# §28.28 Integrals, Integral Representations, and Integral Equations

## §28.28(i) Equations with Elementary Kernels

Let

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

Then

 28.28.2 $\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\mathop{\mathrm{ce}_{n}\/}% \nolimits\!\left(t,h^{2}\right)\mathrm{d}t={\mathrm{i}^{n}}\mathop{\mathrm{ce}% _{n}\/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{Mc}^{(1)}_{n}}\/}% \nolimits\!\left(z,h\right),$
 28.28.3 $\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\mathop{\mathrm{se}_{n}\/}% \nolimits\!\left(t,h^{2}\right)\mathrm{d}t={\mathrm{i}^{n}}\mathop{\mathrm{se}% _{n}\/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{Ms}^{(1)}_{n}}\/}% \nolimits\!\left(z,h\right),$
 28.28.4 $\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{% d}t={\mathrm{i}^{n}}\mathop{\mathrm{ce}_{n}\/}\nolimits'\!\left(\alpha,h^{2}% \right)\mathop{{\mathrm{Mc}^{(1)}_{n}}\/}\nolimits\!\left(z,h\right),$
 28.28.5 $\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{% d}t={\mathrm{i}^{n}}\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(\alpha,h^{2}% \right)\mathop{{\mathrm{Ms}^{(1)}_{n}}\/}\nolimits\!\left(z,h\right).$

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

 28.28.6 $\mathcal{L}_{1}\mbox{ : from }-\eta_{1}+\mathrm{i}\infty\mbox{ to }2\pi-\eta_{% 1}+\mathrm{i}\infty,$ $\mathcal{L}_{3}\mbox{ : from }-\eta_{1}+\mathrm{i}\infty\mbox{ to }\eta_{2}-% \mathrm{i}\infty,$ $\mathcal{L}_{4}\mbox{ : from }\eta_{2}-\mathrm{i}\infty\mbox{ to }2\pi-\eta_{1% }+\mathrm{i}\infty,$ Defines: $\mathcal{L}_{j}$: integration paths (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $j$: integer 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 See also: Annotations for 28.28(i)

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

 28.28.7 $\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}e^{2\mathrm{i}hw}\mathop{\mathrm{me}_{\nu}% \/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\mathop{% \mathrm{me}_{\nu}\/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{M}^{(% j)}_{\nu}}\/}\nolimits\!\left(z,h\right),$ $j=3,4,$
 28.28.8 $\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}2\mathrm{i}h\frac{\partial w}{\partial% \alpha}e^{2\mathrm{i}hw}\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}% \right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\mathop{\mathrm{me}_{\nu}\/}% \nolimits'\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}% \nolimits\!\left(z,h\right),$ $j=3,4,$
 28.28.9 $\dfrac{1}{2\pi}\int_{\mathcal{L}_{1}}e^{2\mathrm{i}hw}\mathop{\mathrm{me}_{\nu% }\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\mathop% {\mathrm{me}_{\nu}\/}\nolimits\!\left(\alpha,h^{2}\right)\mathop{{\mathrm{M}^{% (1)}_{\nu}}\/}\nolimits\!\left(z,h\right).$

In (28.28.11)–(28.28.14)

