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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix FresnelS function using MatrixFunction:

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:

MellinTransform:

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:

TraditionalForm formatting:

Applications (5)

Intensity of a wave diffracted by a half‐plane:

Plot a Cornu spiral:

A solution of the time‐dependent 1D Schrödinger equation for a sudden opening of a shutter:

Check the Schrödinger equation:

Plot the time‐dependent solution:

Plot of FresnelS along a circle in the complex plane:

Fractional derivative of Sin:

Derivative of order of Sin:

Plot a smooth transition between the derivative and integral of Sin:

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 moderate‐size arguments:

Some references use a different convention for the Fresnel integrals:

Neat Examples (1)

Nested integrals:

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