# arccsch

Inverse of the hyperbolic cosecant function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```arccsch(`x`)
```

## Description

`arccsch(x)` represents the inverse of the hyperbolic cosecant function.

`arccsch` 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 inverse hyperbolic cosecant function is multivalued. MuPAD® rewrites `arccsch` as ```arccsch(x) = arcsinh(1/x)```. The MuPAD implementation for `arcsinh` returns values on the main branch defined by the following restriction of the imaginary part. For any finite complex x, $-\frac{\pi }{2}\le \Im \left(\mathrm{arcsinh}\left(x\right)\right)\le \frac{\pi }{2}$

The inverse hyperbolic cosecant function is implemented according to the following relation to the logarithm function: ```arccsch(x) = ln(1/x + sqrt(1/x^2 + 1))```. See Example 2.

Consequently, the branch cut is the interval (-i, i) on the imaginary axis. The values jump when the argument crosses a branch cut. See Example 3.

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

## Environment Interactions

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

## Examples

### Example 1

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

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

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

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

`arccsch(0.1234), arccsch(5.6 + 7.8*I), arccsch(1.0/10^20)`

Floating-point intervals are computed for interval arguments:

`arccsch(-1.5...-0.5), arccsch(0.1234...0.12345)`

### Example 2

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

`rewrite(arccsch(x), ln)`

### Example 3

The values jump when crossing a branch cut:

`arccsch(0.5*I + 1/10^10), arccsch(0.5*I - 1/10^10)`

On the branch cut, the values of `arccsch` coincide with the limit "from the left" for imaginary arguments ```x = c*i``` where `-1 < c < 0`:

```limit(arccsch(0.5*I - 1/n), n = infinity); limit(arccsch(0.5*I + 1/n), n = infinity); arccsch(0.5*I)```

The values coincide with the limit "from the right" for imaginary arguments `x = c*i` where ```0 < c < 1```:

```limit(arccsch(-0.5*I - 1/n), n = infinity); limit(arccsch(-0.5*I + 1/n), n = infinity); arccsch(-0.5*I)```

### Example 4

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

`diff(arccsch(x^2), x), float(arccsch(3)*arctanh(5 + I))`

`limit(x/arccsch(1/x), x = 0)`
```1 ```
`taylor(arccsch(1/x), x = 0)`

`series(arccsch(x), x = 0, Right)`

## Parameters

 `x`

## Return Values

Arithmetical expression or floating-point interval

`x`