Köbe Function

Author

Eric W. Weisstein
September 8, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/ComplexAnalysis/KoebeFunction.nb.

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

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

Code

KoebeFunction[x_]:=x/(1-x)^2

Plots

2-D

<<Utilities`Plot`

PlotDiscontinuous[KoebeFunction[x],{x,-5,1,5},PlotRange->{Automatic,{-1,10}},AxesLabel->TraditionalForm/@{x,KoebeFunction[x]}]

3-D

<<Utilities`Typesetting`

ComplexPlot3D[KoebeFunction[z],{z,-3-3I,5+3I},
ComplexPlotLabel->InlineFraction[z,(1-z)^2]
]

ComplexContourPlot[KoebeFunction[z],{z,-3-3I,5+3I},
ComplexPlotLabel->InlineFraction[z,(1-z)^2]
]

Sums

z+Sum[j Exp[I θ] z^j,{j,2,Infinity}]

Simplify[z+Sum[j  z^j,{j,2,Infinity}]]

Derivatives and Extrema

Simplify[D[KoebeFunction[x],x]]

Simplify[D[KoebeFunction[x],{x,2}]]