28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.27 Addition Theorems 28.29 Definitions and Basic Properties

§28.28 Integrals, Integral Representations, and Integral Equations

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Referenced by:: §28.10(iii), §28.18
Permalink:: http://dlmf.nist.gov/28.28
See also:: Annotations for Ch.28

§28.28(i) Equations with Elementary Kernels

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Keywords:: Mathieu functions, integral equations, integral representations, integrals, modified Mathieu functions, radial Mathieu functions, with elementary kernels
Notes:: See Arscott (1964b, pp. 80-86) and Meixner and Schäfke (1954, §2.68).
Permalink:: http://dlmf.nist.gov/28.28.i
See also:: Annotations for §28.28 and Ch.28

Let

28.28.1		$w=\cosh z\cos t\cos\alpha+\sinh z\sin t\sin\alpha.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\sin\NVar{z}$ : 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		$\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{ce}_{n}\left(\alpha,h^{2% }\right){\operatorname{Mc}^{(1)}_{n}}\left(z,h\right),$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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		$\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}\operatorname{se}_{n}\left(t,h^% {2}\right)\,\mathrm{d}t={\mathrm{i}}^{n}\operatorname{se}_{n}\left(\alpha,h^{2% }\right){\operatorname{Ms}^{(1)}_{n}}\left(z,h\right),$
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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		$\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\operatorname{ce}_{n}\left(t,h^{2}\right)\,\mathrm{d}t={\mathrm{i% }}^{n}\operatorname{ce}_{n}'\left(\alpha,h^{2}\right){\operatorname{Mc}^{(1)}_% {n}}\left(z,h\right),$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : 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		$\dfrac{\mathrm{i}h}{\pi}\int_{0}^{2\pi}\frac{\partial w}{\partial\alpha}e^{2% \mathrm{i}hw}\operatorname{se}_{n}\left(t,h^{2}\right)\,\mathrm{d}t={\mathrm{i% }}^{n}\operatorname{se}_{n}'\left(\alpha,h^{2}\right){\operatorname{Ms}^{(1)}_% {n}}\left(z,h\right).$
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : 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 $\mathcal{L}_{j}$ 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		$\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}e^{2\mathrm{i}hw}\operatorname{me}_{\nu}% \left(t,h^{2}\right)\,\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\operatorname{me}_{% \nu}\left(\alpha,h^{2}\right){\operatorname{M}^{(j)}_{\nu}}\left(z,h\right),$
$j=3,4$ ,
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $w(z)$ : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : 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		$\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}2\mathrm{i}h\frac{\partial w}{\partial% \alpha}e^{2\mathrm{i}hw}\operatorname{me}_{\nu}\left(t,h^{2}\right)\,\mathrm{d% }t=e^{\mathrm{i}\nu\pi/{2}}\operatorname{me}_{\nu}'\left(\alpha,h^{2}\right){% \operatorname{M}^{(j)}_{\nu}}\left(z,h\right),$
$j=3,4$ ,
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $h$ : parameter, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $w(z)$ : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : 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		$\dfrac{1}{2\pi}\int_{\mathcal{L}_{1}}e^{2\mathrm{i}hw}\operatorname{me}_{\nu}% \left(t,h^{2}\right)\,\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}\operatorname{me}_{% \nu}\left(\alpha,h^{2}\right){\operatorname{M}^{(1)}_{\nu}}\left(z,h\right).$
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter, $w(z)$ : Mathieu’s equation solution, $\alpha$ : parameter and $\mathcal{L}_{j}$ : 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		$0<\operatorname{ph}\left(h(\cosh z\pm 1)\right)<\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\operatorname{ph}$ : 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		$\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}\operatorname{Ce}_{\nu}\left(t,% h^{2}\right)\,\mathrm{d}t=\tfrac{1}{2}\pi\mathrm{i}e^{\mathrm{i}\nu\pi}% \operatorname{ce}_{\nu}\left(0,h^{2}\right){\operatorname{M}^{(3)}_{\nu}}\left% (z,h\right),$
ⓘ 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, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : 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		$\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}\sinh z\sinh t\operatorname{Se}% _{\nu}\left(t,h^{2}\right)\,\mathrm{d}t=-\dfrac{\pi}{4h}e^{\mathrm{i}\nu\pi/{2% }}\operatorname{se}_{\nu}'\left(0,h^{2}\right){\operatorname{M}^{(3)}_{\nu}}% \left(z,h\right),$
ⓘ 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 $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $\operatorname{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : 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		$\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}\sinh z\sinh t\operatorname{Fe}% _{m}\left(t,h^{2}\right)\,\mathrm{d}t=-\dfrac{\pi}{4h}{\mathrm{i}}^{m}% \operatorname{fe}_{m}'\left(0,h^{2}\right){\operatorname{Mc}^{(3)}_{m}}\left(z% ,h\right),$
ⓘ Symbols: $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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		$\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}\operatorname{Ge}_{m}\left(t,h^% {2}\right)\,\mathrm{d}t=\tfrac{1}{2}\pi{\mathrm{i}}^{m+1}\operatorname{ge}_{m}% \left(0,h^{2}\right){\operatorname{Ms}^{(3)}_{m}}\left(z,h\right).$
ⓘ Symbols: $\operatorname{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{Ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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 $\Re z\geq 0$ , $0\leq\Im z\leq\pi$ .