 28.28.10 $0<\mathop{\mathrm{ph}\/}\nolimits\!\left(h(\mathop{\cosh\/}\nolimits z\pm 1)% \right)<\pi.$
 28.28.11 $\int_{0}^{\infty}e^{2\mathrm{i}h\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}% \nolimits t}\mathop{\mathrm{Ce}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)\mathrm% {d}t=\tfrac{1}{2}\pi\mathrm{i}e^{\mathrm{i}\nu\pi}\mathop{\mathrm{ce}_{\nu}\/}% \nolimits\!\left(0,h^{2}\right)\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!% \left(z,h\right),$
 28.28.12 $\int_{0}^{\infty}e^{2\mathrm{i}h\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}% \nolimits t}\mathop{\sinh\/}\nolimits z\mathop{\sinh\/}\nolimits t\mathop{% \mathrm{Se}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi}{4h% }e^{\mathrm{i}\nu\pi/{2}}\mathop{\mathrm{se}_{\nu}\/}\nolimits'\!\left(0,h^{2}% \right)\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z,h\right),$
 28.28.13 $\int_{0}^{\infty}e^{2\mathrm{i}h\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}% \nolimits t}\mathop{\sinh\/}\nolimits z\mathop{\sinh\/}\nolimits t\mathop{% \mathrm{Fe}_{m}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi}{4h}{% \mathrm{i}^{m}}\mathop{\mathrm{fe}_{m}\/}\nolimits'\!\left(0,h^{2}\right)% \mathop{{\mathrm{Mc}^{(3)}_{m}}\/}\nolimits\!\left(z,h\right),$
 28.28.14 $\int_{0}^{\infty}e^{2\mathrm{i}h\mathop{\cosh\/}\nolimits z\mathop{\cosh\/}% \nolimits t}\mathop{\mathrm{Ge}_{m}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d% }t=\tfrac{1}{2}\pi{\mathrm{i}^{m+1}}\mathop{\mathrm{ge}_{m}\/}\nolimits\!\left% (0,h^{2}\right)\mathop{{\mathrm{Ms}^{(3)}_{m}}\/}\nolimits\!\left(z,h\right).$

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

 28.28.15 $\int_{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits y% \mathop{\cosh\/}\nolimits t\right)\mathop{\mathrm{Ce}_{2n}\/}\nolimits\!\left(% t,h^{2}\right)\mathrm{d}t=(-1)^{n+1}\tfrac{1}{2}\pi\mathop{{\mathrm{Mc}^{(2)}_% {2n}}\/}\nolimits\!\left(0,h\right)\mathop{\mathrm{ce}_{2n}\/}\nolimits\!\left% (y,h^{2}\right),$
 28.28.16 $\int_{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(2h\mathop{\cos\/}\nolimits y% \mathop{\cosh\/}\nolimits t\right)\mathop{\mathrm{Ce}_{2n}\/}\nolimits\!\left(% t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi A_{0}^{2n}(h^{2})}{2\mathop{\mathrm{ce}_% {2n}\/}\nolimits\!\left(\frac{1}{2}\pi,h^{2}\right)}\*\left(\mathop{\mathrm{ce% }_{2n}\/}\nolimits\!\left(y,h^{2}\right)\mp\dfrac{2}{\pi C_{2n}(h^{2})}\mathop% {\mathrm{fe}_{2n}\/}\nolimits\!\left(y,h^{2}\right)\right),$

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

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 $\mathcal{C}_{\mu}^{(j)}$, $j=1,2,3,4$,

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

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

 28.28.18 $\displaystyle R(z,t)$ $\displaystyle=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(2z\right)+% \mathop{\cos\/}\nolimits\!\left(2t\right))\right)^{\ifrac{1}{2}},$ $\displaystyle R(z,0)$ $\displaystyle=\mathop{\cosh\/}\nolimits z,$

and

 28.28.19 $\displaystyle e^{2\mathrm{i}\phi}$ $\displaystyle=\dfrac{\mathop{\cosh\/}\nolimits\!\left(z+\mathrm{i}t\right)}{% \mathop{\cosh\/}\nolimits\!\left(z-\mathrm{i}t\right)},$ $\displaystyle\phi(z,0)$ $\displaystyle=0.$

