DLMF:19.5.E1

MaRDI QIDQ6826

• Label: DLMF:19.5.E1

Digital Library of Mathematical Functions ID 19.5.E1

${\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left(\tfrac{1}{2}\right)_{m}}}{m!\;m!}k^{2m}=\frac{\pi}{2}{{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right),}}$

Symbols List

• ${\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$: = F ( a , b ; c ; z ) notation for Gauss’ hypergeometric function
• ${\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}$:
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle K\left(\NVar{k}\right)}}$: Legendre’s complete elliptic integral of the first kind
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle m}}$: nonnegative integer
• ${\displaystyle{\displaystyle k}}$: real or complex modulus
 

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