# Documentation

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# `Shi`

Hyperbolic sine integral function

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## Syntax

```Shi(`x`)
```

## Description

`Shi(x)` represents the hyperbolic sine integral $\underset{0}{\overset{x}{\int }}\frac{\mathrm{sinh}\left(t\right)}{t}dt$.

If `x` is a floating-point number, then `Shi(x)` returns floating-point results. The special values `Shi(0) = 0`, ```Shi(∞) = ∞```, `Shi(- ∞) = -∞` are implemented. For all other arguments, `Shi` returns symbolic function calls.

If `x` is a negative integer or a negative rational number, then `Shi(x) = -Shi(-x)`. The `Shi` function also uses this reflection rule when the argument is a symbolic product involving such a factor. See Example 2.

## 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

Most calls with exact arguments return themselves unevaluated:

`Shi(0), Shi(1), Shi(sqrt(2)), Shi(x + 1), Shi(infinity)`

To approximate exact results with floating-point numbers, use `float`:

`float(Shi(1)), float(Shi(sqrt(2)))`

Alternatively, use a floating-point value as an argument:

`Shi(-5.0), Shi(1.0), Shi(2.0 + 10.0*I)`

### Example 2

For negative real numbers and products involving such numbers, `Shi` applies the reflection rule `Shi(-x) = -Shi(x)`:

`Shi(-3), Shi(-3/7), Shi(-sqrt(2)), Shi(-x/7), Shi(-0.3*x)`

No such “normalization” occurs for complex numbers or arguments that are not products:

`Shi(- 3 - I), Shi(3 + I), Shi(x - 1), Shi(1 - x)`

### Example 3

`diff`, `float`, `limit`, `series`, and other functions handle expressions involving `Shi`:

`diff(Shi(x), x, x, x), float(ln(3 + Shi(sqrt(PI))))`

`limit(Shi(2*I*x^2/(1+x)), x = infinity)`

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

`series(Shi(I*x), x = infinity, 3)`

## Parameters

 `x`

## Return Values

Arithmetical expression.

`x`

## Algorithms

`Si`, `Ssi`, and `Shi` are entire functions.

`i*Si(x) = Shi(i*x)` for all `x` in the complex plane.

Reference: M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions”, Dover Publications Inc., New York (1965).