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# digamma

> [Digamma][digamma-function] function.

<section class="intro">

The [digamma function][digamma-function] `ψ` is the logarithmic derivative of the [gamma function][gamma-function], i.e.

<!-- <equation class="equation" label="eq:digamma_function" align="center" raw="\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma\,'(x)}{\Gamma(x)}" alt="Digamma function"> -->

<div class="equation" align="center" data-raw-text="\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma\,&#39;(x)}{\Gamma(x)}" data-equation="eq:digamma_function">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/digamma/docs/img/equation_digamma_function.svg" alt="Digamma function">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

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

#### digamma( x )

Evaluates the [digamma function][digamma-function].

```javascript
var v = digamma( -2.5 );
// returns ~1.103

v = digamma( 1.0 );
// returns ~-0.577

v = digamma( 10.0 );
// returns ~2.252
```

If `x` is `0` or a negative `integer`, the function returns `NaN`.

```javascript
var v = digamma( 0.0 );
// returns NaN

v = digamma( -1.0 );
// returns NaN

v = digamma( -2.0 );
// returns NaN
```

If provided `NaN`, the function returns `NaN`.

```javascript
var v = digamma( NaN );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var uniform = require( '@stdlib/random/array/uniform' );
var logEachMap = require( '@stdlib/console/log-each-map' );
var digamma = require( '@stdlib/math/base/special/digamma' );

var opts = {
    'dtype': 'float64'
};
var x = uniform( 100, -5.0, 5.0, opts );

logEachMap( 'x: %0.4f, f(x): %0.4f', x, digamma );
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/digamma.h"
```

#### stdlib_base_digamma( x )

Evaluates the [digamma function][digamma-function].

```c
double out = stdlib_base_digamma( -2.5 );
// returns ~1.103

out = stdlib_base_digamma( 1.0 );
// returns ~-0.577
```

The function accepts the following arguments:

-   **x**: `[in] double` input value.

```c
double stdlib_base_digamma( const double x );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/digamma.h"
#include <stdlib.h>
#include <stdio.h>

int main( void ) {
    const double x[] = { 4.0, -1.5, -0.5, 0.5 };

    double y;
    int i;
    for ( i = 0; i < 4; i++ ) {
        y = stdlib_base_digamma( x[ i ] );
        printf( "digamma(%lf) = %lf\n", x[ i ], y );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/gamma`][@stdlib/math/base/special/gamma]</span><span class="delimiter">: </span><span class="description">gamma function.</span>
-   <span class="package-name">[`@stdlib/math/base/special/trigamma`][@stdlib/math/base/special/trigamma]</span><span class="delimiter">: </span><span class="description">trigamma function.</span>

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[digamma-function]: https://en.wikipedia.org/wiki/Digamma_function

[gamma-function]: https://en.wikipedia.org/wiki/Gamma_function

<!-- <related-links> -->

[@stdlib/math/base/special/gamma]: https://github.com/stdlib-js/math/tree/main/base/special/gamma

[@stdlib/math/base/special/trigamma]: https://github.com/stdlib-js/math/tree/main/base/special/trigamma

<!-- </related-links> -->

</section>

<!-- /.links -->
