§28.28 Integrals, Integral Representations, and Integral Equations

Let

28.28.1		$$w=\cosh z\cos t\cos \alpha +\sinh z\sin t\sin \alpha .$$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\sin z$: sine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter Permalink: http://dlmf.nist.gov/28.28.E1 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

Then

28.28.2		$$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: 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 See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.3		$$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h),$$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter Permalink: http://dlmf.nist.gov/28.28.E3 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.4		$$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n^{\prime }(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter Permalink: http://dlmf.nist.gov/28.28.E4 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.5		$$\frac{\mathrm{i}h}{\pi }{\int }_0^{2\pi }\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n^{\prime }(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h).$$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter Permalink: http://dlmf.nist.gov/28.28.E5 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

In (28.28.7)–(28.28.9) the paths of integration ${ℒ}_j$ are given by

28.28.6		${ℒ}_1\text{ : from }-{\eta }_1+\mathrm{i}\mathrm{\infty }\text{ to }2\pi -{\eta }_1+\mathrm{i}\mathrm{\infty },$
		${ℒ}_3\text{ : from }-{\eta }_1+\mathrm{i}\mathrm{\infty }\text{ to }{\eta }_2-\mathrm{i}\mathrm{\infty },$
		${ℒ}_4\text{ : from }{\eta }_2-\mathrm{i}\mathrm{\infty }\text{ to }2\pi -{\eta }_1+\mathrm{i}\mathrm{\infty },$
ⓘ Defines: ${ℒ}_j$: 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		$$\frac{1}{\pi }{\int }_{{ℒ}_j}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$
$j=3,4$,
ⓘ Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: 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 See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.8		$$\frac{1}{\pi }{\int }_{{ℒ}_j}2\mathrm{i}h\frac{\partial w}{\partial \alpha }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }^{\prime }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$
$j=3,4$,
ⓘ Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$, $\partial x$: partial differential of $x$, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths A&S Ref: 20.7.19 (in slightly different notation) Permalink: http://dlmf.nist.gov/28.28.E8 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.9		$$\frac{1}{2\pi }{\int }_{{ℒ}_1}{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{me}}_{\nu }(t,h^2)dt={\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{me}}_{\nu }(\alpha ,h^2){\mathrm{M}}_{\nu }^{(1)}(z,h).$$
ⓘ Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable, $\nu$: complex parameter, $w(z)$: Mathieu’s equation solution, $\alpha$: parameter and ${ℒ}_j$: integration paths Referenced by: §28.28(i) Permalink: http://dlmf.nist.gov/28.28.E9 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

In (28.28.11)–(28.28.14)

28.28.10
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cosh z$: hyperbolic cosine function, $\mathrm{ph}$: phase, $h$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.28.E10 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.11		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}{\mathrm{Ce}}_{\nu }(t,h^2)dt=\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi }{\mathrm{ce}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(3)}(z,h),$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.28(i) Permalink: http://dlmf.nist.gov/28.28.E11 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.12		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}\sinh z\sinh t{\mathrm{Se}}_{\nu }(t,h^2)dt\\ =-\frac{\pi }{4h}{\mathrm{e}}^{\mathrm{i}\nu \pi /2}{\mathrm{se}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(3)}(z,h),\end{array}$$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, ${\mathrm{Se}}_{\nu }(z,q)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.28.E12 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.13		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}\sinh z\sinh t{\mathrm{Fe}}_m(t,h^2)dt\\ =-\frac{\pi }{4h}{\mathrm{i}}^m{\mathrm{fe}}_m^{\prime }(0,h^2){\mathrm{Mc}}_m^{(3)}(z,h),\end{array}$$
ⓘ Symbols: ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Fe}}_n(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter and $z$: complex variable Permalink: http://dlmf.nist.gov/28.28.E13 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.14		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}{\mathrm{Ge}}_m(t,h^2)dt=\frac{1}{2}\pi {\mathrm{i}}^{m+1}{\mathrm{ge}}_m(0,h^2){\mathrm{Ms}}_m^{(3)}(z,h).$$
ⓘ Symbols: ${\mathrm{ge}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ge}}_n(z,q)$: modified Mathieu function, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter and $z$: complex variable Referenced by: §28.28(i) Permalink: http://dlmf.nist.gov/28.28.E14 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

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

28.28.15		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}\cos \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt\\ ={(-1)}^{n+1}\frac{1}{2}\pi {\mathrm{Mc}}_{2n}^{(2)}(0,h){\mathrm{ce}}_{2n}(y,h^2),\end{array}$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $y$: real variable Permalink: http://dlmf.nist.gov/28.28.E15 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

28.28.16		$$\begin{array}{l}{\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt\\ =-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),\end{array}$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor Permalink: http://dlmf.nist.gov/28.28.E16 Encodings: TeX, pMML, png See also: Annotations for §28.28(i), §28.28 and Ch.28

where the upper or lower sign is taken according as $0\le y\le \pi$ or $\pi \le y\le 2\pi$. 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).

