# fresnels

Fresnel sine integral function

## Syntax

``fresnels(z)``

## Description

example

````fresnels(z)` returns the Fresnel sine integral of `z`.```

## Examples

### Fresnel Sine Integral Function for Numeric and Symbolic Arguments

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

`fresnels([-2 0.001 1.22+0.31i])`
```ans = -0.3434 + 0.0000i 0.0000 + 0.0000i 0.7697 + 0.2945i```

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

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

Use `vpa` to approximate the results:

`vpa(y)`
```ans = [ -0.34341567836369824219530081595807, 0.00000000052359877559820659249174920261227,... 0.76969209233306959998384249252902 + 0.29449530344285433030167256417637i]```

### Fresnel Sine Integral for Special Values

Find the Fresnel sine integral function for special values:

`fresnels([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 Sine Integral for Symbolic Functions

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

```syms x f = symfun(exp(x)+2*x,x); fresnels(f)```
```ans(x) = fresnels(2*x + exp(x))```

### Fresnel Sine Integral for Symbolic Vectors and Arrays

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

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

### Plot Fresnel Sine Integral Function

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

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

### Differentiate and Find Limits of Fresnel Sine Integral

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

Find the fourth derivative of the Fresnel sine integral function:

```syms x diff(fresnels(x),x,4)```
```ans = - 3*x*pi^2*sin((pi*x^2)/2) - x^3*pi^3*cos((pi*x^2)/2)```

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

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

### Taylor Series Expansion of Fresnel Sine Integral

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

```syms x taylor(fresnels(x))```
```ans = (pi*x^3)/6```

### Simplify Expressions Containing fresnels

Use `simplify` to simplify expressions:

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

## Input Arguments

collapse all

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

collapse all

### Fresnel Sine Integral

The Fresnel sine integral of z is

`$\mathrm{fresnels}\left(z\right)={\int }_{0}^{z}\mathrm{sin}\left(\frac{\pi {t}^{2}}{2}\right)dt$`

.

## Algorithms

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

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