American Journal of Analytical Chemistry, 2013, 4, 725-731
Published Online December 2013 (http://www.scirp.org/journal/ajac)
http://dx.doi.org/10.4236/ajac.2013.412087
Open Access AJAC
Calculation of the Voigt Function in the Region of Very
Small Values of the Parameter a Where the Calculation Is
Notoriously Difficult
Hssaïne Amamou1, Belkacem Ferhat2, André Bois1
1Laboratoire PROTEE-ISO, Université du Sud Toulon-Var, La Garde, France
2Laboratoire d’Electronique Quantique, Faculty of Physique, University of Sciences and Technology Houari Boumediene,
Alger, Algérie
Email: amamou@univ-tln.fr
Received September 13, 2013; revised October 25, 2013; accepted November 15, 2013
Copyright © 2013 Hssaïne Amamou et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The Voigt function is the convolution of a Lorentzian and a Guaussian density. The computation of these functions is re-
quired in several problems arising in a variety of physicochemical subjects; such as nuclear reactors, atmospheric transmit-
tance and spectroscopy. In this work we suggest using a new formula for the calculation of the Voigt function. Our for-
mula is a new integral representation for the Voigt function that gives the perfect results for the Voigt function calculation
and is easily calculable. We give also a comparison between our results of calculation of Voigt function for the very small
values of the parameter a, where the calculation is notoriously difficult, with those of the various algorithms of other authors.
Keywords: Convolution; Line Profile; Voigt Function; Lorentz Profile; Doppler Profile and Spectral Lines
1. Introduction
The shape of the spectral lines is a subject of great inter-
est in physics and chemistry. Indeed, several important
physical and chemical parameters are directly deducted
from the spectral lines whose shape is approximated by
the Gaussian profile or the Lorentzian profile. Therefore,
the parameters obtained do not correspond exactly to the
real physical conditions for which the spectral lines have
the shape of the distribution of Voigt. For this reason, the
study and the calculation of the Voigt function are very
interesting in many fields of physics and chemistry. In-
deed, for the signal emitted by a plasma, for example, the
phenomena that produce the enlargement of the spectral
lines are Doppler broadening caused by thermal agitation
of the particles, and the enlargement of pressure, which
is due to interactions between the transmitters and the
neighboring particles, the resulting profile of these phy-
sical phenomena is a Voigt profile.
The Voigt function results from the convolution prod-
uct between a Gaussian profile and a profile Lorentzian
and is expressed by the following formula:



2
2
2
exp
,
π
x
a
Vau x
aux




0
2ln2
D
u
represents the relationship between
the distance from the center of the Lorentzian line and
the width of the Gaussian line.

ln 2
L
D
a
determines the importance of Lor-
entzian in the profile, thus if this parameter tends to-
wards 0, the Lorentzian is negligible and if it tends to-
wards, the infinite the Lorentzian is dominant.
: Frequency.
: Lorentz half-width at half maximum in frequency.
D
: Doppler half-width at half maximum in frequency.
0
: Frequency in the center of the line.
This function has been studied recently by several
studies [1-8].
2. Calculation of the Voigt Function in the
Region near the u Axis
d
(1)
We give our new formula (That we have demonstrated in
the article of Amamou et al. [9] (Demonstration given in
Appendix 1 of this article)) of Voigt function in the fol-
lowing formula:
H. AMAMOU ET AL.
726





 
 

 

22
22
0
2
0
2
0
2π
,expcos2exp
2
π
sin 2expexpd
cos2cos2 expd
sin 2sin 2expd
u
a
a
Vauaau u
auux x
auuxx x
auuxx x




(2)
This analytical formula of the Voigt function gives a
solution to the mathematical problem which is due at the
infinite boundaries of the integral which defines the Voigt
function. This is a new integral representation for the
Voigt function that gives a perfect formula of Voigt func-
tion easily calculable and it’s different to the formula
given by Roston and Obaid [10] and gives a solution to
the problem of exponential growth described by Van
Synder [11].
This formula can be used for calculation of the spec-
tral lines whose profile is a convolution of a Lorentzian
profile and a Gaussian profile. This type of profile de-
scribes the actual physical conditions of several phys-
icochemical phenomena and its use is very interesting to
adjust the spectral lines by theoretical models.
In the Figures 1-3, the determination of the Lorentzian
profile and the Gaussian profile we have used the fol-
lowing parameters; 0
,, ,
D
L
 
