# Documentation

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# `fresnelS`

The Fresnel sine integral function

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.

## Syntax

```fresnelS(`z`)
```

## Description

`fresnelS(z)` = $\underset{0}{\overset{z}{\int }}\mathrm{sin}\left(\frac{\pi {t}^{2}}{2}\right)dt$.

The function S = `fresnelS` is analytic throughout the complex plane. It satisfies ```fresnelS(-z) = -fresnelS(z)```, `fresnelS(conjugate(z)) = conjugate(fresnelS(z))`, ```fresnelS(I*z) =-I*fresnelS(z)``` for all complex values of z.

`fresnelS(z)` returns special values for ```z = 0```, `z = ±∞`, and ```z = ±i∞```. Symbolic function calls are returned for all other symbolic values of `z`. In a MuPAD® notebook `fresnelC(z)` appears in a typeset notation as $S\left(z\right)$.

For floating-point arguments, `fresnelS` returns floating-point values.

`simplify` and `Simplify`, `fresnelS` uses the reflection rule `fresnelS(-z) = -fresnelS(z)` to create a “normal form” of symbolic function calls. See Example 3.

## Environment Interactions

When called with floating-point arguments, these functions are sensitive to the environment variable `DIGITS` which determines the numerical working precision.

## Examples

### Example 1

Call the Fresnel sine integral function with various arguments:

```fresnelS(0), fresnelS(1), fresnelS(PI + I), fresnelS(z), fresnelS(infinity)```

For floating-point arguments, `fresnelS` returns floating-point values:

```fresnelS(1.0), fresnelS(float(PI)), fresnelS(-3.45 + 0.75*I)```

### Example 2

`diff`, `float`, `limit`, `series`, and other functions handle expressions involving the Fresnel sine integral function:

`diff(fresnelS(x), x)`

`float(fresnelS(-100))`

`limit(fresnelS(x), x = -infinity)`

`series(fresnelS(x), x = infinity, 4)`

### Example 3

`simplify` use the reflection rule `fresnelS(-z) = -fresnelS(z)` to create a “normal form” of symbolic function calls:

`simplify(3*fresnelS(z) + 2*fresnelS(-z))`

## Parameters

 `z`

## Return Values

Arithmetical expression.

`z`

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