§28.23 Expansions in Series of Bessel Functions

Keywords:: Bessel functions, modified Mathieu functions, radial Mathieu functions
Referenced by:: §28.24
Permalink:: http://dlmf.nist.gov/28.23

We use the following notations:

28.23.1

${\cal C}_{\mu}^{{(1)}}=\mathop{J_{{\mu}}\/}\nolimits,$

${\cal C}_{\mu}^{{(2)}}=\mathop{Y_{{\mu}}\/}\nolimits,$

${\cal C}_{\mu}^{{(3)}}=\mathop{{H^{{(1)}}_{{\mu}}}\/}\nolimits,$

${\cal C}_{\mu}^{{(4)}}=\mathop{{H^{{(2)}}_{{\mu}}}\/}\nolimits;$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), : integer and $\mathcal{C}_{\mu}^{{(j)}}$ : cylinder functions
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

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 $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(0,h^{2}\right)\mathop{{\mathrm{% M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=\sum_{{n=-\infty}}^{{\infty}% }(-1)^{n}c_{{2n}}^{\nu}(h^{2}){\cal C}_{{\nu+2n}}^{{(j)}}(2h\mathop{\cosh\/}% \nolimits z),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine 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
Permalink:: http://dlmf.nist.gov/28.23.E2
Encodings:: TeX, pMML, png

28.23.3 ${\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}\right)% \mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=i\mathop{% \tanh\/}\nolimits z\sum_{{n=-\infty}}^{{\infty}}(-1)^{n}(\nu+2n)c_{{2n}}^{\nu}% (h^{2}){\cal C}_{{\nu+2n}}^{{(j)}}(2h\mathop{\cosh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent 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
Permalink:: http://dlmf.nist.gov/28.23.E3
Encodings:: TeX, pMML, png

valid for all when , and for $\realpart{z}>0$ and $|\mathop{\cosh\/}\nolimits z|>1$ when .

28.23.4 $\mathop{\mathrm{me}_{{\nu}}\/}\nolimits\!\left(\tfrac{1}{2}\pi,h^{2}\right)% \mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=e^{{i\nu% \ifrac{\pi}{2}}}\sum_{{n=-\infty}}^{{\infty}}c_{{2n}}^{\nu}(h^{2}){\cal C}_{{% \nu+2n}}^{{(j)}}(2h\mathop{\sinh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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
Permalink:: http://dlmf.nist.gov/28.23.E4
Encodings:: TeX, pMML, png

28.23.5 ${\mathop{\mathrm{me}_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(\tfrac{1}{2}\pi,h^% {2}\right)\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)=% ie^{{i\nu\ifrac{\pi}{2}}}\mathop{\coth\/}\nolimits z\sum_{{n=-\infty}}^{{% \infty}}(\nu+2n)c_{{2n}}^{\nu}(h^{2}){\cal C}_{{\nu+2n}}^{{(j)}}(2h\mathop{% \sinh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{me}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, : base of exponential function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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
Permalink:: http://dlmf.nist.gov/28.23.E5
Encodings:: TeX, pMML, png

valid for all when , and for $\realpart{z}>0$ and $|\mathop{\sinh\/}\nolimits z|>1$ when .

In the case when $\nu$ is an integer

28.23.6 $\mathop{{\mathrm{Mc}^{{(j)}}_{{2m}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left(\mathop{\mathrm{ce}_{{2m}}\/}\nolimits\!\left(0,h^{2}\right)\right)^{{-1% }}\sum_{{\ell=0}}^{{\infty}}(-1)^{{\ell}}A_{{2\ell}}^{{2m}}(h^{2}){\cal C}_{{2% \ell}}^{{(j)}}(2h\mathop{\cosh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine 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:: 20.6.12
Referenced by:: §28.23
Permalink:: http://dlmf.nist.gov/28.23.E6
Encodings:: TeX, pMML, png

28.23.7 $\mathop{{\mathrm{Mc}^{{(j)}}_{{2m}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left(\mathop{\mathrm{ce}_{{2m}}\/}\nolimits\!\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{{-1}}\sum_{{\ell=0}}^{{\infty}}A_{{2\ell}}^{{2m}}(h^{2}){\cal C% }_{{2\ell}}^{{(j)}}(2h\mathop{\sinh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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
Permalink:: http://dlmf.nist.gov/28.23.E7
Encodings:: TeX, pMML, png

