Editing Lambert W function (section)

== Applications ==

=== Solving equations ===
The Lambert {{mvar|W}} function is used to solve equations in which the unknown quantity occurs both in the base and in the exponent, or both inside and outside of a logarithm. The strategy is to convert such an equation into one of the form {{math|1=''ze''<sup>''z''</sup> = ''w''}} and then to solve for {{mvar|z}} using the {{mvar|W}} function.

For example, the equation
: <math>3^x=2x+2</math>
(where {{mvar|x}} is an unknown real number) can be solved by rewriting it as
: <math>\begin{align} &(x+1)\ 3^{-x}=\frac{1}{2} & (\mbox{multiply by } 3^{-x}/2) \\
\Leftrightarrow\ &(-x-1)\ 3^{-x-1} = -\frac{1}{6} & (\mbox{multiply by } {-}1/3) \\
\Leftrightarrow\ &(\ln 3) (-x-1)\ e^{(\ln 3)(-x-1)} = -\frac{\ln 3}{6} & (\mbox{multiply by } \ln 3)
\end{align}</math>

This last equation has the desired form and the solutions for real ''x'' are:
: <math>(\ln 3) (-x-1) = W_0\left(\frac{-\ln 3}{6}\right) \ \ \ \textrm{ or }\ \ \ (\ln 3) (-x-1) = W_{-1}\left(\frac{-\ln 3}{6}\right) </math>
and thus:
: <math>x= -1-\frac{W_0\left(-\frac{\ln 3}{6}\right)}{\ln 3} = -0.79011\ldots \ \ \textrm{ or }\ \ x= -1-\frac{W_{-1}\left(-\frac{\ln 3}{6}\right)}{\ln 3} = 1.44456\ldots</math>

Generally, the solution to
: <math>x = a+b\,e^{cx}</math>
is:
: <math>x=a-\frac{1}{c}W(-bc\,e^{ac})</math>
where ''a'', ''b'', and ''c'' are complex constants, with ''b'' and ''c'' not equal to zero, and the ''W'' function is of any integer order.

=== Inviscid flows ===
Applying  the unusual  accelerating  [[Traveling wave|traveling-wave]] [[Ansatz]] in the form of   <math>\rho(\eta) = \rho\big(x-\frac{at^2}{2} \big)</math>  (where <math>\rho</math>, <math>\eta</math>, a, x and t are the density, the reduced variable, the acceleration, the spatial and the temporal variables) the fluid [[density]] of the corresponding [[Euler equations (fluid dynamics)|Euler equation]] can be given with the help of the W function.<ref>{{Cite journal |last1=Barna |first1=I.F. |last2=Mátyás |first2=L. |date=2013 |title=Analytic solutions for the one-dimensional compressible Euler equation with heat conduction closed with different kind of equation of states |url=http://mat76.mat.uni-miskolc.hu/mnotes/article/694 |journal=Miskolc Mathematical Notes |volume=13 |issue=3 |pages=785–799 |arxiv=1209.0607 |doi=10.18514/MMN.2013.694}}</ref> 

=== Viscous flows ===
Granular and debris flow fronts and deposits, and the fronts of viscous fluids in natural events and in laboratory experiments can be described by using the Lambert–Euler omega function as follows:
: <math>H(x)= 1 + W \left((H(0) -1) e^{(H(0)-1)-\frac{x}{L}}\right),</math>
where {{math|''H''(''x'')}} is the debris flow height, {{mvar|x}} is the channel downstream position, {{mvar|L}} is the unified model parameter consisting of several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient.

In [[pipe flow]], the Lambert W function is part of the explicit formulation of the [[Colebrook equation]] for finding the [[Darcy friction factor]]. This factor is used to determine the pressure drop through a straight run of pipe when the flow is [[turbulent]].<ref name="AAMore">{{cite journal |title = Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes |author = More, A. A. |journal = Chemical Engineering Science |volume = 61 |pages = 5515–5519 |year = 2006 |issue = 16 |doi = 10.1016/j.ces.2006.04.003 |bibcode = 2006ChEnS..61.5515M }}</ref>

