BUILT-IN MATHEMATICA SYMBOL

FresnelC

FresnelC[z]
gives the Fresnel integral

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MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

FresnelC[z] is given by .

FresnelC[z] is an entire function of z with no branch cut discontinuities.

For certain special arguments, FresnelC automatically evaluates to exact values.

FresnelC can be evaluated to arbitrary numerical precision.

FresnelC automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

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Scope (7)

Evaluate for complex arguments:

Evaluate to high precision:

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

Simple exact values are generated automatically:

Parity transformation is automatically applied:

FresnelC threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (2)

FresnelC can be applied to power series:

Find series expansions at infinity:

Applications (4)

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 FresnelC along a circle in the complex plane:

Properties & Relations (6)

Use FullSimplify to simplify expressions containing Fresnel integrals:

Find a numerical root:

Obtain FresnelC from integrals and sums:

Solve a differential equation:

Calculate the Wronskian:

Compare with Wronskian:

Integrals:

Integral transforms:

Possible Issues (3)

FresnelC can take large values for moderate-size arguments:

A larger setting for $MaxExtraPrecision can be needed:

Different convention can sometimes be seen in books: