28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.23 Expansions in Series of Bessel Functions 28.25 Asymptotic Expansions for Large $\realpart{z}$

§28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions

Notes:: See Meixner and Schäfke (1954, §2.64 and p. 201).
Keywords:: modified Mathieu functions, radial Mathieu functions
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/28.24
Addition (effective with 1.0.5):: The reference to Larsen et al. (2009) has been added at the end of this subsection.
Reported 2012-07-30

Throughout this section $\varepsilon_{0}=2$ and $\varepsilon_{s}=1$ , $s=1,2,3,\ldots$ .

With ${\cal C}_{\mu}^{{(j)}}$ , $c^{{\nu}}_{{n}}(q)$ , $A_{n}^{m}(q)$ , and $B_{n}^{m}(q)$ as in §28.23,

28.24.1 $c_{{2n}}^{{\nu}}(h^{2})\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left% (z,h\right)=\sum_{{\ell=-\infty}}^{{\infty}}(-1)^{\ell}c_{{2\ell}}^{{\nu}}(h^{% 2})\mathop{J_{{\ell-n}}\/}\nolimits\!\left(he^{{-z}}\right)\mathcal{C}_{{\nu+n% +\ell}}^{{(j)}}(he^{z}),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : parameter, : integer, : integer, : complex variable, $\nu$ : complex parameter, $c_{{2m}}(q)$ : coefficients and $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions
A&S Ref:: 20.4.12
Referenced by:: §28.22(ii), §28.24, §28.34(iv)
Permalink:: http://dlmf.nist.gov/28.24.E1
Encodings:: TeX, pMML, png

where and $n\in\Integer$ .

In the case when $\nu$ is an integer,

28.24.2 $\varepsilon_{s}\mathop{{\mathrm{Mc}^{{(j)}}_{{2m}}}\/}\nolimits\!\left(z,h% \right)=(-1)^{m}\sum_{{\ell=0}}^{{\infty}}(-1)^{\ell}\frac{A_{{2\ell}}^{{2m}}(% h^{2})}{A_{{2s}}^{{2m}}(h^{2})}\left(\mathop{J_{{\ell-s}}\/}\nolimits\!\left(% he^{{-z}}\right){\cal C}_{{\ell+s}}^{{(j)}}(he^{z})+\mathop{J_{{\ell+s}}\/}% \nolimits\!\left(he^{{-z}}\right){\cal C}_{{\ell-s}}^{{(j)}}(he^{z})\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions, $\varepsilon_{s}$ : constants and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.6.7
Permalink:: http://dlmf.nist.gov/28.24.E2
Encodings:: TeX, pMML, png

28.24.3 $\mathop{{\mathrm{Mc}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \sum_{{\ell=0}}^{{\infty}}(-1)^{\ell}\frac{A_{{2\ell+1}}^{{2m+1}}(h^{2})}{A_{{% 2s+1}}^{{2m+1}}(h^{2})}\left(\mathop{J_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}% \right){\cal C}_{{\ell+s+1}}^{{(j)}}(he^{z})+\mathop{J_{{\ell+s+1}}\/}% \nolimits\!\left(he^{{-z}}\right){\cal C}_{{\ell-s}}^{{(j)}}(he^{z})\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{{\mathrm{Mc}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.6.8
Permalink:: http://dlmf.nist.gov/28.24.E3
Encodings:: TeX, pMML, png

28.24.4 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \sum_{{\ell=0}}^{{\infty}}(-1)^{\ell}\frac{B_{{2\ell+1}}^{{2m+1}}(h^{2})}{B_{{% 2s+1}}^{{2m+1}}(h^{2})}\left(\mathop{J_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}% \right){\cal C}_{{\ell+s+1}}^{{(j)}}(he^{z})-\mathop{J_{{\ell+s+1}}\/}% \nolimits\!\left(he^{{-z}}\right){\cal C}_{{\ell-s}}^{{(j)}}(he^{z})\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions and $B_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.6.9
Permalink:: http://dlmf.nist.gov/28.24.E4
Encodings:: TeX, pMML, png

