Definition:Elliptic Integral of the First Kind

Complete Elliptical Integral of the First Kind
The integral:
 * $\displaystyle \int \limits_0^{\pi / 2} {\frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi}}}$

is known as the Complete Elliptical Integral of the First Kind, and is a function of $k$, defined on the interval $0 < k < 1$.

It is denoted $K \left({k}\right)$.

Incomplete Elliptical Integral of the First Kind
The integral:


 * $\displaystyle \int_0^{\phi} {\frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi}}}$

is known as 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 $0 \le \phi \le \pi / 2$.

It is denoted $F \left({k, \phi}\right)$.

Note that $\displaystyle F \left({k, \frac \pi 2}\right) = K \left({k}\right)$.