gives the Fresnel integral TemplateBox[{z}, FresnelS].


  • 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. »


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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 (dTemplateBox[{x}, FresnelS])/(dx)=0:

Visualization  (2)

Plot the FresnelS function:

Plot the real part of TemplateBox[{z}, FresnelS]:

Plot the imaginary part of TemplateBox[{z}, FresnelS]:

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 ^(th) 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:

TraditionalForm formatting:

Applications  (5)

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:

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:


Integral transforms:

Possible Issues  (2)

FresnelS can take large values for moderatesize arguments:

Some references use a different convention for the Fresnel integrals:

Neat Examples  (1)

Nested integrals:

Wolfram Research (1996), FresnelS, Wolfram Language function, (updated 2022).


Wolfram Research (1996), FresnelS, Wolfram Language function, (updated 2022).


Wolfram Language. 1996. "FresnelS." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022.


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


@misc{reference.wolfram_2024_fresnels, author="Wolfram Research", title="{FresnelS}", year="2022", howpublished="\url{}", note=[Accessed: 21-June-2024 ]}


@online{reference.wolfram_2024_fresnels, organization={Wolfram Research}, title={FresnelS}, year={2022}, url={}, note=[Accessed: 21-June-2024 ]}