# Documentation

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# `fresnelC`

The Fresnel cosine integral function

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## Syntax

```fresnelC(`z`)
```

## Description

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

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

`fresnelC(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 $C\left(z\right)$.

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

`simplify` and `Simplify`, `fresnelC` uses the reflection rule `fresnelC(-z) = -fresnelC(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 cosine integral function with various arguments:

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

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

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

### Example 2

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

`diff(fresnelC(x), x)`

`float(fresnelC(PI))`

`limit(fresnelC(x), x = infinity)`

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

### Example 3

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

```simplify(fresnelC(1 - x)), Simplify(fresnelC(x - 1))```

## Parameters

 `z`

## Return Values

Arithmetical expression.

`z`