# Polygamma

Polygamma function.

## Usage

``````var polygamma = require( '@stdlib/math/base/special/polygamma' );
``````

#### polygamma( n, x )

Evaluates the polygamma function of order `n`; i.e., the `(n+1)`th derivative of the natural logarithm of the gamma function.

``````var v = polygamma( 3, 1.2 );
// returns ~3.245

v = polygamma( 5, 1.2 );
// returns ~41.39

v = polygamma( 3, -4.9 );
// returns ~60014.239
``````

If `n` is not a nonnegative `integer`, the function returns `NaN`.

``````var v = polygamma( 2.5, -1.2 );
// returns NaN

v = polygamma( -1, 5.3 );
// returns NaN
``````

If `x` is `0` or a negative odd `integer`, the function returns `+Infinity`.

``````var v = polygamma( 2, 0.0 );
// returns +Infinity

v = polygamma( 2, -1.0 );
// returns +Infinity
``````

If `x` on the other hand is a negative even `integer`, the function returns `NaN`.

``````v = polygamma( 2, -4.0 );
// returns NaN

v = polygamma( 2, -2.0 );
// returns NaN
``````

If provided `NaN` as either parameter, the function returns `NaN`.

``````var v = polygamma( NaN, 2.1 );
// returns NaN

v = polygamma( 1, NaN );
// returns NaN

v = polygamma( NaN, NaN );
// returns NaN
``````

## Examples

``````var randu = require( '@stdlib/random/base/randu' );
var round = require( '@stdlib/math/base/special/round' );
var polygamma = require( '@stdlib/math/base/special/polygamma' );

var n;
var x;
var v;
var i;

for ( i = 0; i < 100; i++ ) {
x = (randu()*100.0) - 50.0;
n = round( randu()*50.0 );
v = polygamma( x, n );
console.log( 'x: %d, ψ^(%d)(x): %d', x, n, v );
}
``````