Bessel Function Zeros

Author

Eric W. Weisstein
September 16, 2007

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

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

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

Examples

Zeros

TableForm[Transpose[PaddedForm[#,{6,4}]&/@Table[BesselJZero[#,k]//N,{k,5}]&/@Range[0,5]],TableHeadings->{Range[5],TraditionalForm/@(BesselJ[n,z]&/@Range[0,5])},
TableAlignments->Right]//TraditionalForm

Graphics[{PointSize[.014],MapIndexed[{Hue[#2[[1]]/10.],#1}&,Transpose[Table[Point[{n,#}]&/@(BesselJZero[n,#]&/@Range[20]),{n,0,20}]]]},Axes->True,AspectRatio->1]

<<Utilities`Plot`

PlotRoots[BesselJ[1,x],{x,-15,15},WorkingPrecision->20]

GraphicsGrid[Partition[Table[PlotRoots[BesselJ[n,x],{x,-15,15},WorkingPrecision->20,AxesLabel->{x,BesselJ[n,x]}][[1]],{n,0,3}],2]]

BesselJ[2,0]

BesselJZeros[0,1,WorkingPrecision->20]

Zeros of Derivative

<<NumericalMath`BesselZeros`

BesselJPrimeZeros[0,1]

BesselJPrimeZeros[1.83,1]

Graphics[{PointSize[.014],MapIndexed[{Hue[#2[[1]]/10.],#1}&,Transpose[Table[Point[{n,#}]&/@BesselJPrimeZeros[n,20],{n,0,20}]]]},Axes->True,AspectRatio->1]

Show[GraphicsArray[Block[{$DisplayFunction=Identity},Table[Plot[BesselJPrime[n,x],{x,-15,15},AxesLabel->TraditionalForm/@{x,BesselJ[n,x]},PlotStyle->Blue,Epilog->{Red,PointSize[.03],Point[{#,BesselJ[n,#]}]&/@PlotRoots[BesselJ[n,x],{x,-15,15},WorkingPrecision->20]}],{n,0,3}]
]],GraphicsSpacing->-.02]

MakeBoxes[BesselJPrime[n_,z_],TraditionalForm]:=RowBox[{SubsuperscriptBox["J",MakeBoxes[n,TraditionalForm]," ′"],"(",MakeBoxes[z,TraditionalForm],")"}]

BesselJPrime[n,z]//TraditionalForm

TableForm[Transpose[PaddedForm[#,{6,4}]&/@BesselJPrimeZeros[#,5]&/@Range[0,5]],TableHeadings->{Range[5],TraditionalForm/@(BesselJPrime[n,z]&/@Range[0,5])},
TableAlignments->Right]//TraditionalForm