Pole

Author

Eric W. Weisstein
April 15, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/Pole.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Poles[f_, z_] := 
    Union[z /. {ToRules[Roots[Denominator[Together[D[f, z]]] == 0, z]]}]

Examples

Table[Poles[1/z^n,z],{n,3}]

Poles[z/(z^2+z+1),z]

Plots of 1/(z^n)

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

Show[GraphicsArray[{ComplexContourPlot[#,{z,-1,1},
DisplayFunction->Identity,
ComplexPlotLabel->InlineFraction@@(Through[{Numerator,Denominator}[#]])
]}&/@{1/z,1/z^2}]
]

Show[GraphicsArray[{ComplexPlot3D[#,{z,-1,1},
DisplayFunction->Identity,
PlotPoints->50,Mesh->False,
ClipFill->None,
ComplexPlotLabel->InlineFraction@@(Through[{Numerator,Denominator}[#]])
]}&/@{1/z,1/z^2}]
]