m+1}}ih% \alpha^{{(0)}}_{{\nu,m}}\mathop{\mathrm{D}_{{0}}\/}\nolimits\!\left(\nu,\nu+2m% +1,z\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E25
Encodings:: TeX, pMML, png

28.28.26 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \sin\/}\nolimits t\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right% )\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop% {\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{m+1}}ih% \alpha^{{(1)}}_{{\nu,m}}\mathop{\mathrm{D}_{{0}}\/}\nolimits\!\left(\nu,\nu+2m% +1,z\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E26
Encodings:: TeX, pMML, png

where

28.28.27 $\alpha^{{(0)}}_{{\nu,m}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}\mathop{\cos\/}% \nolimits t\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{m}% \dfrac{2i}{\pi}\dfrac{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(0,h^{2}% \right)\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(0,h^{2}\right)}{h% \mathop{\mathrm{D}_{{0}}\/}\nolimits\!\left(\nu,\nu+2m+1,0\right)},$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, : integer, : parameter, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E27
Encodings:: TeX, pMML, png

28.28.28 $\alpha^{{(1)}}_{{\nu,m}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}\mathop{\sin\/}% \nolimits t\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{{m% +1}}\dfrac{2i}{\pi}\dfrac{{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}% \!\left(0,h^{2}\right)\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(0,h% ^{2}\right)}{h\mathop{\mathrm{D}_{{1}}\/}\nolimits\!\left(\nu,\nu+2m+1,0\right% )}.$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E28
Encodings:: TeX, pMML, png

28.28.29 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \sin\/}\nolimits t{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(% t,h^{2}\right)\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}% \right)}{{\mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}% dt=(-1)^{{m+1}}ih\alpha^{{(0)}}_{{\nu,m}}\mathop{\mathrm{D}_{{1}}\/}\nolimits% \!\left(\nu,\nu+2m+1,z\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E29
Encodings:: TeX, pMML, png

28.28.30 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \cos\/}\nolimits t{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(% t,h^{2}\right)\mathop{\mathrm{me}_{{-\nu-2m-1}}\/}\nolimits\!\left(t,h^{2}% \right)}{{\mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}% dt=(-1)^{m}ih\alpha^{{(1)}}_{{\nu,m}}\mathop{\mathrm{D}_{{1}}\/}\nolimits\!% \left(\nu,\nu+2m+1,z\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E30
Encodings:: TeX, pMML, png

28.28.31 $\dfrac{2}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{\cos\/}\nolimits t\mathop{% \sin\/}\nolimits t\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(t,h^{2}\right% )\mathop{\mathrm{me}_{{-\nu-2m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{m}i\gamma% _{{\nu,m}}\mathop{\mathrm{D}_{{0}}\/}\nolimits\!\left(\nu,\nu+2m,z\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\gamma_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E31
Encodings:: TeX, pMML, png

28.28.32 $\dfrac{\mathop{\sinh\/}\nolimits\!\left(2z\right)}{\pi^{2}}\int_{0}^{{2\pi}}% \dfrac{{\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(t,h^{2}% \right)\mathop{\mathrm{me}_{{-\nu-2m}}\/}\nolimits\!\left(t,h^{2}\right)}{{% \mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{% m+1}}i\gamma_{{\nu,m}}\mathop{\mathrm{D}_{{1}}\/}\nolimits\!\left(\nu,\nu+2m,z% \right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $\gamma_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E32
Encodings:: TeX, pMML, png

where

28.28.33 $\gamma_{{\nu,m}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}{\mathop{\mathrm{me}_{{\nu}}% \/}\nolimits^{{\prime}}}\!\left(t\right)\mathop{\mathrm{me}_{{-\nu-2m}}\/}% \nolimits\!\left(t\right)dt=(-1)^{{m}}\dfrac{4i}{\pi}\frac{{\mathop{\mathrm{me% }_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(0\right)\mathop{\mathrm{me}_{{-\nu-2m% }}\/}\nolimits\!\left(0\right)}{\mathop{\mathrm{D}_{{1}}\/}\nolimits\!\left(% \nu,\nu+2m,0\right)}.$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, : integer, : parameter, $\nu$ : complex parameter and $\gamma_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E33
Encodings:: TeX, pMML, png

