# fresnelC

The Fresnel cosine integral function

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## Syntax

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

## Description

`fresnelC(z)` = .

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, , and . Symbolic function calls are returned for all other symbolic values of z. In the graphical user interface of MuPAD® symbolic function calls are typeset as `fresnelC(z)` = .

When called with floating-point arguments, the function returns floating-point values.

With `simplify` and `Simplify`, the reflection rule `fresnelC(-z) = -fresnelC(z)` is used to create a "normal form" of symbolic function calls. Cf. 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

We call the Fresnel functions with various arguments: Symbolic calls are typeset as `fresnelC(z)` = and `fresnelS(z)` = , respectively:

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

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

Floating point values are returned for floating-point arguments:

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

### Example 2

The functions `diff`, `float`, `limit`, and `series` handle expressions involving the Fresnel functions:

```diff(fresnelC(x), x), diff(fresnelS(x), x)```

```float(fresnelC(PI)), float(fresnelS(-100))```

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

```series(fresnelC(x), x = 0), series(fresnelS(x), x = infinity, 4)```

### Example 3

With `simplify` and `Simplify`, the reflection rules `fresnelC(-z) = -fresnelC(z)` and ```fresnelS(-z) = -fresnelS(z)``` are used to create a "normal form" of symbolic function calls:

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

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

`Simplify(%)`

## Parameters

 `z`

## Return Values

Arithmetical expression.

`z`