§19.6(i) Complete Elliptic Integrals

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

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

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

and

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

Exact values of $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ 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) $\mathop{F\/}\nolimits\!\left(\phi,k\right)$

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

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

§19.6(iii) $\mathop{E\/}\nolimits\!\left(\phi,k\right)$

 19.6.9 $\displaystyle\mathop{E\/}\nolimits\!\left(0,k\right)$ $\displaystyle=0,$ $\displaystyle\mathop{E\/}\nolimits\!\left(\phi,0\right)$ $\displaystyle=\phi,$ $\displaystyle\mathop{E\/}\nolimits\!\left(\tfrac{1}{2}\pi,1\right)$ $\displaystyle=1,$ $\displaystyle\mathop{E\/}\nolimits\!\left(\phi,1\right)$ $\displaystyle=\mathop{\sin\/}\nolimits\phi,$ $\displaystyle\mathop{E\/}\nolimits\!\left(\tfrac{1}{2}\pi,k\right)$ $\displaystyle=\mathop{E\/}\nolimits\!\left(k\right).$
 19.6.10 $\lim_{\phi\to 0}\ifrac{\mathop{E\/}\nolimits\!\left(\phi,k\right)}{\phi}=1.$

§19.6(iv) $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$

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

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

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

§19.6(v) $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$

 19.6.15 $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(x,x\right)$ $\displaystyle=x^{-1/2},$ $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(\lambda x,\lambda y\right)$ $\displaystyle=\lambda^{-1/2}\mathop{R_{C}\/}\nolimits\!\left(x,y\right),$ $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ $\displaystyle\to+\infty,$ $y\to 0+$ or $y\to 0-$, $x>0$, $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(0,y\right)$ $\displaystyle=\tfrac{1}{2}\pi y^{-1/2},$ $|\mathop{\mathrm{ph}\/}\nolimits y|<\pi$, $\displaystyle\mathop{R_{C}\/}\nolimits\!\left(0,y\right)$ $\displaystyle=0,$ $y<0$. Symbols: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{\mathrm{ph}\/}\nolimits$: 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