# §19.6 Special Cases

## §19.6(i) Complete Elliptic Integrals

 19.6.1 $\displaystyle K\left(0\right)$ $\displaystyle=E\left(0\right)={K^{\prime}}\left(1\right)={E^{\prime}}\left(1% \right)=\tfrac{1}{2}\pi,$ $\displaystyle K\left(1\right)$ $\displaystyle={K^{\prime}}\left(0\right)=\infty,$ $\displaystyle E\left(1\right)$ $\displaystyle={E^{\prime}}\left(0\right)=1.$
 19.6.2 $\displaystyle\Pi\left(k^{2},k\right)$ $\displaystyle=E\left(k\right)/{k^{\prime}}^{2},$ $k^{2}<1$, $\displaystyle\Pi\left(-k,k\right)$ $\displaystyle=\tfrac{1}{4}\pi(1+k)^{-1}+\tfrac{1}{2}K\left(k\right),$ $0\leq k^{2}<1$.
 19.6.3 $\Pi\left(\alpha^{2},0\right)=\pi/(2\sqrt{1-\alpha^{2}}),\quad\Pi\left(0,k% \right)=K\left(k\right),$ $-\infty<\alpha^{2}<1$.
 19.6.4 $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle\to+\infty,$ $\alpha^{2}\to 1-$, $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle\to\infty\operatorname{sign}\left(1-\alpha^{2}\right),$ $k^{2}\to 1-$.

If $1<\alpha^{2}<\infty$, then the Cauchy principal value satisfies

 19.6.5 $\Pi\left(\alpha^{2},k\right)=K\left(k\right)-\Pi\left(k^{2}/\alpha^{2},k\right),$

and

 19.6.6 $\displaystyle\Pi\left(\alpha^{2},0\right)$ $\displaystyle=0,$ $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle\to K\left(k\right)-\left(E\left(k\right)/{k^{\prime}}^{2}\right),$ $\alpha^{2}\to 1+$, $\displaystyle\Pi\left(\alpha^{2},k\right)$ $\displaystyle\to-\infty,$ $k^{2}\rightarrow 1-$.

Exact values of $K\left(k\right)$ and $E\left(k\right)$ for various special values of $k$ are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006).

## §19.6(ii) $F\left(\phi,k\right)$

 19.6.7 $\displaystyle F\left(0,k\right)$ $\displaystyle=0,$ $\displaystyle F\left(\phi,0\right)$ $\displaystyle=\phi,$ $\displaystyle F\left(\tfrac{1}{2}\pi,1\right)$ $\displaystyle=\infty,$ $\displaystyle F\left(\tfrac{1}{2}\pi,k\right)$ $\displaystyle=K\left(k\right),$ $\displaystyle\lim_{\phi\to 0}\ifrac{F\left(\phi,k\right)}{\phi}$ $\displaystyle=1.$
 19.6.8 $F\left(\phi,1\right)=(\sin\phi)R_{C}\left(1,{\cos}^{2}\phi\right)={% \operatorname{gd}^{-1}}\left(\phi\right).$

For the inverse Gudermannian function ${\operatorname{gd}^{-1}}\left(\phi\right)$ see §4.23(viii). Compare also (19.10.2).

## §19.6(iii) $E\left(\phi,k\right)$

 19.6.9 $\displaystyle E\left(0,k\right)$ $\displaystyle=0,$ $\displaystyle E\left(\phi,0\right)$ $\displaystyle=\phi,$ $\displaystyle E\left(\tfrac{1}{2}\pi,1\right)$ $\displaystyle=1,$ $\displaystyle E\left(\phi,1\right)$ $\displaystyle=\sin\phi,$ $\displaystyle E\left(\tfrac{1}{2}\pi,k\right)$ $\displaystyle=E\left(k\right).$
 19.6.10 $\lim_{\phi\to 0}\ifrac{E\left(\phi,k\right)}{\phi}=1.$ ⓘ Symbols: $E\left(\NVar{\phi},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the second kind, $\phi$: real or complex argument and $k$: real or complex modulus Permalink: http://dlmf.nist.gov/19.6.E10 Encodings: TeX, pMML, png See also: Annotations for §19.6(iii), §19.6 and Ch.19

## §19.6(iv) $\Pi\left(\phi,\alpha^{2},k\right)$

Circular and hyperbolic cases, including Cauchy principal values, are unified by using $R_{C}\left(x,y\right)$. Let $c={\csc}^{2}\phi\neq\alpha^{2}$ and $\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}$. Then

 19.6.11 $\displaystyle\Pi\left(0,\alpha^{2},k\right)$ $\displaystyle=0,$ $\displaystyle\Pi\left(\phi,0,0\right)$ $\displaystyle=\phi,$ $\displaystyle\Pi\left(\phi,1,0\right)$ $\displaystyle=\tan\phi.$
 19.6.12 $\displaystyle\Pi\left(\phi,\alpha^{2},0\right)$ $\displaystyle=R_{C}\left(c-1,c-\alpha^{2}\right),$ $\displaystyle\Pi\left(\phi,\alpha^{2},1\right)$ $\displaystyle=\frac{1}{1-\alpha^{2}}\left(R_{C}\left(c,c-1\right)-\alpha^{2}R_% {C}\left(c,c-\alpha^{2}\right)\right),$ $\displaystyle\Pi\left(\phi,1,1\right)$ $\displaystyle=\tfrac{1}{2}(R_{C}\left(c,c-1\right)+\sqrt{c}(c-1)^{-1}).$
 19.6.13 $\displaystyle\Pi\left(\phi,0,k\right)$ $\displaystyle=F\left(\phi,k\right),$ $\displaystyle\Pi\left(\phi,k^{2},k\right)$ $\displaystyle=\frac{1}{{k^{\prime}}^{2}}\left(E\left(\phi,k\right)-\frac{k^{2}% }{\Delta}\sin\phi\cos\phi\right),$ $\displaystyle\Pi\left(\phi,1,k\right)$ $\displaystyle=F\left(\phi,k\right)-\frac{1}{{k^{\prime}}^{2}}(E\left(\phi,k% \right)-\Delta\tan\phi).$
 19.6.14 $\displaystyle\Pi\left(\tfrac{1}{2}\pi,\alpha^{2},k\right)$ $\displaystyle=\Pi\left(\alpha^{2},k\right),$ $\displaystyle\lim_{\phi\to 0}\ifrac{\Pi\left(\phi,\alpha^{2},k\right)}{\phi}$ $\displaystyle=1.$

For the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>c$, see §19.7(iii).

## §19.6(v) $R_{C}\left(x,y\right)$

 19.6.15 $\displaystyle R_{C}\left(x,x\right)$ $\displaystyle=x^{-1/2},$ $\displaystyle R_{C}\left(\lambda x,\lambda y\right)$ $\displaystyle=\lambda^{-1/2}R_{C}\left(x,y\right),$ $\displaystyle R_{C}\left(x,y\right)$ $\displaystyle\to+\infty,$ $y\to 0+$ or $y\to 0-$, $x>0$, $\displaystyle R_{C}\left(0,y\right)$ $\displaystyle=\tfrac{1}{2}\pi y^{-1/2},$ $|\operatorname{ph}y|<\pi$, $\displaystyle R_{C}\left(0,y\right)$ $\displaystyle=0,$ $y<0$. ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter and $\operatorname{ph}$: phase Referenced by: §19.2(iv), §19.2(iv), §19.20(iii), §19.6(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.6.E15 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png See also: Annotations for §19.6(v), §19.6 and Ch.19