Identity Function

Author

Eric W. Weisstein
September 19, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Real Axis

Plot[x,{x,-10,10},AspectRatio->Automatic,
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,id[x]},ImageSize->200]

Complex Plane

ComplexPlot3D[Identity[z],{z,-5,5},ComplexPlotLabel->id[z]]

ComplexContourPlot[Identity[z],{z,-5,5},ComplexPlotLabel->id[z]]//Timing

Approximation

Series

Series[((3Sin[x])/(2+Cos[x])),{x,0,14}]

Series[(1/(Sqrt[x]))((3Sin[(Sqrt[x])])/(2+Cos[(Sqrt[x])])),{x,0,7}]

l=Last/@Partition[CoefficientList[Series[((3Sin[x])/(2+Cos[x])),{x,0,20}],x],2]

Through[{Numerator,Denominator}[l]]

Series Term

V5

<<DiscreteMath`RSolve`

SeriesTerm[((3Sin[x])/(2+Cos[x])),{x,0,2n-1}]

...wrong...

V7

SeriesCoefficient[((3Sin[x])/(2+Cos[x])),{x,0,n}]//Timing

SeriesCoefficient[(1/(Sqrt[x]))((3Sin[(Sqrt[x])])/(2+Cos[(Sqrt[x])])),{x,0,n}]//Timing

SeriesCoefficient[TrigToExp[((3 Sin[x])/(2+Cos[x]))],{x,0,n}]//Timing

FindGeneratingFunction[l,x]

Plots

<<Utilities`Typesetting`

Plot[((3Sin[x])/(2+Cos[x])),{x,-10,10},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,((3Sin[x])/(2+Cos[x]))}]

eqns=Table[{x,((3Sin[x])/(2+Cos[x]))},{x,0,π/4,π/16}]//FunctionExpand//FullSimplify

Solve[#,p]&/@(Equal@@@(eqns/.π->p))

N[p/.%]-Pi

Plot[x-((3Sin[x])/(2+Cos[x])),{x,0,1},PlotStyle->List/@{Red,Blue},PlotRange->All,AxesLabel->TraditionalForm/@{x,x-((3Sin[x])/(2+Cos[x]))}]

ComplexPlot3D[z-((3Sin[z])/(2+Cos[z])),{z,-7,7}]

ComplexContourPlot[z-((3Sin[z])/(2+Cos[z])),{z,-7,7}]