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

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

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

 28.28.21 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\mathop{\cos\/}% \nolimits\!\left((2\ell+1)\phi\right)\mathop{\mathrm{ce}_{2m+1}\/}\nolimits\!% \left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}A^{2m+1}_{2\ell+1}(h^{2})\mathop{% {\mathrm{Mc}^{(j)}_{2m+1}}\/}\nolimits\!\left(z,h\right),$ Symbols: $\mathop{\mathrm{ce}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, $\mathcal{C}_{\mu}^{(j)}$: cylinder functions, $\phi(z,t)$: function, $R(z,t)$: function and $A_{m}(q)$: Fourier coefficient A&S Ref: 20.7.26 (in different form and only for $\ell=0$) Referenced by: Equations (28.28.21) and (28.28.22) Permalink: http://dlmf.nist.gov/28.28.E21 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the prefactor and uppper limit of integration were given incorrectly as $\dfrac{2}{\pi}\int_{0}^{\pi}$. Reported 2015-05-20 by Ruslan Kabasayev See also: Annotations for 28.28(ii)
 28.28.22 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\mathop{\sin\/}% \nolimits\!\left((2\ell+1)\phi\right)\mathop{\mathrm{se}_{2m+1}\/}\nolimits\!% \left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{2m+1}_{2\ell+1}(h^{2})\mathop{% {\mathrm{Ms}^{(j)}_{2m+1}}\/}\nolimits\!\left(z,h\right),$ Symbols: $\mathop{\mathrm{se}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, $\mathcal{C}_{\mu}^{(j)}$: cylinder functions, $\phi(z,t)$: function, $R(z,t)$: function and $B_{m}(q)$: Fourier coefficient A&S Ref: 20.7.27 (in different form and only for $\ell=0$) Referenced by: Equations (28.28.21) and (28.28.22) Permalink: http://dlmf.nist.gov/28.28.E22 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the prefactor and uppper limit of integration were given incorrectly as $\dfrac{2}{\pi}\int_{0}^{\pi}$. Reported 2015-05-20 by Ruslan Kabasayev See also: Annotations for 28.28(ii)
 28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\mathop{\sin\/}% \nolimits\!\left((2\ell+2)\phi\right)\mathop{\mathrm{se}_{2m+2}\/}\nolimits\!% \left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{2m+2}_{2\ell+2}(h^{2})\mathop{% {\mathrm{Ms}^{(j)}_{2m+2}}\/}\nolimits\!\left(z,h\right).$

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

With the parameter $h$ suppressed we use the notation

 28.28.24 $\displaystyle\mathop{\mathrm{D}_{0}\/}\nolimits\!\left(\nu,\mu,z\right)$ $\displaystyle=\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z\right)% \mathop{{\mathrm{M}^{(4)}_{\mu}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% M}^{(4)}_{\nu}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{M}^{(3)}_{\mu}}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,\mu,z\right)$ $\displaystyle=\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits'\!\left(z\right)% \mathop{{\mathrm{M}^{(4)}_{\mu}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% M}^{(4)}_{\nu}}\/}\nolimits'\!\left(z\right)\mathop{{\mathrm{M}^{(3)}_{\mu}}\/% }\nolimits\!\left(z\right),$ Defines: $\mathop{\mathrm{D}_{\NVar{j}}\/}\nolimits\!\left(\NVar{\nu},\NVar{\mu},\NVar{z% }\right)$: cross-products of modified Mathieu functions and their derivatives Symbols: $\mathop{{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.28.E24 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.28(iii)

and assume $\nu\notin\mathbb{Z}$ and $m\in\mathbb{Z}$. Then

 28.28.25 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos% \/}\nolimits t\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m+1}% \mathrm{i}h\alpha^{(0)}_{\nu,m}\mathop{\mathrm{D}_{0}\/}\nolimits\!\left(\nu,% \nu+2m+1,z\right),$
 28.28.26 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin% \/}\nolimits t\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m+1}% \mathrm{i}h\alpha^{(1)}_{\nu,m}\mathop{\mathrm{D}_{0}\/}\nolimits\!\left(\nu,% \nu+2m+1,z\right),$

