The complete elliptic integral of the first
kind is

$$[K(m)]={\displaystyle {\int}_{0}^{1}{[(1-{t}^{2})(1-m{t}^{2})]}^{-\frac{1}{2}}}dt.$$

where *m* is the first argument of `ellipke`

.

The complete elliptic integral of the second kind is

$$E(m)={\displaystyle {\int}_{0}^{1}(}1-{t}^{2}{)}^{-\frac{1}{2}}{(1-m{t}^{2})}^{\frac{1}{2}}dt.$$

Some definitions of the elliptic functions use the elliptical
modulus *k* or modular angle *α* instead
of the parameter *m*. They are related by

$${k}^{2}=m={\mathrm{sin}}^{2}\alpha .$$