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

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

 28.25.1 $\mathop{{\mathrm{M}^{(3,4)}_{\nu}}\/}\nolimits\!\left(z,h\right)\sim\frac{e^{% \pm\mathrm{i}\left(2h\mathop{\cosh\/}\nolimits z-\left(\frac{1}{2}\nu+\frac{1}% {4}\right)\pi\right)}}{\left(\pi h(\mathop{\cosh\/}\nolimits z+1)\right)^{% \frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{\left(\mp 4\mathrm{i}h(% \mathop{\cosh\/}\nolimits z+1)\right)^{m}},$ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\mathop{{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: 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

where the coefficients are given by

 28.25.2 $\displaystyle D_{-1}^{\pm}$ $\displaystyle=0,$ $\displaystyle D_{0}^{\pm}$ $\displaystyle=1,$ 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

 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: $m$: 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

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

 28.25.4 $\Re{z}\to+\infty,$ $-\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits h+\Im{z}\leq 2\pi-\delta$,

and for $\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right)$ when

 28.25.5 $\Re{z}\to+\infty,$ $-2\pi+\delta\leq\mathop{\mathrm{ph}\/}\nolimits h+\Im{z}\leq\pi-\delta$,

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

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