Exponential Function

Author

Eric W. Weisstein
April 5, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Plots

Real Axis

Plot[Exp[x],{x,-2,2},
PlotStyle->Red,
AxesLabel->Thread[Style[{x,e^x},Directive[Black,14]]],
BaseStyle->Directive[Black,FontFamily->"Times"]]

Complex Plane

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

ComplexPlot3D[Exp[z],{z,-5,5},PlotPoints->100,ComplexPlotLabel->HoldForm[Exp[z]]]//Timing

ComplexContourPlot[Exp[z],{z,-5,5},
ComplexPlotLabel->HoldForm[Exp[z]]]//Timing

(e^(1/z))

Plots

Real Axis

PlotDiscontinuous[Exp[1/x],{x,-5,0,5},PlotRange->{{-5,5},{-1,5}},
HorizontalAsymptotes->{1},
AxesLabel->TraditionalForm/@{x,Exp[InlineFraction[1,x]]}
]

Complex Plane

ComplexPlot3D[Exp[1/z],{z,-5,5},
PlotPoints->100,
ComplexPlotLabel->HoldForm[Exp[InlineFraction[1,z]]]]//Timing

ComplexContourPlot[If[Abs[z]<10^-3,0,Exp[1/z]],{z,-.1,.1},
ComplexPlotLabel->HoldForm[Exp[InlineFraction[1,z]]]]//Timing

Use these two

Show[GraphicsArray[ComplexContourPlot[Exp[1/z],{z,-#,#},
DisplayFunction->Identity,
ComplexPlotLabel->HoldForm[Exp[InlineFraction[1,z]]]
][[1]]&/@{.5,5}],ImageWidth->500]//Timing

Limits

Limit[Exp[1/x],x->0,Direction->#]&/@{-1,1}

Limit[Exp[1/x],x->#]&/@{-∞,∞}

Integrals

Integrate[Exp[Exp[I t]],{t,0,2π}]

Integrate[Exp[Exp[I t]-I t],{t,0,2π}]

Series

Series[Exp[x],{x,0,10}]

SeriesCoefficient[Exp[x],{x,0,n}]

GeneratingFunction[%,n,x]

Sum

(∑_(n=0)^∞)((x^n)/(n!))==exp(x)

Sum[((x^n)/(n!)),{n,0,∞}]

Continued Fraction

With[{z=1.1},{(1/(1+ContinuedFractionK[(((-1))^n)z,If[Mod[n,2]==1,n,2],{n,35}])),Exp[z]}]

(1/(1+ContinuedFractionK[(((-1))^n)z,If[Mod[n,2]==1,n,2],{n,∞}]))