28.23.8 $\mathop{{\mathrm{Mc}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left(\mathop{\mathrm{ce}_{{2m+1}}\/}\nolimits\!\left(0,h^{2}\right)\right)^{{% -1}}\sum_{{\ell=0}}^{{\infty}}(-1)^{{\ell}}A_{{2\ell+1}}^{{2m+1}}(h^{2}){\cal C% }_{{2\ell+1}}^{{(j)}}(2h\mathop{\cosh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine 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
Permalink:: http://dlmf.nist.gov/28.23.E8
Encodings:: TeX, pMML, png

28.23.9 $\mathop{{\mathrm{Mc}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{{m+% 1}}\left({\mathop{\mathrm{ce}_{{2m+1}}\/}\nolimits^{{\prime}}}\!\left(\tfrac{1% }{2}\pi,h^{2}\right)\right)^{{-1}}\mathop{\coth\/}\nolimits z\sum_{{\ell=0}}^{% {\infty}}(2\ell+1)A_{{2\ell+1}}^{{2m+1}}(h^{2}){\cal C}_{{2\ell+1}}^{{(j)}}(2h% \mathop{\sinh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{ce}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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
Permalink:: http://dlmf.nist.gov/28.23.E9
Encodings:: TeX, pMML, png

28.23.10 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left({\mathop{\mathrm{se}_{{2m+1}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}% \right)\right)^{{-1}}\mathop{\tanh\/}\nolimits z\sum_{{\ell=0}}^{{\infty}}(-1)% ^{{\ell}}(2\ell+1)B_{{2\ell+1}}^{{2m+1}}(h^{2}){\cal C}_{{2\ell+1}}^{{(j)}}(2h% \mathop{\cosh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent 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
Permalink:: http://dlmf.nist.gov/28.23.E10
Encodings:: TeX, pMML, png

28.23.11 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+1}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left(\mathop{\mathrm{se}_{{2m+1}}\/}\nolimits\!\left(\tfrac{1}{2}\pi,h^{2}% \right)\right)^{{-1}}\sum_{{\ell=0}}^{{\infty}}B_{{2\ell+1}}^{{2m+1}}(h^{2}){% \cal C}_{{2\ell+1}}^{{(j)}}(2h\mathop{\sinh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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:: 20.6.13
Permalink:: http://dlmf.nist.gov/28.23.E11
Encodings:: TeX, pMML, png

28.23.12 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+2}}}\/}\nolimits\!\left(z,h\right)=(-1)^{m}% \left({\mathop{\mathrm{se}_{{2m+2}}\/}\nolimits^{{\prime}}}\!\left(0,h^{2}% \right)\right)^{{-1}}\mathop{\tanh\/}\nolimits z\sum_{{\ell=0}}^{{\infty}}(-1)% ^{{\ell}}(2\ell+2)B_{{2\ell+2}}^{{2m+2}}(h^{2}){\cal C}_{{2\ell+2}}^{{(j)}}(2h% \mathop{\cosh\/}\nolimits z),$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\tanh\/}\nolimits z$ : hyperbolic tangent 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
Permalink:: http://dlmf.nist.gov/28.23.E12
Encodings:: TeX, pMML, png

28.23.13 $\mathop{{\mathrm{Ms}^{{(j)}}_{{2m+2}}}\/}\nolimits\!\left(z,h\right)=(-1)^{{m+% 1}}\left({\mathop{\mathrm{se}_{{2m+2}}\/}\nolimits^{{\prime}}}\!\left(\tfrac{1% }{2}\pi,h^{2}\right)\right)^{{-1}}\mathop{\coth\/}\nolimits z\sum_{{\ell=0}}^{% {\infty}}(2\ell+2)B_{{2\ell+2}}^{{2m+2}}(h^{2}){\cal C}_{{2\ell+2}}^{{(j)}}(2h% \mathop{\sinh\/}\nolimits z).$

Symbols:: $\mathop{\mathrm{se}_{{n}}\/}\nolimits\!\left(z,q\right)$ : Mathieu function, $\mathop{\coth\/}\nolimits z$ : hyperbolic cotangent function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine 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
Referenced by:: §28.23
Permalink:: http://dlmf.nist.gov/28.23.E13
Encodings:: TeX, pMML, png

When , each of the series (28.23.6)–(28.23.13) converges for all . When the series in the even-numbered equations converge for $\realpart{z}>0$ and $|\mathop{\cosh\/}\nolimits z|>1$ , and the series in the odd-numbered equations converge for $\realpart{z}>0$ and $|\mathop{\sinh\/}\nolimits z|>1$ .

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