Definition:Incomplete Elliptic Integral of the Second Kind/Definition 2

Special Function

$\ds \map E {k, \phi} = \int \limits_0^x \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \rd v$

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

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

$x = \sin \phi$, where $\phi$ is defined on the interval $0 \le \phi \le \pi / 2$.

Also see

• Equivalence of Definitions of Incomplete Elliptic Integral of the Second Kind

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 34$: Elliptic Functions: Incomplete Elliptic Integral of the Second Kind: $34.3$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): elliptic integral
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): elliptic integral
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 35$: Elliptic Functions: Incomplete Elliptic Integral of the Second Kind: $35.3.$