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 ${𝒞}_{\mu }^{(j)}$, $j=1,2,3,4$,

28.28.17		$$\frac{1}{\pi }{\int }_0^{\pi }{𝒞}_{\nu +2s}^{(j)}(2hR){\mathrm{e}}^{-\mathrm{i}(\nu +2s)\varphi }{\mathrm{me}}_{\nu }(t,h^2)dt={(-1)}^s{c}_{2s}^{\nu }(h^2){\mathrm{M}}_{\nu }^{(j)}(z,h),$$
$s\in \mathbb{Z}$,
ⓘ Symbols: ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\int$: integral, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function and $R(z,t)$: function A&S Ref: 20.7.12 (in different form) Permalink: http://dlmf.nist.gov/28.28.E17 Encodings: TeX, pMML, png See also: Annotations for §28.28(ii), §28.28 and Ch.28

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

28.28.18		$R(z,t)$	$={\left(\frac{1}{2}(\cosh \left(2z\right)+\cos \left(2t\right))\right)}^{1/2},$
		$R(z,0)$	$=\cosh z,$
ⓘ Symbols: $\cos z$: cosine function, $\cosh z$: 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		${\mathrm{e}}^{2\mathrm{i}\varphi }$	$={\displaystyle \frac{\cosh \left(z+\mathrm{i}t\right)}{\cosh \left(z-\mathrm{i}t\right)}},$
		$\varphi (z,0)$	$=0.$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\varphi (z,t)$: 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 $\mathrm{\ell }=0,1,2,\mathrm{\dots }$,

28.28.20		$$\begin{array}{l}\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }}^{(j)}(2hR)\cos \left(2\mathrm{\ell }\varphi \right){\mathrm{ce}}_{2m}(t,h^2)dt\\ ={\epsilon }_{\mathrm{\ell }}{(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }}^{2m}(h^2){\mathrm{Mc}}_{2m}^{(j)}(z,h),\end{array}$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function, ${\epsilon }_{\mathrm{\ell }}$: constants and $A_m(q)$: Fourier coefficient A&S Ref: 20.7.26 (in different form and only for $\mathrm{\ell }=0$) Permalink: http://dlmf.nist.gov/28.28.E20 Encodings: TeX, pMML, png See also: Annotations for §28.28(ii), §28.28 and Ch.28

where again ${\epsilon }_0=2$ and ${\epsilon }_{\mathrm{\ell }}=1$, $\mathrm{\ell }=1,2,3,\mathrm{\dots }$.

28.28.21		$$\begin{array}{l}\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\cos \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{ce}}_{2m+1}(t,h^2)dt\\ ={(-1)}^{\mathrm{\ell }+m}{A}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Mc}}_{2m+1}^{(j)}(z,h),\end{array}$$
ⓘ Symbols: ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $A_m(q)$: Fourier coefficient A&S Ref: 20.7.26 (in different form and only for $\mathrm{\ell }=0$) 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 ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$. Reported 2015-05-20 by Ruslan Kabasayev See also: Annotations for §28.28(ii), §28.28 and Ch.28

28.28.22		$$\begin{array}{l}\frac{4}{\pi }{\int }_0^{\pi /2}{𝒞}_{2\mathrm{\ell }+1}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+1)\varphi \right){\mathrm{se}}_{2m+1}(t,h^2)dt\\ ={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+1}^{2m+1}(h^2){\mathrm{Ms}}_{2m+1}^{(j)}(z,h),\end{array}$$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient A&S Ref: 20.7.27 (in different form and only for $\mathrm{\ell }=0$) 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 ${\displaystyle \frac{2}{\pi }}{\int }_0^{\pi }$. Reported 2015-05-20 by Ruslan Kabasayev See also: Annotations for §28.28(ii), §28.28 and Ch.28

28.28.23		$$\begin{array}{l}\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt\\ ={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).\end{array}$$
ⓘ Symbols: ${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient A&S Ref: 20.7.28 (in different form and only for $\mathrm{\ell }=0$) Permalink: http://dlmf.nist.gov/28.28.E23 Encodings: TeX, pMML, png See also: Annotations for §28.28(ii), §28.28 and Ch.28

