§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 Ch.28

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

 28.28.7 $\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}e^{2\mathrm{i}hw}\mathrm{me}_{\nu}\left(t,% h^{2}\right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\mathrm{me}_{\nu}\left(\alpha,% h^{2}\right){\mathrm{M}^{(j)}_{\nu}}\left(z,h\right),$ $j=3,4$,
 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 Ch.28

and

 28.28.19 $\displaystyle e^{2\mathrm{i}\phi}$ $\displaystyle=\dfrac{\cosh\left(z+\mathrm{i}t\right)}{\cosh\left(z-\mathrm{i}t% \right)},$ $\displaystyle\phi(z,0)$ $\displaystyle=0.$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\cosh\NVar{z}$: 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 Ch.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: item 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 $\dfrac{4}{\pi}\int_{0}^{\pi/2}\mathcal{C}^{(j)}_{2\ell+1}(2hR)\sin\left((2\ell% +1)\phi\right)\mathrm{se}_{2m+1}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B% ^{2m+1}_{2\ell+1}(h^{2}){\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right),$ ⓘ 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: item 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 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\mathrm{se}_{2m+2}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{% 2m+2}_{2\ell+2}(h^{2}){\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right).$

§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 Ch.28

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

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