# Polygamma
> [Polygamma][polygamma-function] function.
## Usage
```javascript
var polygamma = require( '@stdlib/math/base/special/polygamma' );
```
#### polygamma( n, x )
Evaluates the [polygamma function][polygamma-function] of order `n`; i.e., the `(n+1)`th derivative of the [natural logarithm][@stdlib/math/base/special/ln] of the [gamma function][@stdlib/math/base/special/gamma].
```javascript
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`.
```javascript
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`.
```javascript
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`.
```javascript
v = polygamma( 2, -4.0 );
// returns NaN
v = polygamma( 2, -2.0 );
// returns NaN
```
If provided `NaN` as either parameter, the function returns `NaN`.
```javascript
var v = polygamma( NaN, 2.1 );
// returns NaN
v = polygamma( 1, NaN );
// returns NaN
v = polygamma( NaN, NaN );
// returns NaN
```
## Examples
```javascript
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 );
}
```
[polygamma-function]: https://en.wikipedia.org/wiki/Polygamma_function
[@stdlib/math/base/special/ln]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/special/ln
[@stdlib/math/base/special/gamma]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/special/gamma