# Documentation

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# sinh

Hyperbolic sine function

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## Syntax

```sinh(`x`)
```

## Description

`sinh(x)` represents the hyperbolic sine function. 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 $\frac{i\pi }{2}$ 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. See Example 2.

The special values `sinh(0) = 0`, ```sinh(∞) = ∞```, and `sinh(–∞) = –∞` are implemented.

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

You can rewrite other hyperbolic functions in terms of `sinh` and `cosh`. For example, `csch(x)` is rewritten as `1/sinh(x)`. Use `expand` or `rewrite` to rewrite expressions involving `tanh` and `coth` in terms of `sinh` and `cosh`. See Example 4.

The inverse function is implemented by `arcsinh`. 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 `sinh` with the following exact and symbolic input arguments:

`sinh(I*PI), sinh(1), sinh(5 + I), sinh(PI), sinh(1/11), sinh(8)`
``` ```
`sinh(x), sinh(x + I*PI), sinh(x^2 - 4)`
``` ```

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

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

Floating-point intervals are computed for interval arguments:

`sinh(-1...1), sinh(0...1/2)`
``` ```

### Example 2

Simplifications are implemented for arguments that are integer multiples of $\frac{i\pi }{2}$:

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

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

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

### Example 3

The `expand` function implements the addition theorems:

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

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

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

### Example 5

The inverse function is implemented as `arcsinh`:

```sinh(arcsinh(x)), arcsinh(sinh(x))```
``` ```

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

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

### Example 6

`diff`, `float`, `limit`, `taylor`, `series` and other system functions 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)`
``` ```
`taylor(sinh(x), x = 0)`
``` ```
`series((tanh(sinh(x)) - sinh(tanh(x)))/sinh(x^7), x = 0)`
``` ```

## Parameters

 `x`

## Return Values

Arithmetical expression or a floating-point interval

`x`