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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.26 Asymptotic Approximations for Large q

Contents

§28.26(i) Goldstein’s Expansions

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Notes:
See Goldstein (1927) and National Bureau of Standards (1967, §IV).
Defines:
\mathop{\mathrm{Fc}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function, \mathop{\mathrm{Fs}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function, \mathop{\mathrm{Gc}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function and \mathop{\mathrm{Gs}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function
Permalink:
http://dlmf.nist.gov/28.26.i

Denote

where

28.26.3\phi=2h\mathop{\sinh\/}\nolimits z-\left(m+\tfrac{1}{2}\right)\mathop{\mathrm{% arctan}\/}\nolimits\!\left(\mathop{\sinh\/}\nolimits z\right).
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Symbols:
\mathop{\sinh\/}\nolimits z: hyperbolic sine function, \mathop{\mathrm{arctan}\/}\nolimits z: arctangent function, m: integer, h: parameter, z: complex variable and \phi
A&S Ref:
20.9.8 (in slightly different notation)
Permalink:
http://dlmf.nist.gov/28.26.E3
Encodings:
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Then as h\to+\infty with fixed z in \realpart{z}>0 and fixed s=2m+1,

28.26.4\mathop{\mathrm{Fc}_{{m}}\/}\nolimits\!\left(z,h\right)\sim 1+\dfrac{s}{8h{% \mathop{\cosh\/}\nolimits^{{2}}}z}+\dfrac{1}{2^{{11}}h^{2}}\left(\dfrac{s^{4}+% 86s^{2}+105}{{\mathop{\cosh\/}\nolimits^{{4}}}z}-\dfrac{s^{4}+22s^{2}+57}{{% \mathop{\cosh\/}\nolimits^{{2}}}z}\right)+\dfrac{1}{2^{{14}}h^{3}}\left(-% \dfrac{s^{5}+14s^{3}+33s}{{\mathop{\cosh\/}\nolimits^{{2}}}z}-\dfrac{2s^{5}+12% 4s^{3}+1122s}{{\mathop{\cosh\/}\nolimits^{{4}}}z}+\dfrac{3s^{5}+290s^{3}+1627s% }{{\mathop{\cosh\/}\nolimits^{{6}}}z}\right)+\cdots,
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Symbols:
\mathop{\mathrm{Fc}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function, \mathop{\cosh\/}\nolimits z: hyperbolic cosine function, \sim: asymptotic equality, m: integer, h: parameter and z: complex variable
A&S Ref:
20.9.9 (in slightly different notation)
Referenced by:
§28.26(i)
Permalink:
http://dlmf.nist.gov/28.26.E4
Encodings:
TeX, pMML, png
28.26.5\mathop{\mathrm{Gc}_{{m}}\/}\nolimits\!\left(z,h\right)\sim\dfrac{\mathop{% \sinh\/}\nolimits z}{{\mathop{\cosh\/}\nolimits^{{2}}}z}\left(\dfrac{s^{2}+3}{% 2^{5}h}+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\mathop{\cosh% \/}\nolimits^{{2}}}z}\right)+\dfrac{1}{2^{{14}}h^{3}}\left(5s^{4}+34s^{2}+9-% \dfrac{s^{6}-47s^{4}+667s^{2}+2835}{12{\mathop{\cosh\/}\nolimits^{{2}}}z}+% \dfrac{s^{6}+505s^{4}+12139s^{2}+10395}{12{\mathop{\cosh\/}\nolimits^{{4}}}z}% \right)\right)+\cdots.
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Symbols:
\mathop{\mathrm{Gc}_{{m}}\/}\nolimits\!\left(z,h\right): Mathieu function, \mathop{\cosh\/}\nolimits z: hyperbolic cosine function, \mathop{\sinh\/}\nolimits z: hyperbolic sine function, \sim: asymptotic equality, m: integer, h: parameter and z: complex variable
A&S Ref:
20.9.10 (in slightly different notation)
Referenced by:
§28.26(i)
Permalink:
http://dlmf.nist.gov/28.26.E5
Encodings:
TeX, pMML, png

The asymptotic expansions of \mathop{\mathrm{Fs}_{{m}}\/}\nolimits\!\left(z,h\right) and \mathop{\mathrm{Gs}_{{m}}\/}\nolimits\!\left(z,h\right) in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively.

For additional terms see Goldstein (1927).

§28.26(ii) Uniform Approximations

See §28.8(iv). For asymptotic approximations for \mathop{{\mathrm{M}^{{(3,4)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right) see also Naylor (1984, 1987, 1989).

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