# fresnelc

Fresnel cosine integral function

## Syntax

``fresnelc(z)``

## Description

example

````fresnelc(z)` returns the Fresnel cosine integral of `z`.```

## Examples

### Fresnel Cosine Integral Function for Numeric and Symbolic Input Arguments

Find the Fresnel cosine integral function for these numbers. Since these are not symbolic objects, you receive floating-point results.

`fresnelc([-2 0.001 1.22+0.31i])`
```ans = -0.4883 + 0.0000i 0.0010 + 0.0000i 0.8617 - 0.2524i ```

Find the Fresnel cosine integral function symbolically by converting the numbers to symbolic objects:

`y = fresnelc(sym([-2 0.001 1.22+0.31i]))`
```y = [ -fresnelc(2), fresnelc(1/1000), fresnelc(61/50 + 31i/100)]```

Use `vpa` to approximate results:

`vpa(y)`
```ans = [ -0.48825340607534075450022350335726, 0.00099999999999975325988997279422003,... 0.86166573430841730950055370401908 - 0.25236540291386150167658349493972i]```

### Fresnel Cosine Integral Function for Special Values

Find the Fresnel cosine integral function for special values:

`fresnelc([0 Inf -Inf i*Inf -i*Inf])`
```ans = 0.0000 + 0.0000i 0.5000 + 0.0000i -0.5000 + 0.0000i... 0.0000 + 0.5000i 0.0000 - 0.5000i```

### Fresnel Cosine Integral for Symbolic Functions

Find the Fresnel cosine integral for the function ```exp(x) + 2*x```:

```syms f(x) f = exp(x)+2*x; fresnelc(f)```
```ans = fresnelc(2*x + exp(x))```

### Fresnel Cosine Integral for Symbolic Vectors and Arrays

Find the Fresnel cosine integral for elements of vector `V` and matrix `M`:

```syms x V = [sin(x) 2i -7]; M = [0 2; i exp(x)]; fresnelc(V) fresnelc(M)```
```ans = [ fresnelc(sin(x)), fresnelc(2i), -fresnelc(7)] ans = [ 0, fresnelc(2)] [ fresnelc(1i), fresnelc(exp(x))]```

### Plot Fresnel Cosine Integral Function

Plot the Fresnel cosine integral function from `x = -5` to `x = 5`.

```syms x fplot(fresnelc(x),[-5 5]) grid on```

### Differentiate and Find Limits of Fresnel Cosine Integral

The functions `diff` and `limit` handle expressions containing `fresnelc`.

Find the third derivative of the Fresnel cosine integral function:

```syms x diff(fresnelc(x),x,3)```
```ans = - pi*sin((pi*x^2)/2) - x^2*pi^2*cos((pi*x^2)/2)```

Find the limit of the Fresnel cosine integral function as x tends to infinity:

```syms x limit(fresnelc(x),Inf)```
```ans = 1/2```

### Taylor Series Expansion of Fresnel Cosine Integral

Use `taylor` to expand the Fresnel cosine integral in terms of the Taylor series:

```syms x taylor(fresnelc(x))```
```ans = x - (x^5*pi^2)/40```

### Simplify Expressions Containing fresnelc

Use `simplify` to simplify expressions:

```syms x simplify(3*fresnelc(x)+2*fresnelc(-x))```
```ans = fresnelc(x)```

## Input Arguments

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Upper limit on the Fresnel cosine integral, specified as a numeric value, vector, matrix, or as a multidimensional array, or a symbolic variable, expression, vector, matrix, or function.

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### Fresnel Cosine Integral

The Fresnel cosine integral of z is

`$\mathrm{fresnelc}\left(z\right)={\int }_{0}^{z}\mathrm{cos}\left(\frac{\pi {t}^{2}}{2}\right)\text{\hspace{0.17em}}dt.$`

## Algorithms

`fresnelc` is analytic throughout the complex plane. It satisfies fresnelc(-z) = -fresnelc(z), conj(fresnelc(z)) = fresnelc(conj(z)), and fresnelc(i*z) = i*fresnelc(z) for all complex values of z.

`fresnelc` returns special values for z = 0, z = ±∞, and z = ±i∞ which are 0, ±5, and ±0.5i. `fresnelc(z)` returns symbolic function calls for all other symbolic values of `z`.

## Version History

Introduced in R2014a