Also,

28.28.34 $\dfrac{1}{\pi^{2}}\pvint_{0}^{{2\pi}}\dfrac{{\mathop{\mathrm{me}_{{\nu}}\/}% \nolimits^{{\prime}}}\!\left(t,h^{2}\right)\mathop{\mathrm{me}_{{-\nu-2m-1}}\/% }\nolimits\!\left(t,h^{2}\right)}{\mathop{\sin\/}\nolimits t}dt=(-1)^{{m+1}}ih% \alpha^{{(0)}}_{{\nu,m}}\mathop{\mathrm{D}_{{1}}\/}\nolimits\!\left(\nu,\nu+2m% +1,0\right),$

Symbols:: $\mathop{\mathrm{D}_{{j}}\/}\nolimits\!\left(\nu,\mu,z\right)$ : cross-products of modified Mathieu functions and their derivatives, $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\pvint_{a}^{b}$ : Cauchy principal value, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, $\nu$ : complex parameter and $\alpha^{{(s)}}_{{\nu,m}}$
Permalink:: http://dlmf.nist.gov/28.28.E34
Encodings:: TeX, pMML, png

where the integral is a Cauchy principal value (§1.4(v)).

§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

Notes:: See Schäfke (1983). There is a sign error on p. 158 of this reference.
Keywords:: Mathieu functions, integrals, modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.28.iv

Again with the parameter suppressed, let

28.28.35

$\mathop{\mathrm{Ds}_{{0}}\/}\nolimits\!\left(n,m,z\right)=\mathop{{\mathrm{Ms}% ^{{(3)}}_{{n}}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Ms}^{{(4)}}_{{m}}}% \/}\nolimits\!\left(z\right)-\mathop{{\mathrm{Ms}^{{(4)}}_{{n}}}\/}\nolimits\!% \left(z\right)\mathop{{\mathrm{Ms}^{{(3)}}_{{m}}}\/}\nolimits\!\left(z\right),$

$\mathop{\mathrm{Ds}_{{1}}\/}\nolimits\!\left(n,m,z\right)={\mathop{{\mathrm{Ms% }^{{(3)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Ms}^{% {(4)}}_{{m}}}\/}\nolimits\!\left(z\right)-{\mathop{{\mathrm{Ms}^{{(4)}}_{{n}}}% \/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Ms}^{{(3)}}_{{m}}}\/}% \nolimits\!\left(z\right),$

$\mathop{\mathrm{Ds}_{{2}}\/}\nolimits\!\left(n,m,z\right)={\mathop{{\mathrm{Ms% }^{{(3)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right){\mathop{{\mathrm{Ms}^% {{(4)}}_{{m}}}\/}\nolimits^{{\prime}}}\!\left(z\right)-{\mathop{{\mathrm{Ms}^{% {(4)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right){\mathop{{\mathrm{Ms}^{{(% 3)}}_{{m}}}\/}\nolimits^{{\prime}}}\!\left(z\right).$

Defines:: $\mathop{\mathrm{Ds}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives
Symbols:: $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E35
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

Then

28.28.36 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \cos\/}\nolimits t\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{p+1}}ih\widehat{% \alpha}_{{n,m}}^{{(s)}}\mathop{\mathrm{Ds}_{{0}}\/}\nolimits\!\left(n,m,z% \right),$

Symbols:: $\mathop{\mathrm{Ds}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E36
Encodings:: TeX, pMML, png

28.28.37 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \sin\/}\nolimits t{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(t,% h^{2}\right)\mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{% \mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{% p+1}}ih\widehat{\alpha}_{{n,m}}^{{(s)}}\mathop{\mathrm{Ds}_{{1}}\/}\nolimits\!% \left(n,m,z\right),$

Symbols:: $\mathop{\mathrm{Ds}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E37
Encodings:: TeX, pMML, png

where , $p\in\Integer$ ; $m,n=1,2,3,\dots$ . Also,

28.28.38 $\widehat{\alpha}_{{n,m}}^{{(s)}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}\mathop{\cos% \/}\nolimits t\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{se}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{p}\dfrac{2% }{i\pi}\dfrac{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2% }\right){\mathop{\mathrm{se}_{{m}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}% \right)}{h\mathop{\mathrm{Ds}_{{2}}\/}\nolimits\!\left(n,m,0\right)}.$

