Formula:7786

Digital Library of Mathematical Functions ID 22.14.E18

${\displaystyle{\displaystyle\int_{0}^{K\left(k\right)}\ln\left(\operatorname{dn}\left(t,k\right)\right)\mathrm{d}t=\tfrac{1}{2}K\left(k\right)\ln k^{\prime}.}}$

Symbols List

• ${\displaystyle{\displaystyle\operatorname{dn}\left(\NVar{z},\NVar{k}\right)}}$: Jacobian elliptic function
• ${\displaystyle{\displaystyle K\left(\NVar{k}\right)}}$: 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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