=== Time-dependent flow in simple branch hydraulic systems ===
The principal branch of the Lambert {{mvar|W}} function is employed in the field of [[mechanical engineering]], in the study of time dependent transfer of [[Newtonian fluid]]s between two reservoirs with varying free surface levels, using centrifugal pumps.<ref>{{Cite journal |last1=Pellegrini |first1=C. C. |last2=Zappi |first2=G. A. |last3=Vilalta-Alonso |first3=G. |date=2022-05-12 |title=An Analytical Solution for the Time-Dependent Flow in Simple Branch Hydraulic Systems with Centrifugal Pumps |url=https://link.springer.com/10.1007/s13369-022-06864-9 |journal=Arabian Journal for Science and Engineering |volume=47 |issue=12 |pages=16273–16287 |language=en |doi=10.1007/s13369-022-06864-9 |s2cid=248762601 |issn=2193-567X}}</ref> The Lambert {{mvar|W}} function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes: 
<math display="block">\begin{align}
Q_\text{turb} &= \frac{Q_i}{\zeta_i} W_0\left[\zeta_i \, e^{(\zeta_i+\beta t/b)}\right]\\
Q_\text{lam} &= \frac{Q_i}{\xi_i} W_0\left[\xi_i \, e^{\left(\xi_i+\beta t/(b-\Gamma_1)\right)}\right]
\end{align}</math>
where <math>Q_i</math> is the initial flow rate and <math>t</math> is time.

=== Neuroimaging ===
The Lambert {{mvar|W}} function is employed in the field of neuroimaging for linking cerebral blood flow and oxygen consumption changes within a brain [[voxel]], to the corresponding blood oxygenation level dependent (BOLD) signal.<ref>{{cite journal |last1=Sotero |first1=Roberto C. |last2=Iturria-Medina |first2=Yasser |title=From Blood oxygenation level dependent (BOLD) signals to brain temperature maps |journal=Bull Math Biol |volume=73 |issue=11 |pages=2731–47 |year=2011 |doi=10.1007/s11538-011-9645-5 |pmid=21409512|s2cid=12080132 |url=http://precedings.nature.com/documents/4772/version/1 |type=Submitted manuscript }}</ref>

=== Chemical engineering ===
The Lambert {{mvar|W}} function is employed in the field of chemical engineering for modeling the porous electrode film thickness in a [[glassy carbon]] based [[supercapacitor]] for electrochemical energy storage. The Lambert {{mvar|W}} function provides an exact solution for a gas phase thermal activation process where growth of carbon film and combustion of the same film compete with each other.<ref>{{cite journal |last1=Braun |first1=Artur |last2=Wokaun |first2=Alexander |last3=Hermanns |first3=Heinz-Guenter |title=Analytical Solution to a Growth Problem with Two Moving Boundaries |journal=Appl Math Model |volume=27 |issue=1 |pages=47–52 |year=2003 |doi=10.1016/S0307-904X(02)00085-9|doi-access=free }}</ref><ref>{{cite journal |last1=Braun |first1=Artur |last2=Baertsch |first2=Martin |last3=Schnyder |first3=Bernhard |last4=Koetz |first4=Ruediger |title=A Model for the film growth in samples with two moving boundaries – An Application and Extension of the Unreacted-Core Model. |journal=Chem Eng Sci |volume=55 |issue=22 |pages=5273–5282 |year=2000 |doi=10.1016/S0009-2509(00)00143-3}}</ref>

=== Crystal growth ===
In the crystal growth, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient, <math display="inline">k</math>, and solute concentration in the melt, <math display="inline">C_L</math>,<ref>{{cite journal |last1=Asadian |first1=M |last2=Saeedi |first2=H |last3=Yadegari |first3=M |last4=Shojaee |first4=M |title=Determinations of equilibrium segregation, effective segregation and diffusion coefficients for Nd+3 doped in molten YAG |journal=Journal of Crystal Growth |date=June 2014 |volume=396 |issue=15 |pages=61–65|doi=10.1016/j.jcrysgro.2014.03.028 |bibcode=2014JCrGr.396...61A }} https://doi.org/10.1016/j.jcrysgro.2014.03.028</ref><ref>{{cite journal |last1=Asadian |first1=M |last2=Zabihi |first2=F |last3=Saeedi |first3=H |title=Segregation and constitutional supercooling in Nd:YAG Czochralski crystal growth |journal=Journal of Crystal Growth |date=March 2024 |volume=630 |issue= |pages=127605 |doi=10.1016/j.jcrysgro.2024.127605 |bibcode=2024JCrGr.63027605A |s2cid=267414096 }} https://doi.org/10.1016/j.jcrysgro.2024.127605</ref> from the [[Scheil equation]]:

<!--
k=W(Z)/ln(1-fs)
CL=(C0/(1-fs))exp(W(Z))
Z=(Cs/C0)(1-fs)ln(1-fs)

-->
: <math>\begin{align}
& k = \frac{W_0(Z)}{\ln(1-fs)} \\
& C_L=\frac{C_0}{(1-fs)} e^{W_0(Z)}\\
& Z = \frac{C_S}{C_0} (1-fs) \ln(1-fs) 
\end{align}
</math>

=== Materials science ===
The Lambert {{mvar|W}} function is employed in the field of [[Epitaxy|epitaxial film growth]] for the determination of the critical [[dislocation]] onset film thickness. This is the calculated thickness of an epitaxial film, where due to thermodynamic principles the film will develop crystallographic dislocations in order to minimise the elastic energy stored in the films. Prior to application of Lambert {{mvar|W}} for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert {{mvar|W}} turns it in an explicit equation for analytical handling with ease.<ref>{{cite journal |last1=Braun |first1=Artur |last2=Briggs |first2=Keith M. |last3=Boeni |first3=Peter |title=Analytical solution to Matthews' and Blakeslee's critical dislocation formation thickness of epitaxially grown thin films |journal=J Cryst Growth |volume=241 |issue=1–2 |pages=231–234 |year=2003 |doi=10.1016/S0022-0248(02)00941-7 |bibcode=2002JCrGr.241..231B}}</ref>

=== Semiconductor ===
It was shown that a W-function describes the relation between voltage, current and resistance in a diode.<ref>{{Cite journal |last1=Banwell |first1=T.C. |last2=Jayakumar |first2=A. |date=2000 |title=Exact analytical solution for current flow through diode with series resistance |url=https://digital-library.theiet.org/doi/10.1049/el%3A20000301 |journal=Elect. Letters |volume=36 |issue=1 |pages=29–33 |bibcode=2000ElL....36..291B |doi=10.1049/el:20000301}}</ref> 

=== Porous media ===
The Lambert {{mvar|W}} function has been employed in the field of fluid flow in porous media to model the tilt of an interface separating two gravitationally segregated fluids in a homogeneous tilted porous bed of constant dip and thickness where the heavier fluid, injected at the bottom end, displaces the lighter fluid that is produced at the same rate from the top end. The principal branch of the solution corresponds to stable displacements while the −1 branch applies if the displacement is unstable with the heavier fluid running underneath the lighter fluid.<ref>{{cite journal |last1=Colla |first1=Pietro |year=2014 |title=A New Analytical Method for the Motion of a Two-Phase Interface in a Tilted Porous Medium |journal=PROCEEDINGS, Thirty-Eighth Workshop on Geothermal Reservoir Engineering, Stanford University |volume=SGP-TR-202}}([https://pangea.stanford.edu/ERE/pdf/IGAstandard/SGW/2014/Colla.pdf])</ref>

=== Bernoulli numbers and Todd genus ===
The equation (linked with the generating functions of [[Bernoulli number]]s and [[Genus of a multiplicative sequence|Todd genus]]):
: <math> Y = \frac{X}{1-e^X}</math>
can be solved by means of the two real branches {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}}:
: <math> X(Y) = \begin{cases}
W_{-1}\left( Y e^Y\right) - W_0\left( Y e^Y\right) = Y - W_0\left( Y e^Y\right) &\text{for }Y < -1,\\
W_0\left( Y e^Y\right) - W_{-1}\left( Y e^Y\right) = Y - W_{-1}\left(Y e^Y\right) &\text{for }-1 < Y < 0.
\end{cases}</math>

This application shows that the branch difference of the {{mvar|W}} function can be employed in order to solve other transcendental equations.<ref>[https://web.archive.org/web/20150212084155/http://www.apmaths.uwo.ca/~djeffrey/Offprints/SYNASC2014.pdf D. J. Jeffrey and J. E. Jankowski, "Branch differences and Lambert ''W''"]</ref>

