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

Special Function

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

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

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

$n \in \Z$

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

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 34$: Elliptic Functions: Incomplete Elliptic Integral of the Third Kind: $34.5$
• 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 Third Kind: $35.5.$