Jinc Function

Author

Eric W. Weisstein
September 28, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Axis

Plot[Jinc[x],{x,-15,15},
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,Jinc[x]},PlotRange->All
]

Complex Plane

ComplexPlot3D[Jinc[z],{z,-15-2I,15+2I}]//Timing

ComplexContourPlot[Jinc[z],{z,-15-2I,15+2I}]//Timing

ComplexPlot3D[Jinc[z],{z,-15,15},PlotPoints->100]//Timing

ComplexContourPlot[Jinc[z],{z,-15,15}]//Timing

Derivative

D[Jinc[x],x]//FullSimplify

Integral

Integrate[Jinc[x],x]//FullSimplify

Critical Points

Maximum

Maximize[Jinc[x],x]

Maximize[((Jinc[x])/x),x,Reals]

Maximize[{((Jinc[x])/x),-2<x<2},x,Reals]

Inflection Points

Numerator[D[Jinc[x],{x,2}]//FullSimplify]

Plot[-3 x BesselJ[0,x]-(-6+(x^2)) BesselJ[1,x],{x,-3,3}]

Plot[-3 x BesselJ[0,x]-(-6+(x^2)) BesselJ[1,x],{x,-10,10}]

FindRoot[-3 x BesselJ[0,x]-(-6+(x^2)) BesselJ[1,x]==0,{x,2}]

Reduce[-3 x BesselJ[0,x]-(-6+(x^2)) BesselJ[1,x]==0&&x>0,x]

Reduce[-3 x BesselJ[0,x]-(-6+(x^2)) BesselJ[1,x]==0&&1<x<3,x]

Fixed Points

Plot[{Jinc[x],x},{x,-10,10},PlotStyle->{Red,Black},AxesLabel->{x,Jinc[x]},PlotRange->{-1,1}]

FindRoot[Jinc[x]==x,{x,1}]

Reduce[Jinc[x]==x,x]

Reduce[Jinc[x]==x&&x>0,x]

Reduce[Jinc[x]==x&&0<x<2,x]