=== Statistics ===
The centroid of a set of histograms defined with respect to the symmetrized [[Kullback–Leibler divergence]] (also called the Jeffreys divergence <ref>{{cite journal |author=Flavia-Corina Mitroi-Symeonidis |author2=Ion Anghel |author3=Shigeru Furuichi |title=Encodings for the calculation of the permutation hypoentropy and their applications on full-scale compartment fire data |journal=Acta Technica Napocensis |date=2019 |volume=62, IV |pages=607–616}}</ref>) has a closed form using the Lambert {{mvar|W}} function.<ref>[https://ieeexplore.ieee.org/document/6509930 F. Nielsen, "Jeffreys Centroids: A Closed-Form Expression for Positive Histograms and a Guaranteed Tight Approximation for Frequency Histograms"]</ref>

=== Pooling of tests for infectious diseases ===
Solving for the optimal group size to pool tests so that at least one individual is infected involves the Lambert {{math|''W''}} function.<ref>https://arxiv.org/abs/2005.03051 J. Batson et al., "A COMPARISON OF GROUP TESTING ARCHITECTURES FOR COVID-19
TESTING".</ref><ref>[https://arxiv.org/abs/2004.01684 A.Z. Broder, "A Note on Double Pooling Tests"].</ref><ref>
{{cite journal
 |last1=Rudolf Hanel, Stefan Thurner
 |title=Boosting test-efficiency by pooled testing for SARS-CoV-2—Formula for optimal pool size
 |journal=PLOS ONE |date=2020 |volume=15, 11 |issue=11
 |pages=e0240652|doi=10.1371/journal.pone.0240652
 |pmid=33147228
 |pmc=7641378
 |bibcode=2020PLoSO..1540652H
 |doi-access=free
}}</ref>

=== Exact solutions of the Schrödinger equation ===
The Lambert {{mvar|W}} function appears in a quantum-mechanical potential, which affords the fifth – next to those of the [[harmonic oscillator]] plus centrifugal, the Coulomb plus inverse square, the Morse, and the [[inverse square root potential]] – exact solution to the stationary one-dimensional Schrödinger equation in terms of the confluent hypergeometric functions. The potential is given as
: <math> V = \frac{V_0}{1+W \left(e^{-\frac{x}{\sigma}}\right)}.</math>
A peculiarity of the solution is that each of the two fundamental solutions that compose the general solution of the Schrödinger equation is given by a combination of two confluent hypergeometric functions of an argument proportional to<ref>[https://arxiv.org/abs/1509.00846 A.M. Ishkhanyan, "The Lambert ''W'' barrier – an exactly solvable confluent hypergeometric potential"].</ref>
: <math> z = W \left(e^{-\frac{x}{\sigma}}\right).</math>

The Lambert {{mvar|W}} function also appears in the exact solution for the bound state energy of the one dimensional Schrödinger equation with a [[Delta potential#Double delta potential|Double Delta Potential]].

=== Exact solution of QCD coupling constant ===
In [[Quantum chromodynamics]], the [[quantum field theory]] of the [[Strong interaction]], the [[coupling constant]] <math>\alpha_\text{s}</math> is computed perturbatively, the order n corresponding to [[Feynman diagrams]] including n quantum loops.<ref name=PPNG_review_2016>{{cite journal | arxiv=1604.08082 | doi=10.1016/j.ppnp.2016.04.003 | title=The QCD running coupling | year=2016 | last1=Deur | first1=Alexandre | last2=Brodsky | first2=Stanley J. | last3=De Téramond | first3=Guy F. | journal=Progress in Particle and Nuclear Physics | volume=90 | pages=1–74 | bibcode=2016PrPNP..90....1D | s2cid=118854278 }}</ref> The first order, {{math|1=''n'' = 1}}, solution is exact (at that order) and analytical. At higher orders, {{math|1=''n'' > 1}}, there is no exact and analytical solution and one typically uses an [[iterative method]] to furnish an approximate solution. However, for second order, {{math|1=''n'' = 2}}, the Lambert function provides an exact (if non-analytical) solution.<ref name=PPNG_review_2016 />

=== Exact solutions of the Einstein vacuum equations ===
In the [[Schwarzschild metric]] solution of the Einstein vacuum equations, the {{mvar|W}} function is needed to go from the [[Eddington–Finkelstein coordinates]] to the Schwarzschild coordinates. For this reason, it also appears in the construction of the [[Kruskal–Szekeres coordinates]].

