# Si

Sine integral function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```Si(`x`)
```

## Description

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

If `x` is a floating-point number, then `Si(x)` returns floating-point results. The special values `Si(0) = 0`, ```Si(∞) = π/2```, and `Si(-∞) = -π/2` are implemented. For all other arguments, `Si` returns symbolic function calls.

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

The float attribute of `Si` is a kernel function, 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

Most calls with exact arguments return themselves unevaluated:

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

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

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

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

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

### Example 2

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

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

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

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

### Example 3

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

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

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

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

`series(Si(x), x = infinity, 3)`

## Parameters

 `x`

## Return Values

Arithmetical expression.

`x`

## Algorithms

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

`Ssi(x) = Si(x) - π` for all `x` in the complex plane.

`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).

## See Also

### MuPAD Functions

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