# Dirichlet Eta Function
> [Dirichlet eta][eta-function] function.
The [Dirichlet eta][eta-function] function is defined by the [Dirichlet series][dirichlet-series]
where `s` is a complex variable equal to `σ + ti`. The series is convergent for all complex numbers having a real part greater than `0`.
Note that the [Dirichlet eta][eta-function] function is also known as the **alternating zeta function** and denoted `ζ*(s)`. The series is an alternating sum corresponding to the Dirichlet series expansion of the [Riemann zeta][@stdlib/math/base/special/riemann-zeta] function. Accordingly, the following relation holds:
where `ζ(s)` is the [Riemann zeta][@stdlib/math/base/special/riemann-zeta] function.
## Usage
```javascript
var eta = require( '@stdlib/math/base/special/dirichlet-eta' );
```
#### eta( s )
Evaluates the [Dirichlet eta][eta-function] function as a function of a real variable `s`.
```javascript
var v = eta( 0.0 ); // Abel sum of 1-1+1-1+...
// returns 0.5
v = eta( -1.0 ); // Abel sum of 1-2+3-4+...
// returns 0.25
v = eta( 1.0 ); // alternating harmonic series => ln(2)
// returns 0.6931471805599453
v = eta( 3.14 );
// returns ~0.9096
v = eta( NaN );
// returns NaN
```
## Examples
```javascript
var linspace = require( '@stdlib/array/linspace' );
var eta = require( '@stdlib/math/base/special/dirichlet-eta' );
var s = linspace( -50.0, 50.0, 200 );
var v;
var i;
for ( i = 0; i < s.length; i++ ) {
v = eta( s[ i ] );
console.log( 's: %d, η(s): %d', s[ i ], v );
}
```
[eta-function]: https://en.wikipedia.org/wiki/Dirichlet_eta_function
[dirichlet-series]: https://en.wikipedia.org/wiki/Dirichlet_series
[@stdlib/math/base/special/riemann-zeta]: https://www.npmjs.com/package/@stdlib/math/tree/main/base/special/riemann-zeta
