# §12.9 Asymptotic Expansions for Large Variable

## §12.9(i) Poincaré-Type Expansions

Throughout this subsection $\delta$ is an arbitrary small positive constant.

As $z\to\infty$

 12.9.1 $\mathop{U\/}\nolimits\!\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1% }{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z% ^{2})^{s}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{3}{4}\pi-\delta(<\tfrac{3}{4}\pi)$ ,
 12.9.2 $\mathop{V\/}\nolimits\!\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z% ^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{\left(\frac{1}{2}-a\right)_{2s% }}}{s!(2z^{2})^{s}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$ .
 12.9.3 $\mathop{U\/}\nolimits\!\left(a,z\right)\sim e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1% }{2}}\sum_{s=0}^{\infty}(-1)^{s}\frac{{\left(\frac{1}{2}+a\right)_{2s}}}{s!(2z% ^{2})^{s}}\pm i\frac{\sqrt{2\pi}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{% 2}+a\right)}e^{\mp i\pi a}e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{% \infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2s}}}{s!(2z^{2})^{s}},$ $\tfrac{1}{4}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{5}{4}% \pi-\delta$ ,
 12.9.4 $\mathop{V\/}\nolimits\!\left(a,z\right)\sim\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z% ^{2}}z^{a-\frac{1}{2}}\sum_{s=0}^{\infty}\frac{{\left(\tfrac{1}{2}-a\right)_{2% s}}}{s!(2z^{2})^{s}}\pm\frac{i}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}% -a\right)}e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}\sum_{s=0}^{\infty}(-1)^{s}% \frac{{\left(\tfrac{1}{2}+a\right)_{2s}}}{s!(2z^{2})^{s}},$ $-\tfrac{1}{4}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{3}{4% }\pi-\delta$.

To obtain approximations for $\mathop{U\/}\nolimits\!\left(a,-z\right)$ and $\mathop{V\/}\nolimits\!\left(a,-z\right)$ as $z\to\infty$ combine the results above with (12.2.15) and (12.2.16). See also Temme (2015, Chapter 11).

## §12.9(ii) Bounds and Re-Expansions for the Remainder Terms

Bounds and re-expansions for the error term in (12.9.1) can be obtained by use of (12.7.14) and §§13.7(ii), 13.7(iii). Corresponding results for (12.9.2) can be obtained via (12.2.20).