Formula:7785

Digital Library of Mathematical Functions ID 22.14.E17

${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{cn}\left(t,k\right)\right)\mathrm{d}t=-\tfrac{1}{4}{K^{\prime}}\left(k\right)+\tfrac{1}{2}K\left(k\right)\ln\left(k^{\prime}/k\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\operatorname{cn}\left(\NVar{z},\NVar{k}\right)}}$: Jacobian elliptic function
• ${\displaystyle{\displaystyle{K^{\prime}}\left(\NVar{k}\right)}}$: Legendre’s complementary complete elliptic integral of the first kind
• ${\displaystyle{\displaystyle K\left(\NVar{k}\right)}}$: Legendre’s complete elliptic integral of the first kind
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle k}}$: modulus
• ${\displaystyle{\displaystyle k^{\prime}}}$: complementary modulus
 

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