28.24.5 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+2}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \sum_{{\ell=0}}^{{\infty}}(-1)^{\ell}\frac{B_{{2\ell+2}}^{{2m+2}}(h^{2})}{B_{{% 2s+2}}^{{2m+2}}(h^{2})}\left(\mathop{J_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}% \right){\cal C}_{{\ell+s+2}}^{{(j)}}(he^{z})-\mathop{J_{{\ell+s+2}}\/}% \nolimits\!\left(he^{{-z}}\right){\cal C}_{{\ell-s}}^{{(j)}}(he^{z})\right),$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, : base of exponential function, $\mathop{{\mathrm{Ms}^{{(j)}}_{{n}}}\/}\nolimits\!\left(z,h\right)$ : radial Mathieu function, : integer, : parameter, : integer, : complex variable, $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions and $B_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.6.10
Permalink:: http://dlmf.nist.gov/28.24.E5
Encodings:: TeX, pMML, png

where and $s=0,1,2,\dots$ .

Also, with $\mathop{I_{{n}}\/}\nolimits$ and $\mathop{K_{{n}}\/}\nolimits$ denoting the modified Bessel functions (§10.25(ii)), and again with $s=0,1,2,\dots$ ,

28.24.6 $\varepsilon_{s}\mathop{\mathrm{Ie}_{{2m}}\/}\nolimits\!\left(z,h\right)=(-1)^{% s}\sum_{{\ell=0}}^{{\infty}}(-1)^{\ell}\dfrac{A_{{2\ell}}^{{2m}}(h^{2})}{A_{{2% s}}^{{2m}}(h^{2})}\left(\mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}% \right)\mathop{I_{{\ell+s}}\/}\nolimits\!\left(he^{{z}}\right)+\mathop{I_{{% \ell+s}}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{I_{{\ell-s}}\/}\nolimits\!% \left(he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ie}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable, $\varepsilon_{s}$ : constants and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.8.8
Permalink:: http://dlmf.nist.gov/28.24.E6
Encodings:: TeX, pMML, png

28.24.7 $\mathop{\mathrm{Io}_{{2m+2}}\/}\nolimits\!\left(z,h\right)=(-1)^{s}\sum_{{\ell% =0}}^{{\infty}}(-1)^{\ell}\dfrac{B_{{2\ell+2}}^{{2m+2}}(h^{2})}{B_{{2s+2}}^{{2% m+2}}(h^{2})}\left(\mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)% \mathop{I_{{\ell+s+2}}\/}\nolimits\!\left(he^{{z}}\right)-\mathop{I_{{\ell+s+2% }}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{I_{{\ell-s}}\/}\nolimits\!\left(% he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Io}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.24.E7
Encodings:: TeX, pMML, png

28.24.8 $\mathop{\mathrm{Ie}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=(-1)^{s}\sum_{{\ell% =0}}^{{\infty}}(-1)^{\ell}\dfrac{B_{{2\ell+1}}^{{2m+1}}(h^{2})}{B_{{2s+1}}^{{2% m+1}}(h^{2})}\left(\mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)% \mathop{I_{{\ell+s+1}}\/}\nolimits\!\left(he^{{z}}\right)+\mathop{I_{{\ell+s+1% }}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{I_{{\ell-s}}\/}\nolimits\!\left(% he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ie}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.24.E8
Encodings:: TeX, pMML, png

28.24.9 $\mathop{\mathrm{Io}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=(-1)^{s}\sum_{{\ell% =0}}^{{\infty}}(-1)^{\ell}\frac{A_{{2\ell+1}}^{{2m+1}}(h^{2})}{A_{{2s+1}}^{{2m% +1}}(h^{2})}\left(\mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)% \mathop{I_{{\ell+s+1}}\/}\nolimits\!\left(he^{{z}}\right)-\mathop{I_{{\ell+s+1% }}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{I_{{\ell-s}}\/}\nolimits\!\left(% he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Io}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $A_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.24.E9
Encodings:: TeX, pMML, png

