# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English verison of the page.

# `arccoth`

Inverse of the hyperbolic cotangent function

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

## Syntax

```arccoth(`x`)
```

## Description

`arccoth(x)` represents the inverse of the hyperbolic cotangent function.

`arccoth` is defined for complex arguments.

Floating-point values are returned for floating-point arguments. Floating-point intervals are returned for floating-point interval arguments. Unevaluated function calls are returned for most exact arguments.

The following special value is implemented: $\mathrm{arccoth}\left(0\right)=\frac{i\pi }{2}$.

The inverse hyperbolic cotangent function is multivalued. The MuPAD® implementation returns values on the main branch defined by the following restriction of the imaginary part. For any finite complex x, $-\frac{\pi }{2}<\Im \left(\mathrm{arccoth}\left(x\right)\right)\le \frac{\pi }{2}$.

`arccoth` is defined by ```arccoth(x) = arctanh(1/x)```. However, MuPAD does not automatically rewrite it in terms of `arctanh`.

The inverse hyperbolic tangent function is implemented according to the following relation to the logarithm function: ```arccoth(x) = (ln(1 + 1/x) - ln(1 - 1/x))/2```. See Example 2.

Consequently, the branch cut is the real interval [-1, 1]. The values jump when the argument crosses a branch cut. See Example 3.

The float attributes are kernel functions, thus, floating-point evaluation is fast.

## Environment Interactions

When called with a floating-point argument, `arccoth` is sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Call `arccoth` with the following exact and symbolic input arguments:

```arccoth(-3), arccoth(3/sqrt(3)), arccoth(5 + I), arccoth(1/3), arccoth(I), arccoth(2)```

`arccoth(-x), arccoth(x + 1), arccoth(1/x)`

Floating-point values are computed for floating-point arguments:

`arccoth(-1.1234), arccoth(5.6 + 7.8*I), arccoth(1.0/10^20)`

Floating-point intervals are computed for interval arguments:

`arccoth(-1.5...-1.1), arccoth(1.1234...1.12345)`

### Example 2

The inverse hyperbolic cotangent function can be rewritten in terms of the logarithm function:

`rewrite(arccoth(x), ln)`

### Example 3

The values jump when crossing a branch cut:

`arccoth(0.5 + I/10^10), arccoth(0.5 - I/10^10)`

On the branch cut, the values of `arccoth` coincide with the limit “from above” for real arguments ```0 < x < 1```:

```limit(arccoth(0.5 - I/n), n = infinity); limit(arccoth(0.5 + I/n), n = infinity); arccoth(0.5)```

The values coincide with the limit “from below” for real `-1 < x < 0`:

```limit(arccoth(-0.5 - I/n), n = infinity); limit(arccoth(-0.5 + I/n), n = infinity); arccoth(-0.5)```

### Example 4

`diff`, `float`, `limit`, `taylor`, `series`, and other system functions handle expressions involving the inverse hyperbolic functions:

`diff(arccoth(x^2), x), float(arccosh(3)*arccoth(5 + I))`

`limit(1/arccoth(sin(x)/x), x = 0)`

`taylor(arccoth(1/x), x = 0)`

`series(arccoth(x), x = 0)`

## Parameters

 `x`

## Return Values

Arithmetical expression or floating-point interval

`x`