where

 28.28.27 $\alpha^{(0)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\cos\/}\nolimits t% \mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)\mathop{\mathrm{me}% _{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=(-1)^{m}\dfrac{2% \mathrm{i}}{\pi}\dfrac{\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(0,h^{2}% \right)\mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(0,h^{2}\right)}{h% \mathop{\mathrm{D}_{0}\/}\nolimits\!\left(\nu,\nu+2m+1,0\right)},$
 28.28.28 $\alpha^{(1)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\sin\/}\nolimits t% \mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)\mathop{\mathrm{me}% _{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=(-1)^{m+1}\dfrac{2% \mathrm{i}}{\pi}\dfrac{\mathop{\mathrm{me}_{\nu}\/}\nolimits'\!\left(0,h^{2}% \right)\mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(0,h^{2}\right)}{h% \mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,\nu+2m+1,0\right)}.$
 28.28.29 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin% \/}\nolimits t\mathop{\mathrm{me}_{\nu}\/}\nolimits'\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m+1}% \mathrm{i}h\alpha^{(0)}_{\nu,m}\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,% \nu+2m+1,z\right),$
 28.28.30 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos% \/}\nolimits t\mathop{\mathrm{me}_{\nu}\/}\nolimits'\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m}% \mathrm{i}h\alpha^{(1)}_{\nu,m}\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,% \nu+2m+1,z\right),$
 28.28.31 $\dfrac{2}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos\/}\nolimits t\mathop{\sin% \/}\nolimits t\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(t,h^{2}\right)% \mathop{\mathrm{me}_{-\nu-2m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{% \sinh\/}\nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m}% \mathrm{i}\gamma_{\nu,m}\mathop{\mathrm{D}_{0}\/}\nolimits\!\left(\nu,\nu+2m,z% \right),$
 28.28.32 $\dfrac{\mathop{\sinh\/}\nolimits\!\left(2z\right)}{\pi^{2}}\int_{0}^{2\pi}% \dfrac{\mathop{\mathrm{me}_{\nu}\/}\nolimits'\!\left(t,h^{2}\right)\mathop{% \mathrm{me}_{-\nu-2m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{m+1}\mathrm{% i}\gamma_{\nu,m}\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,\nu+2m,z\right),$

where

 28.28.33 $\gamma_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\mathrm{me}_{\nu}\/}% \nolimits'\!\left(t\right)\mathop{\mathrm{me}_{-\nu-2m}\/}\nolimits\!\left(t% \right)\mathrm{d}t=(-1)^{m}\dfrac{4\mathrm{i}}{\pi}\frac{\mathop{\mathrm{me}_{% \nu}\/}\nolimits'\!\left(0\right)\mathop{\mathrm{me}_{-\nu-2m}\/}\nolimits\!% \left(0\right)}{\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,\nu+2m,0\right)}.$

Also,

 28.28.34 $\dfrac{1}{\pi^{2}}\pvint_{0}^{2\pi}\dfrac{\mathop{\mathrm{me}_{\nu}\/}% \nolimits'\!\left(t,h^{2}\right)\mathop{\mathrm{me}_{-\nu-2m-1}\/}\nolimits\!% \left(t,h^{2}\right)}{\mathop{\sin\/}\nolimits t}\mathrm{d}t=(-1)^{m+1}\mathrm% {i}h\alpha^{(0)}_{\nu,m}\mathop{\mathrm{D}_{1}\/}\nolimits\!\left(\nu,\nu+2m+1% ,0\right),$