28.28.15		$\int_{0}^{\infty}\cos\left(2h\cos y\cosh t\right)\operatorname{Ce}_{2n}\left(t% ,h^{2}\right)\,\mathrm{d}t=(-1)^{n+1}\tfrac{1}{2}\pi{\operatorname{Mc}^{(2)}_{% 2n}}\left(0,h\right)\operatorname{ce}_{2n}\left(y,h^{2}\right),$
ⓘ 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 $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : 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		$\int_{0}^{\infty}\sin\left(2h\cos y\cosh t\right)\operatorname{Ce}_{2n}\left(t% ,h^{2}\right)\,\mathrm{d}t=-\dfrac{\pi A_{0}^{2n}(h^{2})}{2\operatorname{ce}_{% 2n}\left(\frac{1}{2}\pi,h^{2}\right)}\*\left(\operatorname{ce}_{2n}\left(y,h^{% 2}\right)\mp\dfrac{2}{\pi C_{2n}(h^{2})}\operatorname{fe}_{2n}\left(y,h^{2}% \right)\right),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : second solution, Mathieu’s equation, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\operatorname{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$ : modified Mathieu function, $\sin\NVar{z}$ : 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 $0\leq y\leq\pi$ 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

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Keywords:: Mathieu functions, integral representations, integrals, modified Mathieu functions, of products with Bessel functions, radial Mathieu functions
Notes:: See Arscott (1964b, §4.6).
Permalink:: http://dlmf.nist.gov/28.28.ii
See also:: Annotations for §28.28 and Ch.28

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		$\dfrac{1}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{\nu+2s}(2hR)e^{-\mathrm{i}(\nu+% 2s)\phi}\operatorname{me}_{\nu}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{s}c^{% \nu}_{2s}(h^{2}){\operatorname{M}^{(j)}_{\nu}}\left(z,h\right),$
$s\in\mathbb{Z}$ ,
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\int$ : integral, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions, $\phi(z,t)$ : 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 $\phi=\phi(z,t)$ are analytic functions for $\Re z>0$ and real $t$ with

28.28.18	$\displaystyle R(z,t)$	$\displaystyle=\left(\tfrac{1}{2}(\cosh\left(2z\right)+\cos\left(2t\right))% \right)^{\ifrac{1}{2}},$
28.28.18	$\displaystyle R(z,0)$	$\displaystyle=\cosh z,$
ⓘ 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	$\displaystyle e^{2\mathrm{i}\phi}$	$\displaystyle=\dfrac{\cosh\left(z+\mathrm{i}t\right)}{\cosh\left(z-\mathrm{i}t% \right)},$
28.28.19	$\displaystyle\phi(z,0)$	$\displaystyle=0.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $z$ : 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		$\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell}(2hR)\cos\left(2\ell\phi% \right)\operatorname{ce}_{2m}\left(t,h^{2}\right)\,\mathrm{d}t=\varepsilon_{% \ell}(-1)^{\ell+m}A^{2m}_{2\ell}(h^{2}){\operatorname{Mc}^{(j)}_{2m}}\left(z,h% \right),$
ⓘ 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, $\varepsilon_{\ell}$ : constants and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.7.26 (in different form and only for $\ell=0$ ) 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 $\varepsilon_{0}=2$ 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		$\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\operatorname{se}_{2m+2}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+2}_{2\ell+2}(h^{2}){\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h% \right).$
ⓘ 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: 20.7.28 (in different form and only for $\ell=0$ ) Permalink: http://dlmf.nist.gov/28.28.E23 Encodings: TeX, pMML, png See also: Annotations for §28.28(ii), §28.28 and Ch.28

