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

Special Function

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

is the incomplete elliptic integral of the second kind, and is a function of the variables:

$k$, defined on the interval $0 < k < 1$

$\phi$, defined on the interval $0 \le \phi \le \pi / 2$.

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:Complete 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): $34.3$: Incomplete Elliptic Integral of the Second Kind
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 1.5$: Falling Bodies and Other Rate Problems