Variation of Argument

Author

Eric W. Weisstein
January 10, 2009

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

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Example

Limit[Arg[(z-1)/z^2/. z->R Exp[I θ]],R->0]

Plot

V5

With[{R=.0001},
Show[GraphicsArray[{
Plot[Arg[(z-1)/z/. z->R Exp[I θ]],{θ,0,2π},
DisplayFunction->Identity,PlotStyle->Red,
AxesLabel->TraditionalForm/@{θ,Arg[(z-1)/z]}],
Plot[-2π UnitStep[θ-π]+Arg[(z-1)/z^2/. z->R Exp[I θ]],{θ,0,2π},
DisplayFunction->Identity,PlotStyle->Red,
AxesLabel->TraditionalForm/@{θ,Arg[(z-1)/z^2]}],
Plot[-2π UnitStep[θ-2π/3]-2π UnitStep[θ-4π/3]+Arg[(z-1)/z^3/. z->R Exp[I θ]],{θ,0,2π},
DisplayFunction->Identity,PlotStyle->Red,
AxesLabel->TraditionalForm/@{θ,Arg[(z-1)/z^3]}]
}]]
]

2Pi Range[3]//N

V7

With[{R=.0001},
GraphicsGrid[{{
Plot[Arg[(z-1)/z/. z->R Exp[I θ]],{θ,0,2π},PlotStyle->Red,AxesLabel->{θ,Arg[(z-1)/z]}],
Plot[-2π UnitStep[θ-π]+Arg[(z-1)/z^2/. z->R Exp[I θ]],{θ,0,2π},PlotStyle->Red,AxesLabel->{θ,Arg[(z-1)/z^2]}],
Plot[-2π UnitStep[θ-2π/3]-2π UnitStep[θ-4π/3]+Arg[(z-1)/z^3/. z->R Exp[I θ]],{θ,0,2π},PlotStyle->Red,AxesLabel->{θ,Arg[(z-1)/z^3]}]
}},ImageSize->600,Spacings->-40]
]