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Appendix A. List of functions

The list of all functions can be obtained using i+ types. Some formulae here have long parameter names (like “height”, “center” and “hwhm”) replaced with a_i

Gaussian:

y = a_0 \exp\left[-\ln(2)\left(\frac{x-a_1}{a_2}\right)^{2}\right]

SplitGaussian:

y(x;a_0,a_1,a_2,a_3) = \begin{cases} \textrm{Gaussian}(x;a_0,a_1,a_2) & x\leq a_1\\ \textrm{Gaussian}(x;a_0,a_1,a_3) & x>a_1\end{cases}

GaussianA:

y = \sqrt{\frac{\ln(2)}{\pi}}\frac{a_0}{a_2} \exp\left[-\ln(2)\left(\frac{x-a_1}{a_2}\right)^{2}\right]

Lorentzian:

y = \frac{a_0}{1+\left(\frac{x-a_1}{a_2}\right)^2}

LorentzianA:

y = \frac{a_0}{\pi a_2\left[1+\left(\frac{x-a_1}{a_2}\right)^2\right]}

Pearson VII (Pearson7):

y = \frac{a_0}{\pi a_2\left[1+\left(\frac{x-a_1}{a_2}\right)^2\right]}

Split-Pearson-VII (SplitPearson7):

y(x;a_{0},a_{1},a_{2},a_{3},a_{4},a_{5}) = \begin{cases} \textrm{Pearson7}(x;a_0,a_1,a_2,a_4) & x\leq a_1\\ \textrm{Pearson7}(x;a_0,a_1,a_3,a_5) & x>a_1\end{cases}

Pearson-VII-Area (Pearson7A):

y = \frac{a_0\Gamma(a_3)\sqrt{2^{\frac{1}{a_3}}-1}} {a_2\Gamma(a_3-\frac{1}{2})\sqrt{\pi} \left[ 1 + \left(\frac{x-a_1}{a_2}\right)^2 \left(2^{\frac{1}{a_3}}-1\right) \right]^{a_3}}

Pseudo-Voigt (PseudoVoigt):

y = a_0 \left[(1-a_3)\exp\left(-\ln(2)\left(\frac{x-a_1}{a_2}\right)^2\right) + \frac{a_3}{1+\left(\frac{x-a_1}{a_2}\right)^2} \right]

Pseudo-Voigt is a name given to the sum of Gaussian and Lorentzian. a_3 parameters in Pearson VII and Pseudo-Voigt are not related.

Pseudo-Voigt-Area (PseudoVoigtA):

y = a_0 \left[\frac{(1-a_3)\sqrt{\ln(2)}}{a_2\sqrt{\pi}} \exp\left(-\ln2\left(\frac{x-a_1}{a_2}\right)^2\right) + \frac{a_3}{\pi a_2 \left[1+\left(\frac{x-a_1}{a_2}\right)^2\right]} \right]

Voigt:

y = \frac {a_0 \int_{-\infty}^{+\infty} \frac{\exp(-t^2)}{a_3^2+(\frac{x-a_1}{a_2}-t)^2} dt} {\int_{-\infty}^{+\infty} \frac{\exp(-t^2)}{a_3^2+t^2} dt}

The Voigt function is a convolution of Gaussian and Lorentzian functions. a_0 = heigth, a_1 = center, a_2 is proportional to the Gaussian width, and a_3 is proportional to the ratio of Lorentzian and Gaussian widths.

Voigt is computed according to R.J.Wells, Rapid approximation to the Voigt/Faddeeva function and its derivatives, Journal of Quantitative Spectroscopy & Radiative Transfer 62 (1999) 29-48. (See also: http://www.atm.ox.ac.uk/user/wells/voigt.html). The approximation is very fast, but not very exact.

VoigtA:

y = \frac{a_0}{\sqrt{\pi}a_2} \int_{-\infty}^{+\infty} \frac{\exp(-t^2)}{a_3^2+(\frac{x-a_1}{a_2}-t)^2} dt

Exponentially Modified Gaussian (EMG):

y = \frac{ac\sqrt{2\pi}}{2d} \exp\left(\frac{b-x}{d}+\frac{c^2}{2d^2}\right) \left[\frac{d}{\left|d\right|} -\textrm{erf}\left(\frac{b-x}{\sqrt{2}c} + \frac{c}{\sqrt{2}d}\right) \right]

LogNormal:

y = h \exp\left\{ -\ln(2) \left[ \frac{\ln\left(1+2b\frac{x-c}{w}\right)}{b} \right]^{2} \right\}

Doniach-Sunjic (DoniachSunjic):

y = \frac{h\left[\frac{\pi a}{2} + (1-a)\arctan\left(\frac{x-E}{F}\right)\right]} {F+(x-E)^2}

Polynomial5:

y = a_0 + a_1 x +a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5

Appendix B. Command shortenings

The pipe symbol (|) shows the minimum length of the command. “def|ine” means that command “define” can be shortened as “def”, “defi” or “defin”. Commands not listed here cannot be shortened. Arguments of “info” command also cannot be shortened (i.e. you must write “i fit”, not “i f”).

  • c|ommands
  • def|ine
  • f|it
  • g|uess
  • i|nfo
  • p|lot
  • s|et
  • undef|ine
  • w|ith

Appendix C. Literature

The following books were helpful when writing the program (from scientific, not programming side).

William Press, Saul Teukolsky, William Vetterling, Brian Flannery. Numerical Recipes in C. http://www.nr.com

Peter Gans. Data Fitting in the Chemical Sciences by the Method of Least Squares. John Wiley & Sons. 1992.

Siegmund Brandt. Data Analysis. Springer Verlag. 1999.

PeakFit 4.0 for Windows User’s Manual. AISN Software. 1997.

Appendix D. License

Fityk is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

Text of the license is distributed with the program in the file COPYING.

Appendix E. About this manual

This manual is written ReStructuredText. All changes, improvements, corrections, etc. are welcome. Use the Show Source link to get the source of the page, save it, edit, and send me either modified version or patch containing changes.

Following people have contributed to this manual (in chronological order): Marcin Wojdyr (maintainer), Stan Gierlotka, Jaap Folmer, Michael Richardson.

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