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

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

Again with the parameter $h$ suppressed, let

 28.28.35 $\displaystyle\mathop{\mathrm{Ds}_{0}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(3)}_{n}}\/}\nolimits\!\left(z\right)% \mathop{{\mathrm{Ms}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Ms}^{(4)}_{n}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Ms}^{(3)}_{m}}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{Ds}_{1}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(3)}_{n}}\/}\nolimits'\!\left(z\right)% \mathop{{\mathrm{Ms}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Ms}^{(4)}_{n}}\/}\nolimits'\!\left(z\right)\mathop{{\mathrm{Ms}^{(3)}_{m}}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{Ds}_{2}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(3)}_{n}}\/}\nolimits'\!\left(z\right)% \mathop{{\mathrm{Ms}^{(4)}_{m}}\/}\nolimits'\!\left(z\right)-\mathop{{\mathrm{% Ms}^{(4)}_{n}}\/}\nolimits'\!\left(z\right)\mathop{{\mathrm{Ms}^{(3)}_{m}}\/}% \nolimits'\!\left(z\right).$ Defines: $\mathop{\mathrm{Ds}_{\NVar{j}}\/}\nolimits\!\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/28.28.E35 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 28.28(iv)

Then

 28.28.36 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos% \/}\nolimits t\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop% {\mathrm{se}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}\nolimits% ^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}h% \widehat{\alpha}_{n,m}^{(s)}\mathop{\mathrm{Ds}_{0}\/}\nolimits\!\left(n,m,z% \right),$
 28.28.37 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin% \/}\nolimits t\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(t,h^{2}\right)% \mathop{\mathrm{se}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{% i}h\widehat{\alpha}_{n,m}^{(s)}\mathop{\mathrm{Ds}_{1}\/}\nolimits\!\left(n,m,% z\right),$

where $m-n=2p+1$, $p\in\mathbb{Z}$; $m,n=1,2,3,\dots$. Also,

 28.28.38 $\widehat{\alpha}_{n,m}^{(s)}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\cos\/}% \nolimits t\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop{% \mathrm{se}_{m}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p}\dfrac{2}% {\mathrm{i}\pi}\dfrac{\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(0,h^{2}% \right)\mathop{\mathrm{se}_{m}\/}\nolimits'\!\left(0,h^{2}\right)}{h\mathop{% \mathrm{Ds}_{2}\/}\nolimits\!\left(n,m,0\right)}.$

Let

 28.28.39 $\displaystyle\mathop{\mathrm{Dc}_{0}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Mc}^{(3)}_{n}}\/}\nolimits\!\left(z\right)% \mathop{{\mathrm{Mc}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Mc}^{(4)}_{n}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Mc}^{(3)}_{m}}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{Dc}_{1}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Mc}^{(3)}_{n}}\/}\nolimits'\!\left(z\right)% \mathop{{\mathrm{Mc}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Mc}^{(4)}_{n}}\/}\nolimits'\!\left(z\right)\mathop{{\mathrm{Mc}^{(3)}_{m}}\/}% \nolimits\!\left(z\right),$ Defines: $\mathop{\mathrm{Dc}_{\NVar{j}}\/}\nolimits\!\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/28.28.E39 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.28(iv)
 28.28.40 $\displaystyle\mathop{\mathrm{Dsc}_{0}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(3)}_{n}}\/}\nolimits\!\left(z\right)% \mathop{{\mathrm{Mc}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Ms}^{(4)}_{n}}\/}\nolimits\!\left(z\right)\mathop{{\mathrm{Mc}^{(3)}_{m}}\/}% \nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{Dsc}_{1}\/}\nolimits\!\left(n,m,z\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(3)}_{n}}\/}\nolimits'\!\left(z\right)% \mathop{{\mathrm{Mc}^{(4)}_{m}}\/}\nolimits\!\left(z\right)-\mathop{{\mathrm{% Ms}^{(4)}_{n}}\/}\nolimits'\!\left(z\right)\mathop{{\mathrm{Mc}^{(3)}_{m}}\/}% \nolimits\!\left(z\right).$ Defines: $\mathop{\mathrm{Dsc}_{\NVar{j}}\/}\nolimits\!\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/28.28.E40 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.28(iv)