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

ⓘ

Notes:: See Schäfke (1983).
Permalink:: http://dlmf.nist.gov/28.28.iii
See also:: Annotations for §28.28 and Ch.28

With the parameter $h$ suppressed we use the notation

28.28.24	$\displaystyle\operatorname{D}_{0}\left(\nu,\mu,z\right)$	$\displaystyle={\operatorname{M}^{(3)}_{\nu}}\left(z\right){\operatorname{M}^{(% 4)}_{\mu}}\left(z\right)-{\operatorname{M}^{(4)}_{\nu}}\left(z\right){% \operatorname{M}^{(3)}_{\mu}}\left(z\right),$
28.28.24	$\displaystyle\operatorname{D}_{1}\left(\nu,\mu,z\right)$	$\displaystyle={\operatorname{M}^{(3)}_{\nu}}'\left(z\right){\operatorname{M}^{% (4)}_{\mu}}\left(z\right)-{\operatorname{M}^{(4)}_{\nu}}'\left(z\right){% \operatorname{M}^{(3)}_{\mu}}\left(z\right),$
ⓘ 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		$\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{me}_{\nu}% \left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}% ^{2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}% \operatorname{D}_{0}\left(\nu,\nu+2m+1,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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		$\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{me}_{\nu}% \left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}% ^{2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(1)}_{\nu,m}% \operatorname{D}_{0}\left(\nu,\nu+2m+1,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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		$\alpha^{(0)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\cos t\operatorname{me}_{% \nu}\left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)\,% \mathrm{d}t=(-1)^{m}\dfrac{2\mathrm{i}}{\pi}\dfrac{\operatorname{me}_{\nu}% \left(0,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(0,h^{2}\right)}{h% \operatorname{D}_{0}\left(\nu,\nu+2m+1,0\right)},$
ⓘ Defines: $\alpha^{(s)}_{\nu,m}$ (locally) Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\mathrm{i}$ : 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		$\alpha^{(1)}_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\sin t\operatorname{me}_{% \nu}\left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)\,% \mathrm{d}t=(-1)^{m+1}\dfrac{2\mathrm{i}}{\pi}\dfrac{\operatorname{me}_{\nu}'% \left(0,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(0,h^{2}\right)}{h% \operatorname{D}_{1}\left(\nu,\nu+2m+1,0\right)}.$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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		$\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{me}_{\nu}'% \left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}% ^{2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}% \operatorname{D}_{1}\left(\nu,\nu+2m+1,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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		$\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{me}_{\nu}'% \left(t,h^{2}\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{{\sinh}% ^{2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m}\mathrm{i}h\alpha^{(1)}_{\nu,m}% \operatorname{D}_{1}\left(\nu,\nu+2m+1,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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		$\dfrac{2}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\sin t\operatorname{me}_{\nu}% \left(t,h^{2}\right)\operatorname{me}_{-\nu-2m}\left(t,h^{2}\right)}{{\sinh}^{% 2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m}\mathrm{i}\gamma_{\nu,m}\operatorname{D}% _{0}\left(\nu,\nu+2m,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : 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		$\dfrac{\sinh\left(2z\right)}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\operatorname{me}_{% \nu}'\left(t,h^{2}\right)\operatorname{me}_{-\nu-2m}\left(t,h^{2}\right)}{{% \sinh}^{2}z+{\sin}^{2}t}\,\mathrm{d}t=(-1)^{m+1}\mathrm{i}\gamma_{\nu,m}% \operatorname{D}_{1}\left(\nu,\nu+2m,z\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : 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		$\gamma_{\nu,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\operatorname{me}_{\nu}'\left(t% \right)\operatorname{me}_{-\nu-2m}\left(t\right)\,\mathrm{d}t=(-1)^{m}\dfrac{4% \mathrm{i}}{\pi}\frac{\operatorname{me}_{\nu}'\left(0\right)\operatorname{me}_% {-\nu-2m}\left(0\right)}{\operatorname{D}_{1}\left(\nu,\nu+2m,0\right)}.$
ⓘ Defines: $\gamma_{\nu,m}$ (locally) Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : 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		$\dfrac{1}{\pi^{2}}\pvint_{0}^{2\pi}\dfrac{\operatorname{me}_{\nu}'\left(t,h^{2% }\right)\operatorname{me}_{-\nu-2m-1}\left(t,h^{2}\right)}{\sin t}\,\mathrm{d}% t=(-1)^{m+1}\mathrm{i}h\alpha^{(0)}_{\nu,m}\operatorname{D}_{1}\left(\nu,\nu+2% m+1,0\right),$
ⓘ Symbols: $\operatorname{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},\NVar{z}\right)$ : cross-products of modified Mathieu functions and their derivatives, $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $\nu$ : complex parameter and $\alpha^{(s)}_{\nu,m}$ 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)).

