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.
Examples
open allclose allBasic Examples (5)
Plot over a subset of the reals:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (32)
Numerical Evaluation (4)
Visualization (2)
Function Properties (3)
Integration (3)
Indefinite integral of FresnelS:
Definite integral of an odd function over an interval centered at the origin is 0:
Series Expansions (5)
Integral Transforms (2)
Function Identities and Simplifications (2)
Function Representations (5)
Relation to the error function Erf:
FresnelS can be represented as a DifferentialRoot:
FresnelS can be represented in terms of MeijerG:
TraditionalForm formatting:
Applications (4)
Intensity of a wave diffracted by a half‐plane:
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:
Properties & Relations (6)
Use FullSimplify to simplify expressions containing Fresnel integrals:
Obtain FresnelS from integrals and sums:
Solve a differential equation:
Compare with Wronskian:
Possible Issues (2)
FresnelS can take large values for moderate‐size arguments:
Text
Wolfram Research (1996), FresnelS, Wolfram Language function, https://reference.wolfram.com/language/ref/FresnelS.html.
BibTeX
BibLaTeX
CMS
Wolfram Language. 1996. "FresnelS." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FresnelS.html.
APA
Wolfram Language. (1996). FresnelS. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FresnelS.html