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

Special Function

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

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

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

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

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): $\S 34$: Elliptic Functions: Incomplete Elliptic Integral of the First Kind: $34.1$
• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 5$: Falling Bodies and Other Rate Problems
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (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 First Kind: $35.1.$