# Talk:Voigt profile

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## CDF

Could we add some references? Particularly concerning the cumulative distribution function, which is mapped but not defined.129.234.4.1 21:40, 8 December 2006 (UTC)

I will try to find some references. Meanwhile, I have entered the CDF. If you can check it in any way, please do. PAR 00:30, 10 December 2006 (UTC)
Hello, do you where I can find the reference for the CDF? — Preceding unsigned comment added by 24.246.56.7 (talk) 02:12, 7 November 2013 (UTC)

## FWHM of Lorentz

Are you sure, that fL is 2γ? I thought, γ is already the FWHM of the Lorentz-Line, so that just fL = γ. ? 149.220.35.153 09:43, 26 January 2007 (UTC)Len

Hi - see my response on the Cauchy distribution page. PAR 15:07, 26 January 2007 (UTC)

## Pseudo-voigt

Could we put in a section on the pseudo-voigt? I feel that it is a necessary topic when discussing the Voigt profile as direct linear convolution is expensive.. The French version of this article has one =P Stephen 14:07, 29 June 2007 (UTC)

I'm going to be fitting a Voigt to data soon, so I am also interested. The article mentions the pseudo-voigt profile(s) in the lead, but doesn't discuss it. Somebody should add that info! Danski14(talk) 03:50, 28 February 2012 (UTC)

## Wrong formula for V(x;sigma,gamma)?

Currently the article says,

"The defining integral can be evaluated as:

$V(x;\sigma,\gamma)=\frac{\textrm{Re}[w(z)]}{\sigma\sqrt{2 \pi}}$

where Re[w(z) ] is the real part of the complex error function of z  and

$z=\frac{x+i\gamma}{\sigma\sqrt{2}}.$"

but I don't think that's right. For one, if γ=0, it should look like a Gaussian, but error function with real parameter looks nothing like a Gaussian.

Evaluating the integral with Mathematica (under assumptions $\gamma > 0$, $\sigma > 0$ and x (i.e. the parameter that we are not integrating over) gives the expression below, and for a few examples I tried it looks correct, so I'm replacing it.

$V(x;\sigma,\gamma) = \frac{e^{-(x+i \gamma)^2 / 2\sigma^2} \Big\{ 1+ e^{2ix\gamma/\sigma^2}\big(1 - \mathrm{erf}[(ix+\gamma)/\sigma\sqrt{2}]\big) + \mathrm{erf}[(ix - \gamma)/\sigma\sqrt{2}]\Big\}}{2\sigma\sqrt{2\pi}}.$

novakyu (talk) 08:24, 7 May 2009 (UTC)

First of all, the complex error function is defined as:
$w(z):= e^{-z^2}(1-\textrm{erf}(-iz))$
Two important properties of the error function are $\textrm{erf}(z)=-\textrm{erf}(-z)$ and $\textrm{erf}(z*)=\textrm{erf}(z)*$ from which it can be easily shown that if z is real, the real part of $\textrm{erf}(-iz)$ is zero. Thus, if $\gamma=0$
$z=\frac{x}{\sigma\sqrt{2}}$
which is real. Thus the real part of w(x) is
$Re[w(z)]= e^{-x^2/2\sigma^2}$
and so the Voigt profile becomes a Gaussian. Using the two above mentioned properties it can also be shown that the more complicated expression derived by novakyu is correct and entirely equivalent to the simpler expression involving the complex error function. I have therefore reverted the more complicated expression to the previous simpler expression. PAR (talk) 04:35, 10 June 2009 (UTC)
I have to respectfully disagree. Not with the correctness of your statement (I must've misunderstood what is meant by "complex error function" at the time), but your assertion that previous expression is simpler. I imagine a situation where a student or an experimentalist (i.e. someone who would be doing measurements in spectroscopy!) is looking for information on Voigt profile (for lines that are both homogeneously and inhomogeneously broadened) and stumbling on this article. For me, "simpler expression" is the expression which is more immediately useful to these people. The previous expression may appear simple, but in order for anyone to use it, he has to navigate through the definition of z, definition of "complex error function", what-the-hell erfc means, etc, etc. Down these lines of references upon references, there is more chance for confusion and misdirection. Given that both formulas fit in one line on most screens, I think added perceived "complexity" is well worth the time saved by the reader.
Nonetheless, I guess if anyone's looking for it, it is available in the article history, so I'll leave it as is. novakyu (talk) 12:49, 7 August 2009 (UTC)

## Minor correction

The Voigt width formula from [1] had a typo in the last significant figure of one of the coefficients. I fixed it.

Adinov (talk) 16:12, 28 August 2010 (UTC)

## Accuracy of FWHM

I think there is a typo (mixing of units from % to fraction).. when giving better approximation to FWHM. Seems to me that 0.000305% should be 0.0305 of 0.02% should be 0.0002%..etc. The current values do not match up.

f 0.02% is given by[1]

   f_\mathrm{V}\approx 0.5346 f_\mathrm{L}+\sqrt{0.2166f_\mathrm{L}^2+f_\mathrm{G}^2}.


This approximation will be exactly correct for a pure Gaussian, but will have an error of about 0.000305% per