# Documentation

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# fresnelS

The Fresnel sine integral function

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

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))

 z

## Return Values

Arithmetical expression.

z