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

Special Function

$\ds \map F {k, \phi} = \int \limits_0^x \frac {\d v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

is the incomplete elliptic integral of the first 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 $0 \le \phi \le \pi / 2$.

Amplitude

The parameter $\phi$ of $u = \map F {k, \phi}$ is called the amplitude of $u$.

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $34.1$: Incomplete Elliptic Integral of the First Kind
• 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