With the parameter $h$ suppressed we use the notation

28.28.24		${\mathrm{D}}_0(\nu ,\mu ,z)$	$={\mathrm{M}}_{\nu }^{(3)}\left(z\right){\mathrm{M}}_{\mu }^{(4)}\left(z\right)-{\mathrm{M}}_{\nu }^{(4)}\left(z\right){\mathrm{M}}_{\mu }^{(3)}\left(z\right),$
		${\mathrm{D}}_1(\nu ,\mu ,z)$	$=\mathrm{M}_{\nu }^{(3)}{}^{\prime }\left(z\right){\mathrm{M}}_{\mu }^{(4)}\left(z\right)-\mathrm{M}_{\nu }^{(4)}{}^{\prime }\left(z\right){\mathrm{M}}_{\mu }^{(3)}\left(z\right),$
ⓘ Defines: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives Symbols: ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: 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		$$\begin{array}{l}\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt\\ ={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E25 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.26		$$\begin{array}{l}\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt\\ ={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_0(\nu ,\nu +2m+1,z),\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E26 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

where

28.28.27		$$\begin{array}{ll}{\alpha }_{\nu ,m}^{(0)}& =\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)dt\\ & ={(-1)}^m\frac{2\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }(0,h^2){\mathrm{me}}_{-\nu -2m-1}(0,h^2)}{h{\mathrm{D}}_0(\nu ,\nu +2m+1,0)},\end{array}$$
ⓘ Defines: ${\alpha }_{\nu ,m}^{(s)}$ (locally) Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.28.E27 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.28		$$\begin{array}{ll}{\alpha }_{\nu ,m}^{(1)}& =\frac{1}{2\pi }{\int }_0^{2\pi }\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)dt\\ & ={(-1)}^{m+1}\frac{2\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{me}}_{-\nu -2m-1}(0,h^2)}{h{\mathrm{D}}_1(\nu ,\nu +2m+1,0)}.\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E28 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.29		$$\begin{array}{l}\frac{\cosh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\sin t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt\\ ={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E29 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.30		$$\begin{array}{l}\frac{\sinh z}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt\\ ={(-1)}^m\mathrm{i}h{\alpha }_{\nu ,m}^{(1)}{\mathrm{D}}_1(\nu ,\nu +2m+1,z),\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E30 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.31		$$\begin{array}{l}\frac{2}{{\pi }^2}{\int }_0^{2\pi }\frac{\cos t\sin t{\mathrm{me}}_{\nu }(t,h^2){\mathrm{me}}_{-\nu -2m}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt\\ ={(-1)}^m\mathrm{i}{\gamma }_{\nu ,m}{\mathrm{D}}_0(\nu ,\nu +2m,z),\end{array}$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\gamma }_{\nu ,m}$ Permalink: http://dlmf.nist.gov/28.28.E31 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

28.28.32		$$\frac{\sinh \left(2z\right)}{{\pi }^2}{\int }_0^{2\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m}(t,h^2)}{{\sinh }^{2}z+{\sin }^{2}t}dt={(-1)}^{m+1}\mathrm{i}{\gamma }_{\nu ,m}{\mathrm{D}}_1(\nu ,\nu +2m,z),$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and ${\gamma }_{\nu ,m}$ Permalink: http://dlmf.nist.gov/28.28.E32 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

where

28.28.33		$${\gamma }_{\nu ,m}=\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{me}}_{\nu }^{\prime }\left(t\right){\mathrm{me}}_{-\nu -2m}\left(t\right)dt={(-1)}^m\frac{4\mathrm{i}}{\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }\left(0\right){\mathrm{me}}_{-\nu -2m}\left(0\right)}{{\mathrm{D}}_1(\nu ,\nu +2m,0)}.$$
ⓘ Defines: ${\gamma }_{\nu ,m}$ (locally) Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $\nu$: complex parameter Permalink: http://dlmf.nist.gov/28.28.E33 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

Also,

28.28.34		$$\frac{1}{{\pi }^2}{⨍}_0^{2\pi }\frac{{\mathrm{me}}_{\nu }^{\prime }(t,h^2){\mathrm{me}}_{-\nu -2m-1}(t,h^2)}{\sin t}dt={(-1)}^{m+1}\mathrm{i}h{\alpha }_{\nu ,m}^{(0)}{\mathrm{D}}_1(\nu ,\nu +2m+1,0),$$
ⓘ Symbols: ${\mathrm{D}}_j(\nu ,\mu ,z)$: cross-products of modified Mathieu functions and their derivatives, ${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{i}$: imaginary unit, ${⨍}_a^b$: Cauchy principal value, $\sin z$: sine function, $m$: integer, $h$: parameter, $\nu$: complex parameter and ${\alpha }_{\nu ,m}^{(s)}$ Permalink: http://dlmf.nist.gov/28.28.E34 Encodings: TeX, pMML, png See also: Annotations for §28.28(iii), §28.28 and Ch.28

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