DLMF:19.7.E9 (Q6204)

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Language	Label	Description	Also known as
English	DLMF:19.7.E9	No description defined

Statements

Defining formula

{\displaystyle{\displaystyle\Pi\left(\phi,\alpha^{2},k\right)+\Pi\left(\phi,% \omega^{2},k\right)(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(k^{2}-% \omega^{2})\Pi\left(\phi,\omega^{2},k\right)=k^{2}F\left(\phi,k\right)-\alpha^% {2}\omega^{2}\sqrt{c-1}R_{C}\left(c(c-k^{2}),(c-\alpha^{2})(c-\omega^{2})% \right),}}

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DLMF:19.7.E9

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Symbols defined

change of variable (locally)

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{\displaystyle{\displaystyle(1-\alpha^{2})(1-\omega^{2})=1-k^{2}}}

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Carlson’s combination of inverse circular and inverse hyperbolic functions

Defining formula

{\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}

C19.S2.E17.m2aadec

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Legendre’s incomplete elliptic integral of the first kind

Defining formula

{\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}

C19.S2.E4.m2addec

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Legendre’s incomplete elliptic integral of the third kind

Defining formula

{\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}% \right)}}

C19.S2.E7.m2addec

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Defining formula

{\displaystyle{\displaystyle\phi}}

C19.S1.XMD4.m1edec

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Defining formula

{\displaystyle{\displaystyle k}}

C19.S1.XMD5.m1hdec

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Defining formula

{\displaystyle{\displaystyle\alpha^{2}}}

C19.S1.XMD7.m1edec

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Defining formula

{\displaystyle{\displaystyle c}}

C19.S7.XMD9.m1adec

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