. These figures shows
the three profiles; Voigt profile, Gaussian profile and
Lorentzian profile for different parameters a and u.
With:
: Wavelength.
: Lorentz half-width at half maximum in wave-
length.
D
: Doppler half-width at half maximum in wave-
length.
0
: Wavelength in the center of the line.
In the Figure 1 the Lorentzian profile, the Gaussian
profile and the Voigt profile are given for the following
parameters:
0
244:0.1:248 nm,246 nm,
0.4 nm,2.3 nm
DL



 
thus the parameter .
4.79a
In the Figure 2 the Lorentzian profile, the Gaussian
profile and the Voigt profile are given for the following
parameters:
0
244:0.1:248 nm,246 nm,
3 nm,0.003 nm
DL



 
thus the parameter .
0.00083255a
In the Figure 3 the Lorentzian profile, the Gaussian
profile and the Voigt profile are given for the following
parameters:
Figure 1. Voigt profile: “black” Voigt, “blue” Gauss and
“red” Lorentz.
Figure 2. Voigt profile: “black” Voigt, “blue” Gauss and
“red” Lorentz.
Figure 3. Voigt profile: “black” Voigt, “blue” Gauss and
“red” Lorentz.
Open Access AJAC
H. AMAMOU ET AL. 727
0
244:0.1:248 nm,246 nm,
0.3 nm,0.3 nm
DL



 
thus the parameter .
0.8326a
Our formula is also a very interesting method for easy
calculation of the Voigt function. For the calculation of
the integrals of Equation (2) the trapezoidal rule method
and the adaptive Simpson’s method give very good re-
sults.
Table A1 (Appendix 2) gives the values of the Voigt
function calculated with the Formula (2) for the very
small values of the parameter a where the calculation is
notoriously difficult [12]. This table gives also the com-
putation time(s) for the values of each column of the
table. This calculation time depends obviously on the
performances of the computer. The computer that we
have used has a processor Intel pentium 2.3 GHz and a
memory (RAM) 4 GHz. This table gives the reference
values of the Voigt function calculated from Equation
(2).
Table 1 gives a comparison between our results of
calculation of Voigt function in the in the region of very
small values of the parameter a with those of the various
algorithms of other authors.
3. Conclusion
The new representation integral for Voigt function that
we have demonstrated and used to adjustment “fitting”
of lines spectral in a precedent article is used in this work
for calculation of Voigt function. Thus, this function is
easily calculable. We also made a comparison between
the results obtained by our formula and those obtained by
the various algorithms of other authors in the region of
Table 1. Comparison between our results and the results of
various algorithms of other authors (D is here for 10).
Calculation of Voigt in the region
of very small values of the parameter a
Author u = 5.4 a = 1010 u = 5.5a = 1014
Amstrong et al. [13] 12
2.260842D 14
7.307387Da
Humliek [14] 12
2.260842D 16
1.966215Da
Humliek [15] 12
2.260845D 14
7.307387Da
Hui [16] 8
2.667847
D
9
9.238980Da
Lether and Wenston [17] 12
2.260845D 14
7.307386Da
Mclean et al. [18] 5
4.89872D
5
4.24886Da

