§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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

28.25.1		$${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: 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		$D_{-1}^{\pm }$	$=0,$
		$D_0^{\pm }$	$=1,$
ⓘ 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		$$(m+1)D_{m+1}^{\pm }+\left({(m+\frac{1}{2})}^2\pm (m+\frac{1}{4})8\mathrm{i}h+2h^2-a\right)D_m^{\pm }\pm (m-\frac{1}{2})\left(8\mathrm{i}hm\right)D_{m-1}^{\pm }=0,$$
$m\ge 0$.
ⓘ 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 ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ and the lower signs to ${\mathrm{M}}_{\nu }^{(4)}(z,h)$. The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when

28.25.4		$$\mathrm{\Re }z\to +\mathrm{\infty },$$
$-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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 ${\mathrm{M}}_{\nu }^{(4)}(z,h)$ when

28.25.5		$$\mathrm{\Re }z\to +\mathrm{\infty },$$
$-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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).