=== Resonances of the delta-shell potential ===
The s-wave resonances of the delta-shell potential can be written exactly in terms of the Lambert {{mvar|W}} function.<ref>{{cite journal |first=R. |last=de la Madrid |year=2017 |title=Numerical calculation of the decay widths, the decay constants, and the decay energy spectra of the resonances of the delta-shell potential |journal=Nucl. Phys. A |volume=962 |pages=24–45 |doi=10.1016/j.nuclphysa.2017.03.006 |arxiv=1704.00047 |bibcode=2017NuPhA.962...24D |s2cid=119218907 }}</ref>

=== Thermodynamic equilibrium ===
If a reaction involves reactants and products having [[heat capacity|heat capacities]] that are constant with temperature then the equilibrium constant {{mvar|K}} obeys
: <math>\ln K=\frac{a}{T}+b+c\ln T</math>
for some constants {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. When {{mvar|c}} (equal to {{math|{{sfrac|Δ''C<sub>p</sub>''|''R''}}}}) is not zero the value or values of {{mvar|T}} can be found where {{mvar|K}} equals a given value as follows, where {{mvar|L}} can be used for {{math|ln ''T''}}.
: <math>\begin{align}
-a&=(b-\ln K)T+cT\ln T\\
&=(b-\ln K)e^L+cLe^L\\[5pt]
-\frac{a}{c}&=\left(\frac{b-\ln K}{c}+L\right)e^L\\[5pt]
-\frac{a}{c}e^\frac{b-\ln K}{c}&=\left(L+\frac{b-\ln K}{c}\right)e^{L+\frac{b-\ln K}{c}}\\[5pt]
L&=W\left(-\frac{a}{c}e^\frac{b-\ln K}{c}\right)+\frac{\ln K-b}{c}\\[5pt]
T&=\exp\left(W\left(-\frac{a}{c}e^\frac{b-\ln K}{c}\right)+\frac{\ln K-b}{c}\right).
\end{align}</math>

If {{mvar|a}} and {{mvar|c}} have the same sign there will be either two solutions or none (or one if the argument of {{mvar|W}} is exactly {{math|−{{sfrac|1|''e''}}}}). (The upper solution may not be relevant.) If they have opposite signs, there will be one solution.

=== Phase separation of polymer mixtures ===
In the calculation of the phase diagram of thermodynamically incompatible polymer mixtures according to the [[Edmond-Ogston model]], the solutions for binodal and tie-lines are formulated in terms of Lambert {{mvar|W}} functions.<ref>{{Cite journal| last1=Bot|first1=A.|last2=Dewi|first2=B.P.C.|last3=Venema|first3=P.|year=2021|title=Phase-separating binary polymer mixtures: the degeneracy of the virial coefficients and their extraction from phase diagrams|journal=ACS Omega| volume=6| issue=11| pages=7862–7878| doi=10.1021/acsomega.1c00450|pmid=33778298|pmc=7992149|doi-access=free}}</ref>

=== Wien's displacement law in a ''D''-dimensional universe ===
Wien's displacement law is expressed as <math>\nu _{\max }/T=\alpha =\mathrm{const}</math>. With <math>x=h\nu _{\max } / k_\mathrm{B}T</math> and <math>d\rho _{T}\left( x\right) /dx=0</math>, where <math>\rho_{T}</math> is the spectral energy energy density, one finds <math>e^{-x}=1-\frac{x}{D}</math>, where <math>D</math> is the number of degrees of freedom for spatial translation. The solution <math>x=D+W\left( -De^{-D}\right)</math> shows that the spectral energy density is dependent on the dimensionality of the universe.<ref>{{cite journal |last1=Cardoso |first1=T. R. |last2=de Castro |first2=A. S. |title=The blackbody radiation in a {{mvar|D}}-dimensional universe |journal=Rev. Bras. Ens. Fis. |year=2005 |volume=27 |issue=4 |pages=559–563 |doi=10.1590/S1806-11172005000400007|doi-access=free |hdl=11449/211894 |hdl-access=free }}</ref>