Symbols:: $\mathop{\mathrm{Ds}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, : integer, : parameter and : integer
Permalink:: http://dlmf.nist.gov/28.28.E38
Encodings:: TeX, pMML, png

Let

28.28.39

$\mathop{\mathrm{Dc}_{{0}}\/}\nolimits\!\left(n,m,z\right)=\mathop{{\mathrm{Mc}% ^{{(3)}}_{{n}}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Mc}^{{(4)}}_{{m}}}% \/}\nolimits\!\left(z\right)-\mathop{{\mathrm{Mc}^{{(4)}}_{{n}}}\/}\nolimits\!% \left(z\right)\mathop{{\mathrm{Mc}^{{(3)}}_{{m}}}\/}\nolimits\!\left(z\right),$

$\mathop{\mathrm{Dc}_{{1}}\/}\nolimits\!\left(n,m,z\right)={\mathop{{\mathrm{Mc% }^{{(3)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Mc}^{% {(4)}}_{{m}}}\/}\nolimits\!\left(z\right)-{\mathop{{\mathrm{Mc}^{{(4)}}_{{n}}}% \/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Mc}^{{(3)}}_{{m}}}\/}% \nolimits\!\left(z\right),$

Defines:: $\mathop{\mathrm{Dc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives
Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E39
Encodings:: TeX, TeX, pMML, pMML, png, png

28.28.40

$\mathop{\mathrm{Dsc}_{{0}}\/}\nolimits\!\left(n,m,z\right)=\mathop{{\mathrm{Ms% }^{{(3)}}_{{n}}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Mc}^{{(4)}}_{{m}}% }\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{Ms}^{{(4)}}_{{n}}}\/}\nolimits% \!\left(z\right)\mathop{{\mathrm{Mc}^{{(3)}}_{{m}}}\/}\nolimits\!\left(z\right),$

$\mathop{\mathrm{Dsc}_{{1}}\/}\nolimits\!\left(n,m,z\right)={\mathop{{\mathrm{% Ms}^{{(3)}}_{{n}}}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Mc}% ^{{(4)}}_{{m}}}\/}\nolimits\!\left(z\right)-{\mathop{{\mathrm{Ms}^{{(4)}}_{{n}% }}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{{\mathrm{Mc}^{{(3)}}_{{m}}}% \/}\nolimits\!\left(z\right).$

Defines:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives
Symbols:: $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E40
Encodings:: TeX, TeX, pMML, pMML, png, png

Then

28.28.41 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \sin\/}\nolimits t\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{p+1}}ih\widehat{% \beta}_{{n,m}}\mathop{\mathrm{Dsc}_{{0}}\/}\nolimits\!\left(n,m,z\right),$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E41
Encodings:: TeX, pMML, png

28.28.42 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \cos\/}\nolimits t{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(t,% h^{2}\right)\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{% \mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{% p}}ih\widehat{\beta}_{{n,m}}\mathop{\mathrm{Dsc}_{{1}}\/}\nolimits\!\left(n,m,% z\right),$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E42
Encodings:: TeX, pMML, png

where , $p\in\Integer$ ; $m=0,1,2,\dots$ , $n=1,2,3,\dots$ . Also,

28.28.43 $\widehat{\beta}_{{n,m}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}\mathop{\sin\/}% \nolimits t\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)\mathop{% \mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{p}\dfrac{2}{i\pi}% \dfrac{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right% )\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)}{h\mathop{\mathrm% {Dsc}_{{1}}\/}\nolimits\!\left(n,m,0\right)}.$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter and : integer
Permalink:: http://dlmf.nist.gov/28.28.E43
Encodings:: TeX, pMML, png

Next,

28.28.44 $\dfrac{1}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{\sin\/}\nolimits\!\left(2t% \right)\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)\mathop{% \mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{p}i\widehat{% \gamma}_{{n,m}}\mathop{\mathrm{Dsc}_{{0}}\/}\nolimits\!\left(n,m,z\right),$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E44
Encodings:: TeX, pMML, png

