28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.25 Asymptotic Expansions for Large

\Re z

§28.26 Asymptotic Approximations for Large $q$

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Keywords:: asymptotic approximations, for large $q$ , modified Mathieu functions
Referenced by:: item (c)
Permalink:: http://dlmf.nist.gov/28.26
See also:: Annotations for Ch.28

§28.26(i) Goldstein’s Expansions
§28.26(ii) Uniform Approximations

§28.26(i) Goldstein’s Expansions

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Notes:: See Goldstein (1927) and National Bureau of Standards (1967, §IV).
Permalink:: http://dlmf.nist.gov/28.26.i
See also:: Annotations for §28.26 and Ch.28

Denote

28.26.1	$\displaystyle{\operatorname{Mc}^{(3)}_{m}}\left(z,h\right)$	$\displaystyle=\dfrac{e^{\mathrm{i}\phi}}{(\pi h\cosh z)^{\ifrac{1}{2}}}\*\left% (\mathrm{Fc}_{m}\left(z,h\right)-\mathrm{i}\mathrm{Gc}_{m}\left(z,h\right)% \right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$ A&S Ref: 20.9.7 (in slightly different notation) Permalink: http://dlmf.nist.gov/28.26.E1 Encodings: TeX, pMML, png See also: Annotations for §28.26(i), §28.26 and Ch.28
28.26.2	$\displaystyle\mathrm{i}{\operatorname{Ms}^{(3)}_{m+1}}\left(z,h\right)$	$\displaystyle=\dfrac{e^{\mathrm{i}\phi}}{(\pi h\cosh z)^{\ifrac{1}{2}}}\*{% \left(\mathrm{Fs}_{m}\left(z,h\right)-\mathrm{i}\mathrm{Gs}_{m}\left(z,h\right% )\right)},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, ${\operatorname{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$ : radial Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable and $\phi$ A&S Ref: 20.9.7 (in slightly different notation) Permalink: http://dlmf.nist.gov/28.26.E2 Encodings: TeX, pMML, png See also: Annotations for §28.26(i), §28.26 and Ch.28

where

28.26.3		$\phi=2h\sinh z-\left(m+\tfrac{1}{2}\right)\operatorname{arctan}\left(\sinh z% \right).$
ⓘ Symbols: $\sinh\NVar{z}$ : hyperbolic sine function, $\operatorname{arctan}\NVar{z}$ : 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: TeX, pMML, png See also: Annotations for §28.26(i), §28.26 and Ch.28

Then as $h\to+\infty$ with fixed $z$ in $\Re z>0$ and fixed $s=2m+1$ ,

28.26.4		$\mathrm{Fc}_{m}\left(z,h\right)\sim 1+\dfrac{s}{8h{\cosh}^{2}z}+\dfrac{1}{2^{1% 1}h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{{\cosh}^{4}z}-\dfrac{s^{4}+22s^{2}+57}% {{\cosh}^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(-\dfrac{s^{5}+14s^{3}+33s}{{% \cosh}^{2}z}-\dfrac{2s^{5}+124s^{3}+1122s}{{\cosh}^{4}z}+\dfrac{3s^{5}+290s^{3% }+1627s}{{\cosh}^{6}z}\right)+\cdots,$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $m$ : 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 See also: Annotations for §28.26(i), §28.26 and Ch.28

28.26.5		$\mathrm{Gc}_{m}\left(z,h\right)\sim\dfrac{\sinh z}{{\cosh}^{2}z}\left(\dfrac{s% ^{2}+3}{2^{5}h}+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cosh}% ^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(5s^{4}+34s^{2}+9-\dfrac{s^{6}-47s^{4% }+667s^{2}+2835}{12{\cosh}^{2}z}+\dfrac{s^{6}+505s^{4}+12139s^{2}+10395}{12{% \cosh}^{4}z}\right)\right)+\cdots.$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $m$ : 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 See also: Annotations for §28.26(i), §28.26 and Ch.28

The asymptotic expansions of $\mathrm{Fs}_{m}\left(z,h\right)$ and $\mathrm{Gs}_{m}\left(z,h\right)$ 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

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Keywords:: modified Mathieu functions, uniform asymptotic approximations for large parameters
Permalink:: http://dlmf.nist.gov/28.26.ii
See also:: Annotations for §28.26 and Ch.28

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

28.25 Asymptotic Expansions for Large

\Re z

28.27 Addition Theorems

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§28.26 Asymptotic Approximations for Large q

Contents

§28.26(i) Goldstein’s Expansions

§28.26(ii) Uniform Approximations

§28.26 Asymptotic Approximations for Large $q$