# §19.12 Asymptotic Approximations

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

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

where

 19.12.3 $\displaystyle d(m)$ $\displaystyle=\psi\left(1+m\right)-\psi\left(\tfrac{1}{2}+m\right),$ $\displaystyle d(m+1)$ $\displaystyle=d(m)-\frac{2}{(2m+1)(2m+2)}$, $m=0,1,\dots$, ⓘ Symbols: $\psi\left(\NVar{z}\right)$: 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 See also: Annotations for §19.12 and Ch.19

with $d(0)=2\ln 2$.

For the asymptotic behavior of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ 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 $(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\frac{4}{k^{\prime}}\right% )\left(1+O\left({k^{\prime}}^{2}\right)\right)-\alpha^{2}R_{C}\left(1,1-\alpha% ^{2}\right),$ $-\infty<\alpha^{2}<1$,
 19.12.5 $(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\left(\frac{4}{k^{\prime}}% \right)-R_{C}\left(1,1-\alpha^{-2}\right)\right)\*\left(1+O\left({k^{\prime}}^% {2}\right)\right),$ $1<\alpha^{2}<\infty$.

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

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

 19.12.6 $R_{C}\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(% \sqrt{\frac{x}{y}}\right)\right),$ $x/y\to 0$, ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$: the ratio of the circumference of a circle to its diameter Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E6 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19
 19.12.7 $R_{C}\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln% \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)\*(1+O\left(y^{2}/x^{2}\right)),$ $y/x\to 0$. ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln\NVar{z}$: principal branch of logarithm function Referenced by: §19.12, §19.2(iv) Permalink: http://dlmf.nist.gov/19.12.E7 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19