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

ⓘ

Keywords:: Mathieu functions, integral representations, integrals, modified Mathieu functions, of cross-products, of products, radial Mathieu functions
Notes:: See Schäfke (1983). There is a sign error on p. 158 of this reference.
Permalink:: http://dlmf.nist.gov/28.28.iv
See also:: Annotations for §28.28 and Ch.28

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		$\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{se}_{n}\left% (t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^% {2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(s)}% \operatorname{Ds}_{0}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Ds}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\alpha}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E36 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

28.28.37		$\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{se}_{n}'% \left(t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{% \sin}^{2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(s)}% \operatorname{Ds}_{1}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Ds}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\alpha}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E37 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

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

28.28.38		$\widehat{\alpha}_{n,m}^{(s)}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\cos t\operatorname% {se}_{n}\left(t,h^{2}\right)\operatorname{se}_{m}\left(t,h^{2}\right)\,\mathrm% {d}t=(-1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{se}_{n}'\left(0,h^{2% }\right)\operatorname{se}_{m}'\left(0,h^{2}\right)}{h\operatorname{Ds}_{2}% \left(n,m,0\right)}.$
ⓘ Symbols: $\operatorname{Ds}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $m$ : integer, $h$ : parameter, $n$ : integer and $\widehat{\alpha}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E38 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

Let

28.28.39	$\displaystyle\operatorname{Dc}_{0}\left(n,m,z\right)$	$\displaystyle={\operatorname{Mc}^{(3)}_{n}}\left(z\right){\operatorname{Mc}^{(% 4)}_{m}}\left(z\right)-{\operatorname{Mc}^{(4)}_{n}}\left(z\right){% \operatorname{Mc}^{(3)}_{m}}\left(z\right),$
28.28.39	$\displaystyle\operatorname{Dc}_{1}\left(n,m,z\right)$	$\displaystyle={\operatorname{Mc}^{(3)}_{n}}'\left(z\right){\operatorname{Mc}^{% (4)}_{m}}\left(z\right)-{\operatorname{Mc}^{(4)}_{n}}'\left(z\right){% \operatorname{Mc}^{(3)}_{m}}\left(z\right),$
ⓘ 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	$\displaystyle\operatorname{Dsc}_{0}\left(n,m,z\right)$	$\displaystyle={\operatorname{Ms}^{(3)}_{n}}\left(z\right){\operatorname{Mc}^{(% 4)}_{m}}\left(z\right)-{\operatorname{Ms}^{(4)}_{n}}\left(z\right){% \operatorname{Mc}^{(3)}_{m}}\left(z\right),$
28.28.40	$\displaystyle\operatorname{Dsc}_{1}\left(n,m,z\right)$	$\displaystyle={\operatorname{Ms}^{(3)}_{n}}'\left(z\right){\operatorname{Mc}^{% (4)}_{m}}\left(z\right)-{\operatorname{Ms}^{(4)}_{n}}'\left(z\right){% \operatorname{Mc}^{(3)}_{m}}\left(z\right).$
ⓘ 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		$\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{se}_{n}\left% (t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^% {2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\beta}_{n,m}\operatorname{Dsc% }_{0}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\beta}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E41 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

28.28.42		$\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{se}_{n}'% \left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{% \sin}^{2}t}\,\mathrm{d}t=(-1)^{p}\mathrm{i}h\widehat{\beta}_{n,m}\operatorname% {Dsc}_{1}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\beta}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E42 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

