28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.22 Connection Formulas 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

§28.23 Expansions in Series of Bessel Functions

ⓘ

Keywords:: Bessel functions, expansions in series of, expansions in series of Bessel functions, modified Mathieu functions, radial Mathieu functions
Referenced by:: §28.24
Permalink:: http://dlmf.nist.gov/28.23
See also:: Annotations for Ch.28

We use the following notations:

28.23.1	$\displaystyle{\cal C}_{\mu}^{(1)}$	$\displaystyle=J_{\mu},$
	$\displaystyle{\cal C}_{\mu}^{(2)}$	$\displaystyle=Y_{\mu},$
	$\displaystyle{\cal C}_{\mu}^{(3)}$	$\displaystyle={H^{(1)}_{\mu}},$
	$\displaystyle{\cal C}_{\mu}^{(4)}$	$\displaystyle={H^{(2)}_{\mu}};$
ⓘ Defines: $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions (locally) Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function) and $j$ : integer A&S Ref: 20.4.7 (in different notation) Referenced by: §28.28(ii) Permalink: http://dlmf.nist.gov/28.23.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §28.23 and Ch.28

compare §10.2(ii). For the coefficients $c^{\nu}_{n}(q)$ see §28.14. For $A_{n}^{m}(q)$ and $B_{n}^{m}(q)$ see §28.4.

28.23.2	$\displaystyle\operatorname{me}_{\nu}\left(0,h^{2}\right){\operatorname{M}^{(j)% }_{\nu}}\left(z,h\right)$	$\displaystyle=\sum_{n=-\infty}^{\infty}(-1)^{n}c_{2n}^{\nu}(h^{2}){\cal C}_{% \nu+2n}^{(j)}(2h\cosh z),$
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $n$ : integer, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients and $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions Permalink: http://dlmf.nist.gov/28.23.E2 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.3	$\displaystyle\operatorname{me}_{\nu}'\left(0,h^{2}\right){\operatorname{M}^{(j% )}_{\nu}}\left(z,h\right)$	$\displaystyle=\mathrm{i}\tanh z\sum_{n=-\infty}^{\infty}(-1)^{n}(\nu+2n)c_{2n}% ^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\cosh z),$
ⓘ Symbols: $\operatorname{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $n$ : integer, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients and $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions Permalink: http://dlmf.nist.gov/28.23.E3 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28

valid for all $z$ when $j=1$ , and for $\Re z>0$ and $|\cosh z|>1$ when $j=2,3,4$ .

28.23.4	$\displaystyle\operatorname{me}_{\nu}\left(\tfrac{1}{2}\pi,h^{2}\right){% \operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$	$\displaystyle=e^{\mathrm{i}\nu\ifrac{\pi}{2}}\sum_{n=-\infty}^{\infty}c_{2n}^{% \nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\sinh z),$
ⓘ 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, $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $n$ : integer, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients and $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions Permalink: http://dlmf.nist.gov/28.23.E4 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.5	$\displaystyle\operatorname{me}_{\nu}'\left(\tfrac{1}{2}\pi,h^{2}\right){% \operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$	$\displaystyle=\mathrm{i}e^{\mathrm{i}\nu\ifrac{\pi}{2}}\coth z\sum_{n=-\infty}% ^{\infty}(\nu+2n)c_{2n}^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\sinh z),$
ⓘ 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, $\mathrm{e}$ : base of natural logarithm, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $h$ : parameter, $n$ : integer, $j$ : integer, $z$ : complex variable, $\nu$ : complex parameter, $c_{2m}(q)$ : coefficients and $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions Permalink: http://dlmf.nist.gov/28.23.E5 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28

valid for all $z$ when $j=1$ , and for $\Re z>0$ and $|\sinh z|>1$ when $j=2,3,4$ .

In the case when $\nu$ is an integer

28.23.6	$\displaystyle{\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{ce}_{2m}\left(0,h^{2}\right)\right)^% {-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell}^{2m}(h^{2}){\cal C}_{2\ell}^{(j% )}(2h\cosh z),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $A_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.12 Referenced by: §28.23 Permalink: http://dlmf.nist.gov/28.23.E6 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.7	$\displaystyle{\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{ce}_{2m}\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{-1}\sum_{\ell=0}^{\infty}A_{2\ell}^{2m}(h^{2}){\cal C}_{2\ell}% ^{(j)}(2h\sinh z),$
ⓘ 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, $\sinh\NVar{z}$ : hyperbolic sine function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.23.E7 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28

28.23.8	$\displaystyle{\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{ce}_{2m+1}\left(0,h^{2}\right)\right% )^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2% \ell+1}^{(j)}(2h\cosh z),$
ⓘ Symbols: $\operatorname{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.23.E8 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.9	$\displaystyle{\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m+1}\left(\operatorname{ce}_{2m+1}'\left(\tfrac{1}{2}\pi,h% ^{2}\right)\right)^{-1}\coth z\sum_{\ell=0}^{\infty}(2\ell+1)A_{2\ell+1}^{2m+1% }(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\sinh z),$
ⓘ 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, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $A_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.23.E9 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.10	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{se}_{2m+1}'\left(0,h^{2}\right)% \right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{\ell}(2\ell+1)B_{2\ell+1}^{2m+1% }(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.23.E10 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.11	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{se}_{2m+1}\left(\tfrac{1}{2}\pi,h^{2% }\right)\right)^{-1}\sum_{\ell=0}^{\infty}B_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2% \ell+1}^{(j)}(2h\sinh z),$
ⓘ 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, $\sinh\NVar{z}$ : hyperbolic sine function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient A&S Ref: 20.6.13 Permalink: http://dlmf.nist.gov/28.23.E11 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.12	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h\right)$	$\displaystyle=(-1)^{m}\left(\operatorname{se}_{2m+2}'\left(0,h^{2}\right)% \right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{\ell}(2\ell+2)B_{2\ell+2}^{2m+2% }(h^{2}){\cal C}_{2\ell+2}^{(j)}(2h\cosh z),$
ⓘ Symbols: $\operatorname{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$ : Mathieu function, $\cosh\NVar{z}$ : hyperbolic cosine function, $\tanh\NVar{z}$ : hyperbolic tangent function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient Permalink: http://dlmf.nist.gov/28.23.E12 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28
28.23.13	$\displaystyle{\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h\right)$	$\displaystyle=(-1)^{m+1}\left(\operatorname{se}_{2m+2}'\left(\tfrac{1}{2}\pi,h% ^{2}\right)\right)^{-1}\coth z\sum_{\ell=0}^{\infty}(2\ell+2)B_{2\ell+2}^{2m+2% }(h^{2}){\cal C}_{2\ell+2}^{(j)}(2h\sinh z).$
ⓘ 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, $\coth\NVar{z}$ : hyperbolic cotangent function, $\sinh\NVar{z}$ : hyperbolic sine function, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $j$ : integer, $z$ : complex variable, $\mathcal{C}_{\mu}^{(j)}$ : cylinder functions and $B_{m}(q)$ : Fourier coefficient Referenced by: §28.23 Permalink: http://dlmf.nist.gov/28.23.E13 Encodings: TeX, pMML, png See also: Annotations for §28.23 and Ch.28

When $j=1$ , each of the series (28.23.6)–(28.23.13) converges for all $z$ . When $j=2,3,4$ the series in the even-numbered equations converge for $\Re z>0$ and $|\cosh z|>1$ , and the series in the odd-numbered equations converge for $\Re z>0$ and $|\sinh z|>1$ .

For proofs and generalizations, see Meixner and Schäfke (1954, §§2.62 and 2.64).

28.22 Connection Formulas 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

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