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# atanh

Inverse hyperbolic tangent

## Syntax

```Y = atanh(X) ```

## Description

The `atanh` function operates element-wise on arrays. The function's domains and ranges include complex values. All angles are in radians.

`Y = atanh(X)` returns the inverse hyperbolic tangent for each element of `X`.

## Examples

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Graph the inverse hyperbolic tangent over the domain $-1.

```x = -0.99:0.01:0.99; plot(x,atanh(x)) grid on xlabel('x') ylabel('y')```

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### Inverse Hyperbolic Tangent

For real values $x$ in the domain $-1, the inverse hyperbolic tangent satisfies

`${\mathrm{tanh}}^{-1}\left(x\right)=\frac{1}{2}\mathrm{log}\left(\frac{1+x}{1-x}\right).$`

For complex numbers $z=x+iy$ as well as real values in the regions $-\text{\hspace{0.17em}}\infty and $1, the call `atanh(z)` returns complex results.