# §28.26 Asymptotic Approximations for Large $q$

## §28.26(i) Goldstein’s Expansions

Denote

 28.26.1 $\displaystyle\mathop{{\mathrm{Mc}^{(3)}_{m}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\dfrac{e^{i\phi}}{(\pi h\mathop{\cosh\/}\nolimits z)^{\ifrac{1}{% 2}}}\*\left(\mathop{\mathrm{Fc}_{m}\/}\nolimits\!\left(z,h\right)-i\mathop{% \mathrm{Gc}_{m}\/}\nolimits\!\left(z,h\right)\right),$ 28.26.2 $\displaystyle i\mathop{{\mathrm{Ms}^{(3)}_{m+1}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\dfrac{e^{i\phi}}{(\pi h\mathop{\cosh\/}\nolimits z)^{\ifrac{1}{% 2}}}\*{\left(\mathop{\mathrm{Fs}_{m}\/}\nolimits\!\left(z,h\right)-i\mathop{% \mathrm{Gs}_{m}\/}\nolimits\!\left(z,h\right)\right)},$

where

 28.26.3 $\phi=2h\mathop{\sinh\/}\nolimits z-\left(m+\tfrac{1}{2}\right)\mathop{\mathrm{% arctan}\/}\nolimits\!\left(\mathop{\sinh\/}\nolimits z\right).$

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

 28.26.4 $\mathop{\mathrm{Fc}_{m}\/}\nolimits\!\left(z,h\right)\sim 1+\dfrac{s}{8h{% \mathop{\cosh\/}\nolimits^{2}}z}+\dfrac{1}{2^{11}h^{2}}\left(\dfrac{s^{4}+86s^% {2}+105}{{\mathop{\cosh\/}\nolimits^{4}}z}-\dfrac{s^{4}+22s^{2}+57}{{\mathop{% \cosh\/}\nolimits^{2}}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(-\dfrac{s^{5}+14s^% {3}+33s}{{\mathop{\cosh\/}\nolimits^{2}}z}-\dfrac{2s^{5}+124s^{3}+1122s}{{% \mathop{\cosh\/}\nolimits^{4}}z}+\dfrac{3s^{5}+290s^{3}+1627s}{{\mathop{\cosh% \/}\nolimits^{6}}z}\right)+\cdots,$
 28.26.5 $\mathop{\mathrm{Gc}_{m}\/}\nolimits\!\left(z,h\right)\sim\dfrac{\mathop{\sinh% \/}\nolimits z}{{\mathop{\cosh\/}\nolimits^{2}}z}\left(\dfrac{s^{2}+3}{2^{5}h}% +\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\mathop{\cosh\/}% \nolimits^{2}}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(5s^{4}+34s^{2}+9-\dfrac{s^% {6}-47s^{4}+667s^{2}+2835}{12{\mathop{\cosh\/}\nolimits^{2}}z}+\dfrac{s^{6}+50% 5s^{4}+12139s^{2}+10395}{12{\mathop{\cosh\/}\nolimits^{4}}z}\right)\right)+\cdots.$

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

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