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

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, $\mathrm{i}$: imaginary unit, $j$: 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 Ch.28

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

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

In (28.28.11)–(28.28.14)

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

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

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

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 Ch.28

and

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

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

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

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

 28.28.21 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\cos\left((2\ell% +1)\phi\right)\operatorname{ce}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}A^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h% \right),$ ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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: Erratum (V1.0.11) for 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 Ch.28
 28.28.22 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\operatorname{se}_{2m+1}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+1}_{2\ell+1}(h^{2}){\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h% \right),$ ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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: Erratum (V1.0.11) for 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 Ch.28
 28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\operatorname{se}_{2m+2}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+2}_{2\ell+2}(h^{2}){\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h% \right).$

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

With the parameter $h$ suppressed we use the notation

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

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

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

where

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

where

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

Also,

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

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

Then

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

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

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

Let

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

Then

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

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

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

Next,

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

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

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

Lastly,

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

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

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

## §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).