28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions 28.26 Asymptotic Approximations for Large

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§28.25 Asymptotic Expansions for Large $\Re z$

ⓘ

Keywords:: asymptotic approximations, for large $\Re z$ , modified Mathieu functions
Referenced by:: item (c)
Permalink:: http://dlmf.nist.gov/28.25
See also:: Annotations for Ch.28

For fixed $h(\neq 0)$ and fixed $\nu$ ,

28.25.1		${\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : 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, ${\operatorname{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$ : modified Mathieu function, $m$ : integer, $h$ : parameter, $z$ : complex variable, $\nu$ : complex parameter and $D_{m}^{\pm}$ : coefficients A&S Ref: 20.9.1 (for 3 and for integer $\nu$ ) 20.9.3 (for 4 and for integer $\nu$ ) Referenced by: §28.25 Permalink: http://dlmf.nist.gov/28.25.E1 Encodings: TeX, pMML, png See also: Annotations for §28.25 and Ch.28

where the coefficients are given by

28.25.2	$\displaystyle D_{-1}^{\pm}$	$\displaystyle=0,$
28.25.2	$\displaystyle D_{0}^{\pm}$	$\displaystyle=1,$
ⓘ Defines: $D_{m}^{\pm}$ : coefficients (locally) Symbols: $m$ : integer Permalink: http://dlmf.nist.gov/28.25.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §28.25 and Ch.28

and

28.25.3		$(m+1)D^{\pm}_{m+1}+{\left((m+\tfrac{1}{2})^{2}\pm(m+\tfrac{1}{4})8\mathrm{i}h+% 2h^{2}-a\right)D^{\pm}_{m}}\pm(m-\tfrac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1% }^{\pm}=0,$
$m\geq 0$ .
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $m$ : integer, $h$ : parameter, $a$ : parameter and $D_{m}^{\pm}$ : coefficients A&S Ref: 20.9.2 (for $+$ ) 20.9.4 (for $-$ ) Permalink: http://dlmf.nist.gov/28.25.E3 Encodings: TeX, pMML, png See also: Annotations for §28.25 and Ch.28

The upper signs correspond to ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$ and the lower signs to ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)$ . The expansion (28.25.1) is valid for ${\operatorname{M}^{(3)}_{\nu}}\left(z,h\right)$ when

28.25.4		$\Re z\to+\infty,$
$-\pi+\delta\leq\operatorname{ph}h+\Im z\leq 2\pi-\delta$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $\Re$ : real part, $h$ : parameter, $z$ : complex variable and $\delta$ : arbitrary small positive constant Permalink: http://dlmf.nist.gov/28.25.E4 Encodings: TeX, pMML, png See also: Annotations for §28.25 and Ch.28

and for ${\operatorname{M}^{(4)}_{\nu}}\left(z,h\right)$ when

28.25.5		$\Re z\to+\infty,$
$-2\pi+\delta\leq\operatorname{ph}h+\Im z\leq\pi-\delta$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Im$ : imaginary part, $\operatorname{ph}$ : phase, $\Re$ : real part, $h$ : parameter, $z$ : complex variable and $\delta$ : arbitrary small positive constant Permalink: http://dlmf.nist.gov/28.25.E5 Encodings: TeX, pMML, png See also: Annotations for §28.25 and Ch.28

where $\delta$ again denotes an arbitrary small positive constant.

For proofs and generalizations see Meixner and Schäfke (1954, §2.63).

28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions 28.26 Asymptotic Approximations for Large

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NIST

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§28.25 Asymptotic Expansions for Large ℜ⁡z

§28.25 Asymptotic Expansions for Large $\Re z$