Fresnel Integrals

Author

Eric W. Weisstein
May 1, 2015

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/FresnelIntegrals.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FresnelIntegrals.html.

©2015 Wolfram Research, Inc. except for portions noted otherwise

Plots

Real Axis

V5

Show[GraphicsArray[
Plot[#[x],{x,-5,5},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,#[x]},
DisplayFunction->Identity]&/@{FresnelC,FresnelS}
]]//Timing

V6

g1=Plot[#[x],{x,-5,5},PlotStyle->Red,AxesLabel->{x,#[x]},LabelStyle->(FontFamily->"Times")]&/@{FresnelC,FresnelS}

Complex Plane

FresnelC

ComplexPlot3D[FresnelC[z],{z,-2,2},PlotPoints->100]//Timing

g2=Plot3D[#[FresnelC[x+I y]],{x,-2,2},{y,-2,2},ClippingStyle->None,Mesh->None,AxesLabel->{x,y},PlotLabel->#[FresnelC[z]],LabelStyle->(FontFamily->"Times")]&/@{Re,Im,Abs}

ComplexContourPlot[FresnelC[z],{z,-2,2},PlotPoints->100]//Timing

ComplexContourPlot[FresnelC[z],{z,-5,5},PlotPoints->100]//Timing

FresnelS

ComplexPlot3D[FresnelS[z],{z,-2,2},PlotPoints->100]//Timing

g3=Plot3D[#[FresnelS[x+I y]],{x,-2,2},{y,-2,2},ClippingStyle->None,Mesh->None,AxesLabel->{x,y},PlotLabel->#[FresnelS[z]],LabelStyle->(FontFamily->"Times")]&/@{Re,Im,Abs}

ComplexContourPlot[FresnelS[z],{z,-2,2},PlotPoints->100]//Timing

ComplexContourPlot[FresnelS[z],{z,-5,5},PlotPoints->100]//Timing

Combined Plot

Column[Row/@{g1,g2,g3},Alignment->Center,Spacings->1.2]

Definitions

Integrate[Cos[π (x^2)/2],{x,0,u}]

Integrate[Sin[π (x^2)/2],{x,0,u}]

Special Values

FresnelC/@{-∞,0,∞}

FresnelS/@{-∞,0,∞}

Related Functions

Sqrt[2/π] Integrate[Cos[t^2],{t,0,z}]

Sqrt[2/π] Integrate[Sin[t^2],{t,0,z}]

1/Sqrt[2π] Integrate[Cos[t]/Sqrt[t],{t,0,z},Assumptions->z>0]

1/Sqrt[2π] Integrate[Sin[t]/Sqrt[t],{t,0,z},Assumptions->z>0]