28.24.10 $\varepsilon_{s}\mathop{\mathrm{Ke}_{{2m}}\/}\nolimits\!\left(z,h\right)=\sum_{% {\ell=0}}^{{\infty}}\frac{A_{{2\ell}}^{{2m}}(h^{2})}{A_{{2s}}^{{2m}}(h^{2})}% \left(\mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{K_{{\ell% +s}}\/}\nolimits\!\left(he^{{z}}\right)+\mathop{I_{{\ell+s}}\/}\nolimits\!% \left(he^{{-z}}\right)\mathop{K_{{\ell-s}}\/}\nolimits\!\left(he^{{z}}\right)% \right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ke}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable, $\varepsilon_{s}$ : constants and $A_{{m}}(q)$ : Fourier coefficient
A&S Ref:: 20.8.9
Permalink:: http://dlmf.nist.gov/28.24.E10
Encodings:: TeX, pMML, png

28.24.11 $\mathop{\mathrm{Ko}_{{2m+2}}\/}\nolimits\!\left(z,h\right)=\sum_{{\ell=0}}^{{% \infty}}\frac{B_{{2\ell+2}}^{{2m+2}}(h^{2})}{B_{{2s+2}}^{{2m+2}}(h^{2})}\left(% \mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{K_{{\ell+s+2}}% \/}\nolimits\!\left(he^{{z}}\right)-\mathop{I_{{\ell+s+2}}\/}\nolimits\!\left(% he^{{-z}}\right)\mathop{K_{{\ell-s}}\/}\nolimits\!\left(he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ko}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.24.E11
Encodings:: TeX, pMML, png

28.24.12 $\mathop{\mathrm{Ke}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=\sum_{{\ell=0}}^{{% \infty}}\frac{B_{{2\ell+1}}^{{2m+1}}(h^{2})}{B_{{2s+1}}^{{2m+1}}(h^{2})}\left(% \mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{K_{{\ell+s+1}}% \/}\nolimits\!\left(he^{{z}}\right)-\mathop{I_{{\ell+s+1}}\/}\nolimits\!\left(% he^{{-z}}\right)\mathop{K_{{\ell-s}}\/}\nolimits\!\left(he^{{z}}\right)\right),$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ke}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $B_{{m}}(q)$ : Fourier coefficient
Permalink:: http://dlmf.nist.gov/28.24.E12
Encodings:: TeX, pMML, png

28.24.13 $\mathop{\mathrm{Ko}_{{2m+1}}\/}\nolimits\!\left(z,h\right)=\sum_{{\ell=0}}^{{% \infty}}\frac{A_{{2\ell+1}}^{{2m+1}}(h^{2})}{A_{{2s+1}}^{{2m+1}}(h^{2})}\left(% \mathop{I_{{\ell-s}}\/}\nolimits\!\left(he^{{-z}}\right)\mathop{K_{{\ell+s+1}}% \/}\nolimits\!\left(he^{{z}}\right)+\mathop{I_{{\ell+s+1}}\/}\nolimits\!\left(% he^{{-z}}\right)\mathop{K_{{\ell-s}}\/}\nolimits\!\left(he^{{z}}\right)\right).$

Symbols:: : base of exponential function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{Ko}_{{n}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, : integer, : parameter, : complex variable and $A_{{m}}(q)$ : Fourier coefficient
Referenced by:: §28.24, §28.34(iv)
Permalink:: http://dlmf.nist.gov/28.24.E13
Encodings:: TeX, pMML, png

The expansions (28.24.1)–(28.24.13) converge absolutely and uniformly on compact sets of the -plane.

For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 28.23 Expansions in Series of Bessel Functions 28.25 Asymptotic Expansions for Large $\realpart{z}$