# §19.12 Asymptotic Approximations

With $\mathop{\psi\/}\nolimits\!\left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series:

 19.12.1 $\mathop{K\/}\nolimits\!\left(k\right)=\sum_{m=0}^{\infty}\frac{\left(\tfrac{1}% {2}\right)_{m}\left(\tfrac{1}{2}\right)_{m}}{m!\;m!}{k^{\prime}}^{2m}\left(% \mathop{\ln\/}\nolimits\left(\frac{1}{k^{\prime}}\right)+d(m)\right),$ $0<|k^{\prime}|<1$,
 19.12.2 $\mathop{E\/}\nolimits\!\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{% \left(\tfrac{1}{2}\right)_{m}\left(\tfrac{3}{2}\right)_{m}}{\left(2\right)_{m}% m!}{k^{\prime}}^{2m+2}\*\left(\mathop{\ln\/}\nolimits\left(\frac{1}{k^{\prime}% }\right)+d(m)-\frac{1}{(2m+1)(2m+2)}\right),$ $|k^{\prime}|<1$,

where

 19.12.3 $\displaystyle d(m)$ $\displaystyle=\mathop{\psi\/}\nolimits\!\left(1+m\right)-\mathop{\psi\/}% \nolimits\!\left(\tfrac{1}{2}+m\right),$ $\displaystyle d(m+1)$ $\displaystyle=d(m)-\frac{2}{(2m+1)(2m+2)}$, $m=0,1,\dots$, Symbols: $\mathop{\psi\/}\nolimits\!\left(z\right)$: psi (or digamma) function, $m$: nonnegative integer and $d(m)$: function Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E3 Encodings: TeX, TeX, pMML, pMML, png, png

with $d(0)=2\mathop{\ln\/}\nolimits 2$.

For the asymptotic behavior of $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ and $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ as $\phi\to\tfrac{1}{2}\pi-$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).

As $k^{2}\to 1-$

 19.12.4 $(1-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\left(\mathop% {\ln\/}\nolimits\frac{4}{k^{\prime}}\right)\left(1+\mathop{O\/}\nolimits\!% \left({k^{\prime}}^{2}\right)\right)-\alpha^{2}\mathop{R_{C}\/}\nolimits\!% \left(1,1-\alpha^{2}\right),$ $-\infty<\alpha^{2}<1$,
 19.12.5 $(1-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\left(\mathop% {\ln\/}\nolimits\left(\frac{4}{k^{\prime}}\right)-\mathop{R_{C}\/}\nolimits\!% \left(1,1-\alpha^{-2}\right)\right)\*\left(1+\mathop{O\/}\nolimits\!\left({k^{% \prime}}^{2}\right)\right),$ $1<\alpha^{2}<\infty$.

Asymptotic approximations for $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$, with different variables, are given in Karp et al. (2007). They are useful primarily when $\ifrac{(1-k)}{(1-\mathop{\sin\/}\nolimits\phi)}$ is either small or large compared with 1.

If $x\geq 0$ and $y>0$, then

 19.12.6 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{% x}}{y}\left(1+\mathop{O\/}\nolimits\!\left(\sqrt{\frac{x}{y}}\right)\right),$ $x/y\to 0,$
 19.12.7 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+% \frac{y}{2x}\right)\mathop{\ln\/}\nolimits\left(\frac{4x}{y}\right)-\frac{y}{2% x}\right)\*(1+\mathop{O\/}\nolimits\!\left(y^{2}/x^{2}\right)),$ $y/x\to 0$.