gives the Fresnel integral .


  • 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.


open allclose all

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  (40)

Numerical Evaluation  (4)

Evaluate numerically to high precision:

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

Evaluate for complex arguments:

Evaluate FresnelC efficiently at high precision:

FresnelC threads elementwise over lists and matrices:

Specific Values  (3)

Value at a fixed point:

Values at infinity:

Find a local maximum as a root of (dTemplateBox[{x}, FresnelC])/(dx)=0:

Visualization  (2)

Plot the FresnelC function:

Plot the real part of TemplateBox[{{x, +, {ⅈ,  , y}}}, FresnelC]:

Plot the imaginary part of TemplateBox[{{x, +, {ⅈ,  , y}}}, FresnelC]:

Function Properties  (10)

FresnelC is defined for all real and complex values:

Approximate function range of FresnelC:

FresnelC is an odd function:

FresnelC is an analytic function of x:

FresnelC is neither non-increasing nor non-decreasing:

FresnelC is not injective:

FresnelC is not surjective:

FresnelC is neither non-negative nor non-positive:

FresnelC has no singularities or discontinuities:

Neither convex nor concave:

Differentiation  (3)

First derivative:

Higher derivatives:

Formula for the ^(th) derivative:

Integration  (3)

Indefinite integral of FresnelC:

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

More integrals:

Series Expansions  (5)

Taylor expansion for FresnelC:

Plot the first three approximations for FresnelC around :

General term in the series expansion of FresnelC:

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction :

FresnelC can be applied to power series:

Integral Transforms  (2)

Compute the Laplace transform using LaplaceTransform:


Function Identities and Simplifications  (3)

Verify an identity relating HypergeometricPFQ to FresnelC:

Simplify an integral to FresnelC:

Argument simplifications:

Function Representations  (5)

Integral representation:

Relation to the error function Erf:

FresnelC can be represented as a DifferentialRoot:

FresnelC can be represented in terms of MeijerG:

TraditionalForm formatting:

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 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:


Integral transforms:

Possible Issues  (3)

FresnelC can take large values for moderatesize arguments:

A larger setting for $MaxExtraPrecision can be needed:

Different convention can sometimes be seen in books:

Wolfram Research (1996), FresnelC, Wolfram Language function, (updated 2014).


Wolfram Research (1996), FresnelC, Wolfram Language function, (updated 2014).


@misc{reference.wolfram_2021_fresnelc, author="Wolfram Research", title="{FresnelC}", year="2014", howpublished="\url{}", note=[Accessed: 22-September-2021 ]}


@online{reference.wolfram_2021_fresnelc, organization={Wolfram Research}, title={FresnelC}, year={2014}, url={}, note=[Accessed: 22-September-2021 ]}


Wolfram Language. 1996. "FresnelC." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014.


Wolfram Language. (1996). FresnelC. Wolfram Language & System Documentation Center. Retrieved from