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# csch

Hyperbolic cosecant function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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.

Arguments that are integer multiples of lead to simplified results. If the argument involves a negative numerical factor of Type::Real, then symmetry relations are used to make this factor positive. Cf. Example 2.

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

csch(x) is rewritten as 1/sinh(x). Cf. Example 4.

The inverse function is implemented by arccsch. Cf. Example 5.

The float attributes are kernel functions, i.e., 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

We demonstrate some calls with exact and symbolic input data:

`sinh(I*PI), cosh(1), tanh(5 + I), csch(PI), sech(1/11), coth(8)`

`sinh(x), cosh(x + I*PI), tanh(x^2 - 4)`

Floating point values are computed for floating-point arguments:

`sinh(123.4), cosh(5.6 + 7.8*I), coth(1.0/10^20)`

For floating-point intervals, intervals enclosing the image are calculated:

`cosh(-1 ... 1), tanh(-1 ... 1)`

For functions with discontinuities, evaluation over an interval may result in a union of intervals:

`coth(-1 ... 1)`

### Example 2

Simplifications are implemented for arguments that are integer multiples of :

```sinh(I*PI/2), cosh(40*I*PI), tanh(-10^100*I*PI),
coth(-17/2*I*PI)```

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

`sinh(-5), cosh(-3/2*x), tanh(-x*PI/12), coth(-12/17*x*y*PI)`

### Example 3

The expand function implements the addition theorems:

`expand(sinh(x + PI*I)), expand(cosh(x + y))`

The combine function uses these theorems in the other direction, trying to rewrite products of hyperbolic functions:

`combine(sinh(x)*sinh(y), sinhcosh)`

### Example 4

Various relations exist between the hyperbolic functions:

`csch(x), sech(x)`

Use rewrite to obtain a representation in terms of a specific target function:

`rewrite(tanh(x)*exp(2*x), sinhcosh), rewrite(sinh(x), tanh)`

`rewrite(sinh(x)*coth(y), exp), rewrite(exp(x), coth)`

### Example 5

The inverse functions are implemented by arcsinh, arccosh etc.:

`sinh(arcsinh(x)), sinh(arccosh(x)), cosh(arctanh(x))`

Note that arcsinh(sinh(x)) does not necessarily yield x, because arcsinh produces values with imaginary parts in the interval :

`arcsinh(sinh(3)), arcsinh(sinh(1.6 + 100*I))`

### Example 6

Various system functions such as diff, float, limit, or series handle expressions involving the hyperbolic functions:

`diff(sinh(x^2), x), float(sinh(3)*coth(5 + I))`

`limit(x*sinh(x)/tanh(x^2), x = 0)`

`series((tanh(sinh(x)) - sinh(tanh(x)))/sinh(x^7), x = 0)`

`series(tanh(x), x = infinity)`

 x

## Return Values

Arithmetical expression or a floating-point interval