Poppe and Wiyers [19] 12
2.260850D 14
7.307805Da
Shippony and Read [1] 12
2.260845D 14
14
7.287724Da
Zaghloul [8] 12
2.260844D 7.287724Da
This Works 12
2.260844D 14
7.307387Da
very small values of the parameter a where the calcula-
tion is notoriously difficult.
REFERENCES
[1] Z. Shippony and W. G. Read, “A Correction to a Highly
Accurate Voigt Function Algorithm,” Journal of Quanti-
tative Spectroscopy & Radiative Transfer, Vol. 78, No. 2,
2003, pp. 255-255.
http://dx.doi.org/10.1016/S0022-4073(02)00169-3
[2] M. R. Zaghloul and A. N. Ali, “Algorithm 916: Comput-
ing the Faddeyeva and Voigt Functions,” ACM Transac-
tions on Mathematical Software, Vol. 38, No. 2, 2011, pp.
1-22. http://dx.doi.org/10.1145/2049673.2049679
[3] J. He and Q. G. Zhang, “An Exact Calculation of the
Voigt Spectral Line Profile in Spectroscopy,” Journal of
Optics A: Pure and Applied Optics, Vol. 9, No. 7, 2007,
pp. 565-568.
http://dx.doi.org/10.1088/1464-4258/9/7/003
[4] S. M. Abrarov and B. M. Quine, “Efficient Algorithmic
Implementation of the Voigt/Complex Error Function
Based on Exponential Series Approximation,” Applied
Mathematics and Computation, Vol. 218, No. 5, 2011, pp.
1894-1902. http://dx.doi.org/10.1016/j.amc.2011.06.072
[5] F. Schreier, “Optimized Implementations of Rational
Approximations for the Voigt and Complex Error Func-
tion,” Journal of Quantitative Spectroscopy and Radia-
tive Transfer, Vol. 112, No. 6, 2011, pp. 1010-1025.
http://dx.doi.org/10.1016/j.jqsrt.2010.12.010
[6] S. P. Limandri, R. D. Bonetto, H. O. Di Rocco and J. C.
Trincavelli, “Fast and Accurate Expression for the Voigt
Function. Application to the Determination of Uranium
M Linewidths,” Spectrochimica Acta Part B: Atomic
Spectroscopy, Vol. 63, No. 9, 2008, pp. 962-967.
http://dx.doi.org/10.1016/j.sab.2008.06.001
[7] S. M. Abrarov, B. M. Quine and R. K. Jagpal, “A Simple
Interpolating Algorithm for the Rapid and Accurate Cal-
culation of the Voigt Function,” Journal of Quantitative
Spectroscopy and Radiative Transfer, Vol. 110, No. 6-7,
2009, pp. 376-383.
http://dx.doi.org/10.1016/j.jqsrt.2009.01.003
[8] M. R. Zaghloul, “On the Calculation of the Voigt Line
Profile: A Single Proper Integral with a Damped Sine In-
tegrand,” Monthly Notices of the Royal Astronomical So-
ciety, Vol. 375, No. 3, 2007, pp. 1043-1048.
http://dx.doi.org/10.1111/j.1365-2966.2006.11377.x
[9] H. Amamou, A. Bois, M. Grimaldi and R. Redon, “Exact
Analytical Formula for Voigt Function which Results
from the Convolution of a Gaussian Profile and a Lor-
entzian Profile,” Physical Chemical News PCN, Vol. 43,
2008, pp. 1-6.
[10] G. D. Roston and F. S. Obaid, “Exact Analytical Formula
for Voigt Spectral Line Profile,” Journal of Quantitative
Spectroscopy & Radiative Transfer, Vol. 94, No. 2, 2005,
pp. 255-263. http://dx.doi.org/10.1016/j.jqsrt.2004.09.007
[11] S. Van, “Comment on ‘Exact Analytical Formula for
Voigt Spectral Line Profile’,” Journal of Quantitative
Spectroscopy & Radiative Transfer, Vol. 95, No. 4, 2005,
pp. 557-558. http://dx.doi.org/10.1016/j.jqsrt.2005.03.001
Open Access AJAC
H. AMAMOU ET AL.
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728
[12] R. J. Wells, “Rapid Approximation to the Voigt/Faddeeva
Function and Its Derivatives,” Journal of Quantitative
Spectroscopy and Radiative Transfer, Vol. 62, No. 1,
1999, pp. 29-48.
http://dx.doi.org/10.1016/S0022-4073(97)00231-8
[13] B. H. Armstrong, “Spectrum Line Profiles: The Voigt
Unction,” Journal of Quantitative Spectroscopy and Ra-
diative Transfer, Vol. 7, No. 1, 1967, pp. 61-88.
http://dx.doi.org/10.1016/0022-4073(67)90057-X
[14] J. Humlicek, “Optimized Computation of the Voigt and
Complex Probability Functions,” Journal of Quantitative
Spectroscopy and Radiative Transfer, Vol. 27, No. 4,
1982, pp. 437-444.
http://dx.doi.org/10.1016/0022-4073(82)90078-4
[15] J. Humlicek, “An Efficient Method for Evaluation of the
Complex Probability Function: The Voigt Function and
Its Derivatives,” Journal of Quantitative Spectroscopy &
Radiative Transfer, Vol. 21, No. 4, 1978, pp. 309-313.
http://dx.doi.org/10.1016/0022-4073(79)90062-1
[16] A. K. Hui, B. H. Armstrong and A. A. Wray, “Rapid
Computation of the Voigt and Complex Error Functions,”
Journal of Quantitative Spectroscopy & Radiative Trans-
fer, Vol. 19, No. 5, 1978, pp. 509-516.
http://dx.doi.org/10.1016/0022-4073(78)90019-5
[17] F. G. Lether and P. R. Wenston, “The Numerical Com-
putation of the Voigt Function by a Corrected Midpoint
Quadrature Rule for (−∞, ),” Journal of Computational
and Applied Mathematics, Vol. 34, No. 1, 1991, pp. 75-
92. http://dx.doi.org/10.1016/0377-0427(91)90149-E
[18] A. B. McLean, C. E. J. Mitchell and D. M. Swanston,
“Implementation of an Efficient Analytical Approxima-
tion to the Voigt Function for Photoemission Lineshape
Analysis,” Journal of Electron Spectroscopy and Related
Phenomena, Vol. 69, No. 2, 1994, pp. 125-132.
http://dx.doi.org/10.1016/0368-2048(94)02189-7
[19] G. P. M. Poppe and C. M. J. Wijers, “More Efficient
Computation of the Complex Error Function,” ACM
Transactions on Mathematical Software (TOMS), Vol. 16,
No. 1, 1990, pp. 38-46.
http://dx.doi.org/10.1145/77626.77629
H. AMAMOU ET AL. 729
Appendix 1
The spectral radiant intensity of Voigt profile is given
by:
 
