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Lorentzian Function

LorentzianFunction

The Lorentzian function is the singly peaked function given by

L(x)=1/pi(1/2Gamma)/((x-x_0)^2+(1/2Gamma)^2),

(1)

where x_0 is the center and Gamma is a parameter specifying the width. The Lorentzian function is normalized so that

int_(-infty)^inftyL(x)=1.

(2)

It has a maximum at x=x_0, where

L^'(x)=-(16(x-x_0)Gamma)/(pi[4(x-x_0)^2+Gamma^2]^2)=0.

(3)

Its value at the maximum is

L(x_0)=2/(piGamma).

(4)

It is equal to half its maximum at

x=(x_0+/-1/2Gamma),

(5)

and so has full width at half maximum Gamma. The function has inflection points at

L^('')(x)=16Gamma(12(x-x_0)^2-Gamma^2)/(pi[4(x-x_0)^2+Gamma^2]^3)=0,

(6)

giving

x_1=x_0-1/6sqrt(3)Gamma,

(7)

where

L(x_1)=3/(2piGamma).

(8)

LorentzianReIm

LorentzianContours

The Lorentzian function extended into the complex plane is illustrated above.

The Lorentzian function gives the shape of certain types of spectral lines and is the distribution function in the Cauchy distribution. The Lorentzian function has Fourier transform

F_x[1/pi(1/2Gamma)/((x-x_0)^2+(1/2Gamma)^2)](k)=e^(-2piikx_0-Gammapi|k|).

(9)

LorentzianApodization

The Lorentzian function can also be used as an apodization function, although its instrument function is complicated to express analytically.