Formula:6808

Digital Library of Mathematical Functions ID 19.2.E16

${\displaystyle{\displaystyle\mathrm{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}\frac{\mathrm{d}\theta}{({\cos^{2}}\theta+p{\sin^{2}}\theta)\sqrt{{\cos^{2}}\theta+k_{c}^{2}{\sin^{2}}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),}}$

Constraint(s)

${\displaystyle{\displaystyle x^{2}\neq-1/p}}$

Symbols List

• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)}}$: Legendre’s incomplete elliptic integral of the third kind
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}$:
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle k}}$: real or complex modulus
• ${\displaystyle{\displaystyle p}}$: real parameter
• ${\displaystyle{\displaystyle k_{c}}}$: real parameter not equal to zero
• ${\displaystyle{\displaystyle x}}$: real or complex variable not equal to zero
 

Resources that cite this formula

No records found.