d
I
GLGtL tt
 


 
(4)
With:
  
2
2
2ln2 4ln2
exp
πD
D
G





(5)
is the Gaussian profile whose
D
the Doppler half-
width at half maximum and
is the frequency.
And


2
2
0
1
2π
2
L
L
L





(6)
is the Lorentzian profile whose
the Lorentz half-
width at half maximum and 0
is the frequency in the
center of the line.
By an adequate change of variables, the convolution
Equation (4) can be put in the following form:
 
ln 22,
D
I
Vau

(7)
With



2
2
2
exp
,d
π
x
a
Vau x
aux



(8)
,Vau is the Voigt function whose parameters are:
We can also put the expression (4) in the following
form:




 
2
2
0
12
π
exp2 πexp πd
4ln 2
DL
I
t
it t
II
 









t
(9)
The two integrals

12
,II
are given by the following relations:










2
2
0
10
2
2
20
0
π
exp2 πexp πd
4ln 2
π
exp2 πexp πd
4ln 2
DL
DL
t
I
it t
t
t
I
itt














t
(10)
which can be also written in the following form:
 


 
 


 
2
2
0
10
2
2
20
0
ππ
expln2exp2 πexpln2d
2ln2
π
expln2exp2 πexpln 2d
2ln2
LD
D D
LD
D D
tL
L
I
it
t
t
I
it














 








 







 

 





 

t
(11)
By making the change of variable according to:


0
ln 2
2ln
L
D
D
a
u

2
(12)
Thus, the preceding relation could be in the following form:










2
0
2
10
2
2
20
0
π
expexp2 πexp d
2ln2
π
expexp2 πexp d
2ln2
D
D
t
I
ait a
t
t
I
aita























t
(13)
Open Access AJAC
H. AMAMOU ET AL.
730
By using a suitable change of variable, the expression (13) can be formulated as follows:
 

 

 

 

22
1
22
2
2ln2expexp2exp 2expd
π
2ln2expexp2exp 2expd
π
a
D
a
D
I
aiauiux xx
I
ai auiuxxx




(14)
Thereafter, we can write the two precedent integrals like:
 

 



 

 



00
22
1
22
200
2ln2expexp2exp 2expdexp 2expd
π
2ln2expexp2exp 2expdexp 2expd
π
a
D
a
D
2
2
I
aiauiuxx xiuxx x
I
aiauiuxx xiuxx x
 


 


(15)
Thereafter:
 

 

 


22
3
00
4ln2expcos 2cos 2expdsin 2sin 2expd
π
D
Iaauuxx xauuxx xI
2
 


 (16)
with:
 

 

 

22
300
4ln2expcos2cos2expdsin 2sin 2expd
π
aa
D
2
I
aauuxxxau uxx


x (17)
By a relatively simple mathematical analysis we can give the following solutions:



 



 
24
22
00
3
22
00 0
22 π
cos 2expd1expdexp
1.2 1.2.3.42
2
sin2expd2expdexpexpd
1.2.3
u
ux ux
uxxxxxu
ux
uxx xuxx xux x
 
 

  




 




 
2
22
(18)
Thus, the Voigt profile can be written in the following form:
 




 

2222
3
0
4ln2 π
expcos2 expsin2 expexpd
π2
u
D
IaauuauuxxI





(19)
From this manner and according to the Equation (7) we express the Voigt function in the following way:





 
 

 

222
0
22
00
π
,expcos2 expsin2 expexpd
2
cos 2cos 2expdsin 2sin2expd
u
aa
Vauaau uau uxx
auuxx xauuxx x






2
(20)
Open Access AJAC
H. AMAMOU ET AL.
Open Access AJAC
731
Appendix 2
Table A1. Calculation of Voigt function very small values of the parameter a. We give also the calculation times of each 12 values of each column (D is here for 10).