# §28.28 Integrals, Integral Representations, and Integral Equations

## §28.28(i) Equations with Elementary Kernels

Let

 28.28.1 $w=\cosh z\cos t\cos\alpha+\sinh z\sin t\sin\alpha.$

Then

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

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), 28.28 and 28

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

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

In (28.28.11)–(28.28.14)

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

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

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

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

 28.28.18 $\displaystyle R(z,t)$ $\displaystyle=\left(\tfrac{1}{2}(\cosh\left(2z\right)+\cos\left(2t\right))% \right)^{\ifrac{1}{2}},$ $\displaystyle R(z,0)$ $\displaystyle=\cosh z,$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\cosh\NVar{z}$: hyperbolic cosine function, $z$: complex variable and $R(z,t)$: function Permalink: http://dlmf.nist.gov/28.28.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.28(ii), 28.28 and 28

and

 28.28.19 $\displaystyle e^{2\mathrm{i}\phi}$ $\displaystyle=\dfrac{\cosh\left(z+\mathrm{i}t\right)}{\cosh\left(z-\mathrm{i}t% \right)},$ $\displaystyle\phi(z,0)$ $\displaystyle=0.$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $\cosh\NVar{z}$: hyperbolic cosine function, $z$: complex variable and $\phi(z,t)$: function Permalink: http://dlmf.nist.gov/28.28.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.28(ii), 28.28 and 28

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

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

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

 28.28.21 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\cos\left((2\ell% +1)\phi\right)\mathrm{ce}_{2m+1}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}A% ^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right),$ ⓘ Symbols: $\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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, ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28
 28.28.22 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\mathrm{se}_{2m+1}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B% ^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right),$ ⓘ Symbols: $\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: radial Mathieu function, $\sin\NVar{z}$: 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 and 28
 28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\mathrm{se}_{2m+2}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{% 2m+2}_{2\ell+2}(h^{2}){\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right).$

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

With the parameter $h$ suppressed we use the notation

 28.28.24 $\displaystyle\mathrm{D}_{0}\left(\nu,\mu,z\right)$ $\displaystyle={\mathrm{M}^{(3)}_{\nu}}\left(z\right){\mathrm{M}^{(4)}_{\mu}}% \left(z\right)-{\mathrm{M}^{(4)}_{\nu}}\left(z\right){\mathrm{M}^{(3)}_{\mu}}% \left(z\right),$ $\displaystyle\mathrm{D}_{1}\left(\nu,\mu,z\right)$ $\displaystyle={\mathrm{M}^{(3)}_{\nu}}'\left(z\right){\mathrm{M}^{(4)}_{\mu}}% \left(z\right)-{\mathrm{M}^{(4)}_{\nu}}'\left(z\right){\mathrm{M}^{(3)}_{\mu}}% \left(z\right),$ ⓘ Defines: $\mathrm{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$: cross-products of modified Mathieu functions and their derivatives Symbols: ${\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$: 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), 28.28 and 28

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

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

where

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

where

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

Also,

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

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 $\displaystyle\mathrm{Ds}_{0}\left(n,m,z\right)$ $\displaystyle={\mathrm{Ms}^{(3)}_{n}}\left(z\right){\mathrm{Ms}^{(4)}_{m}}% \left(z\right)-{\mathrm{Ms}^{(4)}_{n}}\left(z\right){\mathrm{Ms}^{(3)}_{m}}% \left(z\right),$ $\displaystyle\mathrm{Ds}_{1}\left(n,m,z\right)$ $\displaystyle={\mathrm{Ms}^{(3)}_{n}}'\left(z\right){\mathrm{Ms}^{(4)}_{m}}% \left(z\right)-{\mathrm{Ms}^{(4)}_{n}}'\left(z\right){\mathrm{Ms}^{(3)}_{m}}% \left(z\right),$ $\displaystyle\mathrm{Ds}_{2}\left(n,m,z\right)$ $\displaystyle={\mathrm{Ms}^{(3)}_{n}}'\left(z\right){\mathrm{Ms}^{(4)}_{m}}'% \left(z\right)-{\mathrm{Ms}^{(4)}_{n}}'\left(z\right){\mathrm{Ms}^{(3)}_{m}}'% \left(z\right).$ ⓘ Defines: $\mathrm{Ds}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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), 28.28 and 28

Then

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

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

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

Let

 28.28.39 $\displaystyle\mathrm{Dc}_{0}\left(n,m,z\right)$ $\displaystyle={\mathrm{Mc}^{(3)}_{n}}\left(z\right){\mathrm{Mc}^{(4)}_{m}}% \left(z\right)-{\mathrm{Mc}^{(4)}_{n}}\left(z\right){\mathrm{Mc}^{(3)}_{m}}% \left(z\right),$ $\displaystyle\mathrm{Dc}_{1}\left(n,m,z\right)$ $\displaystyle={\mathrm{Mc}^{(3)}_{n}}'\left(z\right){\mathrm{Mc}^{(4)}_{m}}% \left(z\right)-{\mathrm{Mc}^{(4)}_{n}}'\left(z\right){\mathrm{Mc}^{(3)}_{m}}% \left(z\right),$ ⓘ Defines: $\mathrm{Dc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28
 28.28.40 $\displaystyle\mathrm{Dsc}_{0}\left(n,m,z\right)$ $\displaystyle={\mathrm{Ms}^{(3)}_{n}}\left(z\right){\mathrm{Mc}^{(4)}_{m}}% \left(z\right)-{\mathrm{Ms}^{(4)}_{n}}\left(z\right){\mathrm{Mc}^{(3)}_{m}}% \left(z\right),$ $\displaystyle\mathrm{Dsc}_{1}\left(n,m,z\right)$ $\displaystyle={\mathrm{Ms}^{(3)}_{n}}'\left(z\right){\mathrm{Mc}^{(4)}_{m}}% \left(z\right)-{\mathrm{Ms}^{(4)}_{n}}'\left(z\right){\mathrm{Mc}^{(3)}_{m}}% \left(z\right).$ ⓘ Defines: $\mathrm{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$: cross-products of radial Mathieu functions and their derivatives Symbols: ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: radial Mathieu function, ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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), 28.28 and 28

Then

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

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

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

Next,

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

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

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

Lastly,

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

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

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

## §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).