Definition:Elliptic Integral of the Second Kind/Complete/Definition 1

Special Function

$\displaystyle E \left({k}\right) = \int \limits_0^{\pi / 2} \sqrt{1 - k^2 \sin^2 \phi} \, \mathrm d \phi$

is the complete elliptic integral of the second kind, and is a function of $k$, defined on the interval $0 < k < 1$.

Also see

• Equivalence of Definitions of Elliptic Integral of the Second Kind

• Definition:Incomplete Elliptic Integral of the First Kind
• Definition:Complete Elliptic Integral of the First Kind

• Definition:Incomplete Elliptic Integral of the Second Kind

• Definition:Incomplete Elliptic Integral of the Third Kind
• Definition:Complete Elliptic Integral of the Third Kind

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $34.4$: Complete Elliptic Integral of the Second Kind