Then

 28.28.41 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin% \/}\nolimits t\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop% {\mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}\nolimits% ^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}h% \widehat{\beta}_{n,m}\mathop{\mathrm{Dsc}_{0}\/}\nolimits\!\left(n,m,z\right),$
 28.28.42 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos% \/}\nolimits t\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(t,h^{2}\right)% \mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p}\mathrm{i}% h\widehat{\beta}_{n,m}\mathop{\mathrm{Dsc}_{1}\/}\nolimits\!\left(n,m,z\right),$

where $m-n=2p+1$, $p\in\mathbb{Z}$; $m=0,1,2,\dots$, $n=1,2,3,\dots$. Also,

 28.28.43 $\widehat{\beta}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\sin\/}\nolimits t% \mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop{\mathrm{ce}_{% m}\/}\nolimits\!\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p}\dfrac{2}{\mathrm{i}% \pi}\dfrac{\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(0,h^{2}\right)\mathop{% \mathrm{ce}_{m}\/}\nolimits\!\left(0,h^{2}\right)}{h\mathop{\mathrm{Dsc}_{1}\/% }\nolimits\!\left(n,m,0\right)}.$

Next,

 28.28.44 $\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin\/}\nolimits\!\left(2t% \right)\mathop{\mathrm{se}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop{% \mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}\nolimits^% {2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p}\mathrm{i}\widehat{% \gamma}_{n,m}\mathop{\mathrm{Dsc}_{0}\/}\nolimits\!\left(n,m,z\right),$
 28.28.45 $\dfrac{\mathop{\sinh\/}\nolimits\!\left(2z\right)}{\pi^{2}}\int_{0}^{2\pi}% \dfrac{\mathop{\mathrm{se}_{n}\/}\nolimits'\!\left(t,h^{2}\right)\mathop{% \mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}\nolimits^% {2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}\widehat% {\gamma}_{n,m}\mathop{\mathrm{Dsc}_{1}\/}\nolimits\!\left(n,m,z\right),$

where $n-m=2p$, $p\in\mathbb{Z}$; $m=0,1,2,\dots$, $n=1,2,3,\dots$. Also,

 28.28.46 $\widehat{\gamma}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\mathop{\mathrm{se}_{n}\/% }\nolimits'\!\left(t,h^{2}\right)\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(t,% h^{2}\right)\mathrm{d}t=(-1)^{p+1}\dfrac{4}{\mathrm{i}\pi}\dfrac{\mathop{% \mathrm{se}_{n}\/}\nolimits'\!\left(0,h^{2}\right)\mathop{\mathrm{ce}_{m}\/}% \nolimits\!\left(0,h^{2}\right)}{\mathop{\mathrm{Dsc}_{1}\/}\nolimits\!\left(n% ,m,0\right)}.$

Lastly,

 28.28.47 $\dfrac{\mathop{\sinh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\cos% \/}\nolimits t\mathop{\mathrm{ce}_{n}\/}\nolimits\!\left(t,h^{2}\right)\mathop% {\mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}\nolimits% ^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{i}h% \widehat{\alpha}_{n,m}^{(c)}\mathop{\mathrm{Dc}_{0}\/}\nolimits\!\left(n,m,z% \right),$
 28.28.48 $\dfrac{\mathop{\cosh\/}\nolimits z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\mathop{\sin% \/}\nolimits t\mathop{\mathrm{ce}_{n}\/}\nolimits'\!\left(t,h^{2}\right)% \mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(t,h^{2}\right)}{{\mathop{\sinh\/}% \nolimits^{2}}z+{\mathop{\sin\/}\nolimits^{2}}t}\mathrm{d}t=(-1)^{p+1}\mathrm{% i}h\widehat{\alpha}_{n,m}^{(c)}\mathop{\mathrm{Dc}_{1}\/}\nolimits\!\left(n,m,% z\right),$

where $m-n=2p+1$, $p\in\mathbb{Z}$; $m,n=0,1,2,\dots$. Also,

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

## §28.28(v) Compendia

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).