# Ssi

Shifted sine integral function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

```Ssi(`x`)
```

## Description

`Ssi(x)` represents the shifted sine integral $\mathrm{Si}\left(x\right)-\frac{\pi }{2}$.

The special values `Ssi(0) = -π/2`, ```Ssi(∞) = 0```, `Ssi(- ∞) = -π` are implemented.

If `x` is a negative integer or a negative rational number, then `Ssi(x) = -Ssi(-x) - π`. The `Ssi` function also uses this reflection rule when 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:

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

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

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

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

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

### Example 2

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

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

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

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

### Example 3

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

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

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

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

`series(Ssi(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.

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