# Documentation

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# `csch`

Hyperbolic cosecant function

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## Syntax

```csch(`x`)
```

## Description

`csch(x)` represents the hyperbolic cosecant function, `1/sinh(x)`. This function 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 hyperbolic cosecant function simplifies to ${\left(-1\right)}^{n+1/2}$ at the points $\frac{\pi i}{2}+\pi in$, where n is an integer. The hyperbolic cosecant function has singularities at the points $i\pi n$, where n is an integer. If the argument involves a negative numerical factor of `Type::Real`, then symmetry relations are used to make this factor positive. See Example 2.

The functions `expand` and `combine` implement the addition theorems for the hyperbolic functions. See Example 3.

`csch(x)` is rewritten as `1/sinh(x)`. Use `expand` or `rewrite` to rewrite expressions involving `csch` in terms of other functions. See Example 4.

The inverse function is implemented as `arccsch`. See Example 5.

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

## Environment Interactions

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

## Examples

### Example 1

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

`csch(I*PI/2), csch(1), csch(5 + I), csch(PI), csch(1/11), csch(8)`

`csch(x), csch(x + I*PI), csch(x^2 - 4)`

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

`csch(1.234), csch(5.6 + 7.8*I), csch(1.0/10^20)`

Floating-point intervals are computed for interval arguments:

`csch(-1...-1/2), csch(1...10)`

For functions with discontinuities, evaluation over an interval can return in a union of intervals:

`csch(-1...1)`

### Example 2

The hyperbolic cosecant function equals simplifies to ${\left(-1\right)}^{n+1/2}$ at the points , where n is an integer:

`assume(n in Z_)`
`simplify(csch((n - 1/2)*I*PI))`

`delete n`

Negative real numerical factors in the argument are rewritten via symmetry relations:

`csch(-5), csch(-3/2*x), csch(-x*PI/12), csch(-12/17*x*y*PI)`

### Example 3

The `expand` function implements the addition theorems:

`expand(csch(x + PI*I)), expand(csch(x + y))`

### Example 4

`csch(x)` is automatically rewritten as `1/sinh(x)`:

`csch(x)`

Use `rewrite` to obtain a representation in terms of other target functions:

`rewrite(csch(x)*exp(2*x), sinhcosh), rewrite(csch(x), exp)`

`rewrite(csch(x)*coth(y), sincos), rewrite(csch(x), tanh)`

### Example 5

The inverse function is implemented as `arccsch`:

```csch(arccsch(x)), arccsch(csch(x))```

Note that `arccsch(csch(x))` does not necessarily yield `x`, because `arccsch` produces values with imaginary parts in the interval $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$:

`arccsch(csch(3)), arccsch(csch(1.6 + 100*I))`

### Example 6

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

`diff(csch(x), x), float(csch(3)*coth(5 + I))`

`limit(x*csch(x)/cosh(x^2), x = 0)`

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

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

## Parameters

 `x`

## Return Values

Arithmetical expression or a floating-point interval

`x`