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

28.28.43		$\widehat{\beta}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\sin t\operatorname{se}_{n% }\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)\,\mathrm{d}t=(-% 1)^{p}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{se}_{n}'\left(0,h^{2}\right% )\operatorname{ce}_{m}\left(0,h^{2}\right)}{h\operatorname{Dsc}_{1}\left(n,m,0% \right)}.$
ⓘ Defines: $\widehat{\beta}_{n,m}$ (locally) Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter and $n$ : integer Permalink: http://dlmf.nist.gov/28.28.E43 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

Next,

28.28.44		$\dfrac{1}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin\left(2t\right)\operatorname{se}_{% n}\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+% {\sin}^{2}t}\,\mathrm{d}t=(-1)^{p}\mathrm{i}\widehat{\gamma}_{n,m}% \operatorname{Dsc}_{0}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\gamma}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E44 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

28.28.45		$\dfrac{\sinh\left(2z\right)}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\operatorname{se}_{% n}'\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z% +{\sin}^{2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}\widehat{\gamma}_{n,m}% \operatorname{Dsc}_{1}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\gamma}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E45 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

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

28.28.46		$\widehat{\gamma}_{n,m}=\dfrac{1}{2\pi}\int_{0}^{2\pi}\operatorname{se}_{n}'% \left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)\,\mathrm{d}t=(-1% )^{p+1}\dfrac{4}{\mathrm{i}\pi}\dfrac{\operatorname{se}_{n}'\left(0,h^{2}% \right)\operatorname{ce}_{m}\left(0,h^{2}\right)}{\operatorname{Dsc}_{1}\left(% n,m,0\right)}.$
ⓘ Defines: $\widehat{\gamma}_{n,m}$ (locally) Symbols: $\operatorname{Dsc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $m$ : integer, $h$ : parameter and $n$ : integer Permalink: http://dlmf.nist.gov/28.28.E46 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

Lastly,

28.28.47		$\dfrac{\sinh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\cos t\operatorname{ce}_{n}\left% (t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{\sin}^% {2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(c)}% \operatorname{Dc}_{0}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\alpha}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E47 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

28.28.48		$\dfrac{\cosh z}{\pi^{2}}\int_{0}^{2\pi}\dfrac{\sin t\operatorname{ce}_{n}'% \left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)}{{\sinh}^{2}z+{% \sin}^{2}t}\,\mathrm{d}t=(-1)^{p+1}\mathrm{i}h\widehat{\alpha}_{n,m}^{(c)}% \operatorname{Dc}_{1}\left(n,m,z\right),$
ⓘ Symbols: $\operatorname{Dc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer, $h$ : parameter, $n$ : integer, $z$ : complex variable and $\widehat{\alpha}_{n,m}$ Permalink: http://dlmf.nist.gov/28.28.E48 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

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

28.28.49		$\widehat{\alpha}_{n,m}^{(c)}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos t\operatorname{% ce}_{n}\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)\,\mathrm{% d}t=(-1)^{p+1}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{ce}_{n}\left(0,h^{2% }\right)\operatorname{ce}_{m}\left(0,h^{2}\right)}{h\operatorname{Dc}_{0}\left% (n,m,0\right)}.$
ⓘ Defines: $\widehat{\alpha}_{n,m}$ (locally) Symbols: $\operatorname{Dc}_{\NVar{j}}\left(\NVar{n},\NVar{m},\NVar{z}\right)$ : cross-products of radial Mathieu functions and their derivatives, $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : 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 $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $m$ : integer, $h$ : parameter and $n$ : integer Permalink: http://dlmf.nist.gov/28.28.E49 Encodings: TeX, pMML, png See also: Annotations for §28.28(iv), §28.28 and Ch.28

§28.28(v) Compendia

ⓘ

Keywords:: Mathieu functions, compendia, integral representations, integrals, modified Mathieu functions, radial Mathieu functions
Permalink:: http://dlmf.nist.gov/28.28.v
See also:: Annotations for §28.28 and Ch.28

See Prudnikov et al. (1990, pp. 359–368), Gradshteyn and Ryzhik (2015, §§6.91–6.93), Sips (1970), and Meixner et al. (1980, §2.1.1).

28.27 Addition Theorems 28.29 Definitions and Basic Properties

NIST

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§28.28 Integrals, Integral Representations, and Integral Equations

Contents

§28.28(i) Equations with Elementary Kernels

§28.28(ii) Integrals of Products with Bessel Functions

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

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

§28.28(v) Compendia