## §19.11(i) General Formulas

 19.11.1 $F\left(\theta,k\right)+F\left(\phi,k\right)=F\left(\psi,k\right),$
 19.11.2 $E\left(\theta,k\right)+E\left(\phi,k\right)=E\left(\psi,k\right)+k^{2}\sin% \theta\sin\phi\sin\psi.$

Here

 19.11.3 $\displaystyle\sin\psi$ $\displaystyle=\frac{(\sin\theta\cos\phi)\Delta(\phi)+(\sin\phi\cos\theta)% \Delta(\theta)}{1-k^{2}{\sin}^{2}\theta{\sin}^{2}\phi},$ $\displaystyle\Delta(\theta)$ $\displaystyle=\sqrt{1-k^{2}{\sin}^{2}\theta}.$ ⓘ Defines: $\psi$: angle (locally) and $\Delta(\theta)$ (locally) Symbols: $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $\phi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude Referenced by: §19.11(iii) Permalink: http://dlmf.nist.gov/19.11.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(i), §19.11 and Ch.19

Also,

 19.11.4 $\displaystyle\cos\psi$ $\displaystyle=\frac{\cos\theta\cos\phi-(\sin\theta\sin\phi)\Delta(\theta)% \Delta(\phi)}{1-k^{2}{\sin}^{2}\theta{\sin}^{2}\phi},$ $\displaystyle\tan\left(\tfrac{1}{2}\psi\right)$ $\displaystyle=\frac{(\sin\theta)\Delta(\phi)+(\sin\phi)\Delta(\theta)}{\cos% \theta+\cos\phi}.$

Lastly,

 19.11.5 $\Pi\left(\theta,\alpha^{2},k\right)+\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left% (\psi,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),$

where

 19.11.6 $\displaystyle\gamma$ $\displaystyle=(({\csc}^{2}\theta)-\alpha^{2})(({\csc}^{2}\phi)-\alpha^{2})(({% \csc}^{2}\psi)-\alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$ ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc\NVar{z}$: cosecant function, $\phi$: real or complex argument, $k$: real or complex modulus, $\alpha^{2}$: real or complex parameter, $\theta$: amplitude and $\psi$: angle Permalink: http://dlmf.nist.gov/19.11.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(i), §19.11 and Ch.19

In the case of $\theta,\phi\in[0,\pi/2)$ and $0\leq k^{2}\leq\alpha^{2}<\min\left(1,\left(1-\cos\theta\cos\phi\cos\psi\right% )^{-1}\right)$, we can use

 19.11.6_5 $R_{C}\left(\gamma-\delta,\gamma\right)=\frac{-1}{\sqrt{\delta}}\operatorname{% arctan}\left(\frac{\sqrt{\delta}\sin\theta\sin\phi\sin\psi}{\alpha^{2}-1-% \alpha^{2}\cos\theta\cos\phi\cos\psi}\right).$

Hence, care has to be taken with the multivalued functions in (19.11.5).

## §19.11(ii) Case $\psi=\pi/2$

 19.11.7 $F\left(\phi,k\right)=K\left(k\right)-F\left(\theta,k\right),$
 19.11.8 $E\left(\phi,k\right)=E\left(k\right)-E\left(\theta,k\right)+k^{2}\sin\theta% \sin\phi,$

where

 19.11.9 $\tan\theta=1/(k^{\prime}\tan\phi).$ ⓘ Symbols: $\tan\NVar{z}$: tangent function, $\phi$: real or complex argument, $k^{\prime}$: complementary modulus and $\theta$: amplitude Permalink: http://dlmf.nist.gov/19.11.E9 Encodings: TeX, pMML, png See also: Annotations for §19.11(ii), §19.11 and Ch.19
 19.11.10 $\Pi\left(\phi,\alpha^{2},k\right)=\Pi\left(\alpha^{2},k\right)-\Pi\left(\theta% ,\alpha^{2},k\right)-\alpha^{2}R_{C}\left(\gamma-\delta,\gamma\right),$

where

 19.11.11 $\displaystyle\gamma$ $\displaystyle=(1-\alpha^{2})(({\csc}^{2}\theta)-\alpha^{2})(({\csc}^{2}\phi)-% \alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$ ⓘ Defines: $\gamma$ (locally) and $\delta$ (locally) Symbols: $\csc\NVar{z}$: cosecant function, $\phi$: real or complex argument, $k$: real or complex modulus, $\alpha^{2}$: real or complex parameter and $\theta$: amplitude Permalink: http://dlmf.nist.gov/19.11.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(ii), §19.11 and Ch.19

## §19.11(iii) Duplication Formulas

If $\phi=\theta$ in §19.11(i) and $\Delta(\theta)$ is again defined by (19.11.3), then

 19.11.12 $F\left(\psi,k\right)=2F\left(\theta,k\right),$
 19.11.13 $E\left(\psi,k\right)=2E\left(\theta,k\right)-k^{2}{\sin}^{2}\theta\sin\psi,$
 19.11.14 $\sin\psi=(\sin 2\theta)\Delta(\theta)/(1-k^{2}{\sin}^{4}\theta),$ ⓘ Defines: $\psi$: angle (locally) Symbols: $\sin\NVar{z}$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\Delta(\theta)$ Permalink: http://dlmf.nist.gov/19.11.E14 Encodings: TeX, pMML, png See also: Annotations for §19.11(iii), §19.11 and Ch.19
 19.11.15 $\displaystyle\cos\psi$ $\displaystyle=(\cos\left(2\theta\right)+k^{2}{\sin}^{4}\theta)/(1-k^{2}{\sin}^% {4}\theta),$ $\displaystyle\tan\left(\tfrac{1}{2}\psi\right)$ $\displaystyle=(\tan\theta)\Delta(\theta),$ $\displaystyle\sin\theta$ $\displaystyle=(\sin\psi)/\sqrt{(1+\cos\psi)(1+\Delta(\psi))},$ $\displaystyle\cos\theta$ $\displaystyle=\sqrt{\frac{(\cos\psi)+\Delta(\psi)}{1+\Delta(\psi)}},$ $\displaystyle\tan\theta$ $\displaystyle=\tan\left(\tfrac{1}{2}\psi\right)\sqrt{\frac{1+\cos\psi}{(\cos% \psi)+\Delta(\psi)}},$
 19.11.16 $\Pi\left(\psi,\alpha^{2},k\right)=2\Pi\left(\theta,\alpha^{2},k\right)+\alpha^% {2}R_{C}\left(\gamma-\delta,\gamma\right),$
 19.11.17 $\displaystyle\gamma$ $\displaystyle=(({\csc}^{2}\theta)-\alpha^{2})^{2}(({\csc}^{2}\psi)-\alpha^{2}),$ $\displaystyle\delta$ $\displaystyle=\alpha^{2}(1-\alpha^{2})(\alpha^{2}-k^{2}).$ ⓘ Defines: $\delta$ (locally) and $\gamma$ (locally) Symbols: $\csc\NVar{z}$: cosecant function, $k$: real or complex modulus, $\alpha^{2}$: real or complex parameter, $\theta$: amplitude and $\psi$: angle Permalink: http://dlmf.nist.gov/19.11.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.11(iii), §19.11 and Ch.19