# FresnelS

FresnelS[z]

gives the Fresnel integral .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• FresnelS[z] is given by .
• FresnelS[z] is an entire function of z with no branch cut discontinuities.
• For certain special arguments, FresnelS automatically evaluates to exact values.
• FresnelS can be evaluated to arbitrary numerical precision.
• FresnelS automatically threads over lists.
• FresnelS can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(39)

### Numerical Evaluation(5)

Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex arguments:

Evaluate FresnelS efficiently at high precision:

FresnelS threads elementwise over lists and matrices:

FresnelS can be used with Interval and CenteredInterval objects:

### Specific Values(3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of :

### Visualization(2)

Plot the FresnelS function:

Plot the real part of :

Plot the imaginary part of :

### Function Properties(9)

FresnelS is defined for all real and complex values:

Approximate function range of FresnelS:

FresnelS is an odd function:

FresnelS is an analytic function of x:

FresnelS is neither non-increasing nor non-decreasing:

FresnelS is not injective:

Not surjective:

FresnelS is neither non-negative nor non-positive:

FresnelS has no singularities or discontinuities:

Neither convex nor concave:

### Differentiation(3)

First derivative:

Higher derivatives:

Formula for the  derivative:

### Integration(3)

Indefinite integral of FresnelS:

Definite integral of an odd function over an interval centered at the origin is 0:

More integrals:

### Series Expansions(5)

Taylor expansion for FresnelS:

Plot the first three approximations for FresnelS around :

General term in the series expansion of FresnelS:

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction :

FresnelS can be applied to power series:

### Integral Transforms(2)

Compute the Laplace transform using LaplaceTransform:

### Function Identities and Simplifications(2)

Verify an identity relating HypergeometricPFQ to FresnelS:

Argument simplifications:

### Function Representations(5)

Integral representation:

Relation to the error function Erf:

FresnelS can be represented as a DifferentialRoot:

FresnelS can be represented in terms of MeijerG:

## Applications(4)

Intensity of a wave diffracted by a halfplane:

Plot a Cornu spiral:

A solution of the timedependent 1D Schrödinger equation for a sudden opening of a shutter:

Check the Schrödinger equation:

Plot the timedependent solution:

Plot of FresnelS along a circle in the complex plane:

## Properties & Relations(6)

Use FullSimplify to simplify expressions containing Fresnel integrals:

Find a numerical root:

Obtain FresnelS from integrals and sums:

Solve a differential equation:

Calculate the Wronskian:

Compare with Wronskian:

Integrals:

Integral transforms:

## Possible Issues(2)

FresnelS can take large values for moderatesize arguments:

Different convention can sometimes be seen in books:

## Neat Examples(1)

Nested integrals: