Formula:6899

Digital Library of Mathematical Functions ID 19.7.E8

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

Constraint(s)

${\displaystyle{\displaystyle\alpha^{2}\omega^{2}=k^{2}}}$

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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