Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











FresnelC






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelC[z] > Introduction to the Fresnel integrals





The best-known properties and formulas for Fresnel integrals

Real values for real arguments

For real values of argument , the values of the Fresnel integrals and are real.

Simple values at zero and infinity

The Fresnel integrals and have simple values for arguments and :

Analyticity

The Fresnel integrals and are defined for all complex values of , and they are analytical functions of over the whole complex ‐plane and do not have branch cuts or branch points. They are entire functions with an essential singular point at .

Periodicity

The Fresnel integrals and do not have periodicity.

Parity and symmetry

The Fresnel integrals and are odd functions and have mirror symmetry:

Series representations

The Fresnel integrals and have rather simple series representations at the origin:

These series converge at the whole ‐plane and their symbolic forms are the following:

Interestingly, closed-form expressions for the truncated version of the Taylor series at the origin can be expressed through the generalized hypergeometric function , for example:

Asymptotic series expansions

The asymptotic behavior of the Fresnel integrals and can be described by the following formulas (only the main terms of asymptotic expansion are given):

The previous formulas are valid in any directions of approaching point to infinity (). In particular cases when and , the formulas can be simplified to the following relations:

Integral representations

The Fresnel integrals and have the following simple integral representations through sine or cosine that directly follow from the definition of these integrals:

Transformations

The argument of the Fresnel integrals and with square root arguments can sometimes be simplified:

Simple representations of derivatives

The derivatives of the Fresnel integrals and are the sine or cosine functions with simple arguments:

The symbolic derivatives of the order have the following representations:

Simple of differential equations

The Fresnel integrals and satisfy the following third-order linear ordinary differential equation:

They can be represented as partial solutions of the previous equation under the following corresponding initial conditions:





© 1998-2014 Wolfram Research, Inc.