Solving equations General case The Lambert 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 and then to solve for using the function. For example, the equation : 3^x=2x+2 (where is an unknown
real number) can be solved by rewriting it as : \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} This last equation has the desired form and the solutions for real
x are: : (\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) and thus: : 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 Generally, the solution to : x = a+b\,e^{cx} is: : x=a-\frac{1}{c}W(-bc\,e^{ac}) 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.
Super root Along with the topic of the so called
Sophomore's dream the
tetration function f(x) = x^x became a well known function. Its inverse function is a special case of the so called super root and it can be determined and displayed as follows: : x^x = y The power rule gives following expression: : \exp[x\ln(x)] = y The natural logarithm of that is taken: : x\ln(x) = \ln(y) The Lambert W function is used now: : \ln(x) = W_{0}[\ln(y)] And in the final step the second last equation will be divided by the last equation: : x = \frac{\ln(y)}{W_{0}[\ln(y)]} A calculation example is made: : x^x = 2 : x = \ln(2) \div W_{0}[\ln(2)] \approx 1.559610469462369349970388768765
Tree counting and combinatorics Cayley's formula states that the number of
tree graphs on
n labeled vertices is n^{n-2}, so that the number of trees with a designated root vertex is n^{n-1}. The
exponential generating function of this counting sequence is:T(x) = \sum_{n=0}^\infty \frac{n^{n-1}}{n!} x^n.The class of rooted trees has a natural recurrence: a rooted tree is equivalent to a root vertex attached to a set of smaller rooted trees. Using the
exponential formula for labeled combinatorial classes, this translates into the equation:T(x) = x e^{T(x)}, which implies -T(-x)e^{-T(-x)} = x and W_0(x) = -T(-x). Reversing the argument, the
Maclaurin series of W_0(x) around x=0 can be found directly using the
Lagrange inversion theorem: : W_0(x)=\sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}x^n, and this gives the standard
analytic proof of Cayley's formula. But the Maclaurin series radius of convergence is limited to |x| because of the branch point at x=-1/e.
Inviscid flows Applying the unusual accelerating
traveling-wave Ansatz in the form of \rho(\eta) = \rho\big(x-\frac{at^2}{2} \big) (where \rho, \eta, 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 equation can be given with the help of the W function.
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: : H(x)= 1 + W \left((H(0) -1) e^{(H(0)-1)-\frac{x}{L}}\right), where is the debris flow height, is the channel downstream position, 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.
Time-dependent flow in simple branch hydraulic systems The principal branch of the Lambert function is employed in the field of
mechanical engineering, in the study of time dependent transfer of
Newtonian fluids between two reservoirs with varying free surface levels, using centrifugal pumps. The Lambert function provided an exact solution to the flow rate of fluid in both the laminar and turbulent regimes: \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} where Q_i is the initial flow rate and t is time.
Neuroimaging The Lambert 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.
Chemical engineering The Lambert 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 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.
Crystal growth In the
crystal growth, the negative principal of the Lambert W-function can be used to calculate the distribution coefficient, k, and solute concentration in the melt, C_L, from the
Scheil equation: : \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}
Materials science The Lambert function is employed in the field of
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 for this problem, the critical thickness had to be determined via solving an implicit equation. Lambert turns it in an explicit equation for analytical handling with ease.
Semiconductor devices It was shown that a W-function describes the relation between voltage, current and resistance in a diode. The use of the Lambert
W Function to analytically and exactly solve the terminals' current and voltage as explicit functions of each other in a circuit model of a diode with both series and shunt resistances was first reported in the year 2000. The Lambert
W Function was introduced into compact modeling of MOSFETs in 2003 as a useful mathematical tool to explicitly describe the surface potential in undoped channels. The Lambert
W function-based explicit analytic solution of the illuminated photovoltaic solar cell single-diode model with parasitic series and shunt resistance was published in 2004.
Porous media The Lambert 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.
Bernoulli numbers and Todd genus The equation (linked with the generating functions of
Bernoulli numbers and
Todd genus): : Y = \frac{X}{1-e^X} can be solved by means of the two real branches and : : 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 This application shows that the branch difference of the function can be employed in order to solve other transcendental equations.
Statistics The centroid of a set of histograms defined with respect to the symmetrized
Kullback–Leibler divergence (also called the Jeffreys divergence ) has a closed form using the Lambert function.
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 function.
Exact solutions of the Schrödinger equation The Lambert 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 : V = \frac{V_0}{1+W \left(e^{-\frac{x}{\sigma}}\right)}. 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 : z = W \left(e^{-\frac{x}{\sigma}}\right). The Lambert function also appears in the exact solution for the
bound state energy of the one dimensional Schrödinger equation with a
Double Delta Potential.
Exact solution of QCD coupling constant In
Quantum chromodynamics, the
quantum field theory of the
Strong interaction, the
coupling constant \alpha_\text{s} is computed perturbatively, the order n corresponding to
Feynman diagrams including n quantum loops. The first order, , solution is exact (at that order) and analytical. At higher orders, , there is no exact and analytical solution and one typically uses an
iterative method to furnish an approximate solution. However, for second order, , the Lambert function provides an exact (if non-analytical) solution.
Thermodynamic equilibrium If a reaction involves reactants and products having
heat capacities that are constant with temperature then the equilibrium constant obeys : \ln K=\frac{a}{T}+b+c\ln T for some constants , , and . When (equal to ) is not zero the value or values of can be found where equals a given value as follows, where can be used for . : \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} If and have the same sign there will be either two solutions or none (or one if the argument of is exactly ). (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 functions.
Wien's displacement law in a D-dimensional universe Wien's displacement law is expressed as \nu _{\max }/T=\alpha =\mathrm{const}. With x=h\nu _{\max } / k_\mathrm{B}T and d\rho _{T}\left( x\right) /dx=0, where \rho_{T} is the spectral energy energy density, one finds e^{-x}=1-\frac{x}{D}, where D is the number of degrees of freedom for spatial translation. The solution x=D+W\left( -De^{-D}\right) shows that the spectral energy density is dependent on the dimensionality of the universe.
AdS/CFT correspondence The classical finite-size corrections to the dispersion relations of
giant magnons, single spikes and
GKP strings can be expressed in terms of the Lambert function.
Epidemiology In the limit of the
SIR model, the proportion of susceptible and recovered individuals has a solution in terms of the Lambert function.
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
can be determined in exact form by using the Lambert function.
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 (where and clump together the geometrical and physical factors of the problem), which is solved by the Lambert function. The first solution to this problem, due to Sommerfeld
circa 1898, already contained an iterative method to determine the value of the Lambert function.
Orthogonal trajectories of real ellipses The family of ellipses x^2+(1-\varepsilon^2)y^2 =\varepsilon^2 centered at (0, 0) is parameterized by eccentricity \varepsilon. The orthogonal trajectories of this family are given by the differential equation \left ( \frac{1}{y}+y \right )dy=\left ( \frac{1}{x}-x \right )dx whose general solution is the family y^2=W_0(x^2\exp(-2C-x^2)). == Generalizations ==