# ellipk
> Compute the [complete elliptic integral of the first kind][elliptic-integral].
The [complete elliptic integral of the first kind][elliptic-integral] is defined as
where the parameter `m` is related to the modulus `k` by `m = k^2`.
## Usage
```javascript
var ellipk = require( '@stdlib/math/base/special/ellipk' );
```
#### ellipk( m )
Computes the [complete elliptic integral of the first kind][elliptic-integral].
```javascript
var v = ellipk( 0.5 );
// returns ~1.854
v = ellipk( -1.0 );
// returns ~1.311
v = ellipk( 2.0 );
// returns NaN
v = ellipk( Infinity );
// returns NaN
v = ellipk( -Infinity );
// returns NaN
v = ellipk( NaN );
// returns NaN
```
## Notes
- This function is valid for `-∞ < m <= 1`.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var ellipk = require( '@stdlib/math/base/special/ellipk' );
var m;
var i;
for ( i = 0; i < 100; i++ ) {
m = -1.0 + ( randu() * 2.0 );
console.log( 'ellipk(%d) = %d', m, ellipk( m ) );
}
```
* * *
## References
- Fukushima, Toshio. 2009. "Fast computation of complete elliptic integrals and Jacobian elliptic functions." _Celestial Mechanics and Dynamical Astronomy_ 105 (4): 305. doi:[10.1007/s10569-009-9228-z][@fukushima:2009a].
- Fukushima, Toshio. 2015. "Precise and fast computation of complete elliptic integrals by piecewise minimax rational function approximation." _Journal of Computational and Applied Mathematics_ 282 (July): 71–76. doi:[10.1016/j.cam.2014.12.038][@fukushima:2015a].
[elliptic-integral]: https://en.wikipedia.org/wiki/Elliptic_integral
[@fukushima:2009a]: https://doi.org/10.1007/s10569-009-9228-z
[@fukushima:2015a]: https://doi.org/10.1016/j.cam.2014.12.038
