Gaussian Function

Author

Eric W. Weisstein
October 7, 2004

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

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

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

1-D

Plots

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

Real Axis

Plot[Exp[-x^2],{x,-5,5},
PlotStyle->Red,
PlotRange->All,
AxesLabel->TraditionalForm/@{x,Exp[-x^2]}]

Complex Plane

ComplexPlot3D[Exp[-z^2],{z,-3,3},PlotPoints->200,
ComplexPlotLabel->HoldForm[Exp[-z^2]]]//Timing

ComplexContourPlot[Exp[-z^2],{z,-3,3},
ComplexPlotLabel->HoldForm[Exp[-z^2]]]//Timing

Apodization Function

Apodization Function

Gaussian[x,σ]

Maximum

Limit[Gaussian[x,σ],x->0]

Instrument Function

FullSimplify[InstrumentFunction[Gaussian[x,σ],{x,a},k,Assumptions->k>0&&a>0&&σ>0]]

FullSimplify[InstrumentFunction[Gaussian[x,σ],{x,a},k,Assumptions->k>0&&a>0&&σ>0]]

i[k_]:=(E^(-2 (k^2) (π^2) (σ^2))) (Sqrt[(π/2)]) σ (Erf[((a-2 I k π (σ^2))/((Sqrt[2]) σ))]+Erf[((a+2 I k π (σ^2))/((Sqrt[2]) σ))])

N[i[a]/.{σ->1.,k->.5,a->.4},20]//Chop

With[{σ=1,k=.5,a=.4},
NIntegrate[2 Cos[2 Pi k x] Gaussian[x,σ],{x,0,a}]
]

FullSimplify[InstrumentFunction[Gaussian[x,σ],{x,∞},k,Assumptions->k>0&&σ>0]]

Maximum

Limit[i[a],k->0]

Plot

With[{σ=.3},
ApodizationPlot[{Gaussian[x,σ],(E^(-2 (k^2) (π^2) (σ^2))) (Sqrt[(π/2)]) σ (Erf[((a-2 I k π (σ^2))/((Sqrt[2]) σ))]+Erf[((a+2 I k π (σ^2))/((Sqrt[2]) σ))])},{x,k},a]
]//Timing

2-D

WriteLiveForm["gaussian.m",Show[Plot3D[Exp[-x^2-y^2],{x,-3,3},{y,-3,3},PlotRange->All,PlotPoints->20],Boxed->True,opts]]