# 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.

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)

 x

## Return Values

Arithmetical expression or floating-point interval

x