=== AdS/CFT correspondence ===
The classical finite-size corrections to the dispersion relations of [[giant magnon]]s, single spikes and [[GKP string]]s can be expressed in terms of the Lambert {{mvar|W}} function.<ref>{{cite journal |first1=Emmanuel |last1=Floratos |first2=George |last2=Georgiou |first3=Georgios |last3=Linardopoulos |year=2014|title=Large-Spin Expansions of GKP Strings|journal=JHEP |volume=2014|issue=3 |pages=0180|arxiv=1311.5800|doi=10.1007/JHEP03(2014)018|bibcode=2014JHEP...03..018F |s2cid=53355961 }}</ref><ref>{{cite journal |first1=Emmanuel |last1=Floratos |first2=Georgios |last2=Linardopoulos |year=2015|title=Large-Spin and Large-Winding Expansions of Giant Magnons and Single Spikes|journal=Nucl. Phys. B |volume=897|pages=229–275|arxiv=1406.0796|doi= 10.1016/j.nuclphysb.2015.05.021|bibcode=2015NuPhB.897..229F |s2cid=118526569 }}</ref>

=== Epidemiology ===
In the {{math|''t'' → ∞}} limit of the [[Compartmental models in epidemiology#The SIR model|SIR model]], the proportion of susceptible and recovered individuals has a solution in terms of the Lambert {{mvar|W}} function.<ref>{{cite web |author1=Wolfram Research, Inc. |title=Mathematica, Version 12.1 |url=https://www.wolfram.com/mathematica |publisher=Champaign IL, 2020}}</ref>

=== Determination of the time of flight of a projectile ===
The total time of the journey of a projectile which experiences air resistance proportional to its velocity [[Projectile motion#Time of flight with air resistance|can be determined]] in exact form by using the Lambert {{math|''W''}} function.<ref>{{Cite journal |last1=Packel |first1=E. |last2=Yuen |first2=D. |date=2004 |title=Projectile motion with resistance and the Lambert W function |url=https://www.tandfonline.com/doi/abs/10.1080/07468342.2004.11922095?__cf_chl_tk=lMeR7eBE_W6KCd9mRFyIKS4ythC6fdxtjLT._Qtr73E-1735339489-1.0.1.1-x_kLQ7G4BLkGB79BTmnptVAM_5nuIi70PJ_IE47Mmc8 |journal=College Math. J. |volume=35 |issue=5 |pages=337–341 |doi=10.1080/07468342.2004.11922095}}</ref>

=== Electromagnetic surface wave propagation ===
The transcendental equation that appears in the determination of the propagation wave number of an electromagnetic axially symmetric surface wave (a low-attenuation single TM01 mode) propagating in a cylindrical metallic wire gives rise to an equation like {{math|1=''u'' ln ''u'' = ''v''}} (where {{mvar|u}} and {{mvar|v}} clump together the geometrical and physical factors of the problem), which is solved by the Lambert {{mvar|W}} function. The first solution to this problem, due to Sommerfeld ''circa'' 1898, already contained an iterative method to determine the value of the Lambert {{mvar|W}} function.<ref>{{cite journal | first = J. R. G. | last = Mendonça | year = 2019 | title = Electromagnetic surface wave propagation in a metallic wire and the Lambert {{mvar|W}} function | journal = American Journal of Physics | volume = 87 | issue = 6 | pages = 476–484 | arxiv = 1812.07456 | doi = 10.1119/1.5100943| bibcode = 2019AmJPh..87..476M | s2cid = 119661071 }}</ref>

=== Orthogonal trajectories of real ellipses ===
The family of ellipses <math>x^2+(1-\varepsilon^2)y^2 =\varepsilon^2</math> centered at <math>(0, 0)</math> is parameterized by eccentricity <math>\varepsilon</math>. The orthogonal trajectories of this family are given by the differential equation <math>\left ( \frac{1}{y}+y \right )dy=\left ( \frac{1}{x}-x \right )dx</math> whose general solution is the family <math>y^2=</math><math>W_0(x^2\exp(-2C-x^2))</math>.