Formula:6901

Digital Library of Mathematical Functions ID 19.7.E10

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

Constraint(s)

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

Symbols List

• ${\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}$: Carlson’s combination of inverse circular and inverse hyperbolic functions
• ${\displaystyle{\displaystyle F\left(\NVar{\phi},\NVar{k}\right)}}$: Legendre’s incomplete elliptic integral of the first kind
• ${\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)}}$: Legendre’s incomplete elliptic integral of the third kind
• ${\displaystyle{\displaystyle\phi}}$: real or complex argument
• ${\displaystyle{\displaystyle k}}$: real or complex modulus
• ${\displaystyle{\displaystyle\alpha^{2}}}$: real or complex parameter
• ${\displaystyle{\displaystyle c}}$: change of variable
 

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