§28.26 Asymptotic Approximations for Large $q$

Denote

28.26.1		${\mathrm{Mc}}_m^{(3)}(z,h)$	$={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}}\left({\mathrm{Fc}}_m(z,h)-\mathrm{i}{\mathrm{Gc}}_m(z,h)\right),$
ⓘ Symbols: $\pi$: 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, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$ 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		$\mathrm{i}{\mathrm{Ms}}_{m+1}^{(3)}(z,h)$	$={\displaystyle \frac{{\mathrm{e}}^{\mathrm{i}\varphi }}{{(\pi h\cosh z)}^{1/2}}}\left({\mathrm{Fs}}_m(z,h)-\mathrm{i}{\mathrm{Gs}}_m(z,h)\right),$
ⓘ Symbols: $\pi$: 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, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$ 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		$$\varphi =2h\sinh z-\left(m+\frac{1}{2}\right)\arctan \left(\sinh z\right).$$
ⓘ Symbols: $\sinh z$: hyperbolic sine function, $\arctan z$: arctangent function, $m$: integer, $h$: parameter, $z$: complex variable and $\varphi$ 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 $h\to +\mathrm{\infty }$ with fixed $z$ in $\mathrm{\Re }z>0$ and fixed $s=2m+1$,

28.26.4		$${\mathrm{Fc}}_m(z,h)\sim \begin{array}{l}1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)\\ +\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },\end{array}$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\cosh z$: 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		$${\mathrm{Gc}}_m(z,h)\sim \begin{array}{l}\frac{\sinh z}{{\cosh }^{2}z}(\begin{array}{l}\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)\\ +\frac{1}{2^{14}{h}^3}(\begin{array}{l}5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}\\ +\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\left)\right)\end{array}\end{array}\\ +\mathrm{\cdots }.\end{array}$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: 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 ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ 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).

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