Formula:6909

Digital Library of Mathematical Functions ID 19.8.E7

${\displaystyle{\displaystyle\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right),}}$

Constraint(s)

${\displaystyle{\displaystyle-\infty<\alpha^{2}<1}}$

Symbols List

• ${\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)}}$: Legendre’s complete elliptic integral of the third kind
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle k}}$: real or complex modulus
• ${\displaystyle{\displaystyle k^{\prime}}}$: complementary modulus
• ${\displaystyle{\displaystyle\alpha^{2}}}$: real or complex parameter
• ${\displaystyle{\displaystyle Q_{n}}}$: Q n
 

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