# Documentation

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# acoth

Inverse hyperbolic cotangent

## Syntax

`Y = acoth(X)`

## Description

`Y = acoth(X)` returns the inverse hyperbolic cotangent for each element of `X`.

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

## Examples

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Graph the inverse hyperbolic cotangent function over the domains and .

```x1 = -30:0.1:-1.1; x2 = 1.1:0.1:30; plot(x1,acoth(x1),x2,acoth(x2)) grid on xlabel('x') ylabel('y') ```

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### Inverse Hyperbolic Cotangent

For real values $x$ in the domain $-\text{\hspace{0.17em}}\infty and $1, the inverse hyperbolic cotangent satisfies

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

For complex numbers $z=x+iy$ as well as real values in the domain $-1\le z\le 1$, the call `acoth(z)` returns complex results.

### Tall Array Support

This function fully supports tall arrays. For more information, see Tall Arrays.