28.28.45 $\dfrac{\mathop{\sinh\/}\nolimits\!\left(2z\right)}{\pi^{2}}\int_{0}^{{2\pi}}% \dfrac{{\mathop{\mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(t,h^{2}\right% )\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/% }\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{p+1}}i\widehat{% \gamma}_{{n,m}}\mathop{\mathrm{Dsc}_{{1}}\/}\nolimits\!\left(n,m,z\right),$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E45
Encodings:: TeX, pMML, png

where , $p\in\Integer$ ; $m=0,1,2,\dots$ , $n=1,2,3,\dots$ . Also,

28.28.46 $\widehat{\gamma}_{{n,m}}=\dfrac{1}{2\pi}\int_{0}^{{2\pi}}{\mathop{\mathrm{se}_% {{n}}\/}\nolimits^{{\prime}}}\!\left(t,h^{2}\right)\mathop{\mathrm{ce}_{{m}}\/% }\nolimits\!\left(t,h^{2}\right)dt=(-1)^{{p+1}}\dfrac{4}{i\pi}\dfrac{{\mathop{% \mathrm{se}_{{n}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)\mathop{\mathrm% {ce}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)}{\mathop{\mathrm{Dsc}_{{1}}\/}% \nolimits\!\left(n,m,0\right)}.$

Symbols:: $\mathop{\mathrm{Dsc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\int$ : integral, : integer, : parameter and : integer
Permalink:: http://dlmf.nist.gov/28.28.E46
Encodings:: TeX, pMML, png

Lastly,

28.28.47 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \cos\/}\nolimits t\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{p+1}}ih\widehat{% \alpha}_{{n,m}}^{{(c)}}\mathop{\mathrm{Dc}_{{0}}\/}\nolimits\!\left(n,m,z% \right),$

Symbols:: $\mathop{\mathrm{Dc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E47
Encodings:: TeX, pMML, png

28.28.48 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{{2\pi}}\dfrac{\mathop{% \sin\/}\nolimits t{\mathop{\mathrm{ce}_{{n}}\/}\nolimits^{{\prime}}}\!\left(t,% h^{2}\right)\mathop{\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)}{{% \mathop{\sinh\/}\nolimits^{{2}}}z+{\mathop{\sin\/}\nolimits^{{2}}}t}dt=(-1)^{{% p+1}}ih\widehat{\alpha}_{{n,m}}^{{(c)}}\mathop{\mathrm{Dc}_{{1}}\/}\nolimits\!% \left(n,m,z\right),$

Symbols:: $\mathop{\mathrm{Dc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : differential of , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : integer, : parameter, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/28.28.E48
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where , $p\in\Integer$ ; $m,n=0,1,2,\dots$ . Also,

28.28.49 $\widehat{\alpha}_{{n,m}}^{{(c)}}=\frac{1}{2\pi}\int_{0}^{{2\pi}}\mathop{\cos\/% }\nolimits t\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(t,h^{2}\right)\mathop% {\mathrm{ce}_{{m}}\/}\nolimits\!\left(t,h^{2}\right)dt=(-1)^{{p+1}}\dfrac{2}{i% \pi}\dfrac{\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(0,h^{2}\right)\mathop{% \mathrm{ce}_{{m}}\/}\nolimits\!\left(0,h^{2}\right)}{h\mathop{\mathrm{Dc}_{{0}% }\/}\nolimits\!\left(n,m,0\right)}.$

Symbols:: $\mathop{\mathrm{Dc}_{{j}}\/}\nolimits\!\left(n,m,z\right)$ : cross-products of radial Mathieu functions and their derivatives, $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, : integer, : parameter and : integer
Permalink:: http://dlmf.nist.gov/28.28.E49
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§28.28(v) Compendia

Keywords:: Mathieu functions, integrals, modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.28.v

See Prudnikov et al. (1990, pp. 359–368), Gradshteyn and Ryzhik (2000, pp. 755–759), Sips (1970), and Meixner et al. (1980, §2.1.1).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.27 Addition Theorems 28.29 Definitions and Basic Properties

§28.28 Integrals, Integral Representations, and Integral Equations

Contents

§28.28(i) Equations with Elementary Kernels

§28.28(ii) Integrals of Products with Bessel Functions

§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

§28.28(v) Compendia