Lorentzian Function

Author

Eric W. Weisstein
October 7, 2004

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

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

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

Function

Plots

<<Utilities`Plot`

Real Axis

V5

Plot[1/(x^2+1)/Pi,{x,-6,6},PlotStyle->Red,AxesLabel->TraditionalForm/@{x,InlineFraction[1,x^2+1]}]

V6

Plot[1/(x^2+1)/Pi,{x,-6,6},PlotStyle->Red,AxesLabel->{x,Row[{1,"/","(",x^2+1,")"}]},LabelStyle->{FontFamily->"Times"}]

Complex Plane

ComplexPlot3D[1/(z^2+1),{z,-5,5},ComplexPlotLabel->InlineFraction[1,z^2+1],
PlotPoints->100]

ComplexContourPlot[1/(z^2+1),{z,-5,5},ComplexPlotLabel->InlineFraction[1,z^2+1]]//Timing

Properties

Lorentzian[x_,{g_,x0_}]:=1/Pi g/2/((x-x0)^2+(g/2)^2)

Lorentzian[x0,{Γ,x0}]

ComplexExpand[Integrate[Lorentzian[x,{Γ,x0}],{x,-Infinity,Infinity}]]

% /.Γ->1

FullSimplify[D[Lorentzian[x,{Γ,x0}],x]]

x/.Solve[Lorentzian[x,{Γ,x0}]==Lorentzian[0,{Γ,0}]/2,x]

Simplify[Subtract@@%]

FullSimplify[D[Lorentzian[x,{Γ,x0}],{x,2}]]

192/16

Solve[D[Lorentzian[x,{Γ,x0}],{x,2}]==0,x]

FullSimplify[Lorentzian[(1/6) (6 x0-(Sqrt[3]) Γ),{Γ,x0}]]

Apodization Function

Initialization

<<Graphics`Colors`;
<<Plot`;
<<Apodization`;

Apodization Function

Lorentzian[x,a]

Maximum

Limit[Lorentzian[x,a],x->0]

Instrument Function

This integral is probably wrong...

v4.2

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

FullSimplify[Integrate[Lorentzian[x,a] Exp[2π I k x],{x,-a,a},Assumptions->a>0],a>0&&k>0]

q=FullSimplify[Integrate[Lorentzian[x,a]Exp[2π I k x],x],a>0&&k>0]

FunctionExpand[FullSimplify[Limit[q,x->0],a>0]]

FunctionExpand[FullSimplify[Limit[q,x->a],a>0]]

Integrate[Lorentzian[x,a]Cos[2π k x],x]

i[k_]:=a Cosh[2 a k π] (π-I CosIntegral[(2-2 I) a k π]+I CosIntegral[(2+2 I) a k π])-a Sinh[2 a k π] (SinIntegral[(2-2 I) a k π]+SinIntegral[(2+2 I) a k π])

i[k_]:=-(1/2) I a (E^(-2 a k π)) ((E^(4 a k π)) (ExpIntegralEi[(-2-2 I) a k π]-ExpIntegralEi[(-2+2 I) a k π])-ExpIntegralEi[(2-2 I) a k π]+ExpIntegralEi[(2+2 I) a k π])

i[k_]:=NIntegrate[2Lorentzian[x,1]Cos[2π k x],{x,0,1}]

v5.0

Integrate[Lorentzian[x,a]Cos[2π k x],x]

InstrumentFunction[Lorentzian[x,a],{x,a},k]

2Integrate[Lorentzian[x,a]Cos[2π k x],{x,0,a}]

Integrate[Lorentzian[x,a] Exp[2π I k x],{x,-a,a}]

Maximum

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

FullSimplify[Limit[-(1/2) I a (E^(-2 a k π)) ((E^(4 a k π)) (ExpIntegralEi[(-2-2 I) a k π]-ExpIntegralEi[(-2+2 I) a k π])-ExpIntegralEi[(2-2 I) a k π]+ExpIntegralEi[(2+2 I) a k π]),k->0],a>0]

Plots

ApodizationPlot[{Lorentzian[x,a],-(1/2) I a (E^(-2 a k π)) ((E^(4 a k π)) (ExpIntegralEi[(-2-2 I) a k π]-ExpIntegralEi[(-2+2 I) a k π])-ExpIntegralEi[(2-2 I) a k π]+ExpIntegralEi[(2+2 I) a k π])},{x,k},a]

With[{opts=Sequence[PlotStyle->Red,DisplayFunction->Identity]},Show[GraphicsArray[{Plot[Lorentzian[x,1],{x,-1,1},opts,PlotRange->{{-1,1},{0,1}},AxesLabel->TraditionalForm/@{InlineFraction[x,a],A[InlineFraction[x,a]]}],Plot[i[k],{k,-Pi,Pi},opts,PlotRange->All,AxesLabel->TraditionalForm/@{k a,InlineFraction[ii[k a],a]}],Plot[i[k],{k,-5,5},opts,AxesLabel->TraditionalForm/@{k a,InlineFraction[ii[k a],a]}]}],ImageSize->400,GraphicsSpacing->-.1]]//Timing

Instrument Function

2NFWHM[i[k],k]/L

Sidelobes

<<Plot`;
<<NumericalMath`NLimit`;
<<Graphics`Colors`;

i[k_?NumericQ]:=NIntegrate[2(1/(1+(x^2)))Cos[2π k x],{x,0,1}]

f[x_?NumericQ]:=ND[i[k],k,x]

roots=x/.PlotRoot[f[x],{x,0,5},PlotStyle->Red]

extrema=Chop[{#,i[#]/.a->1}&/@roots]

Plot[i[k],{k,0,5},PlotStyle->Red,
Epilog->{PointSize[.02],Red,Point/@extrema}]