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