The Lambert *W* function, defined implicitly as the inverse of the nonlinear transcendental equation

$W\left(z\right){e}^{W\left(z\right)}=z$

is notoriously difficult to handle as one would a typical analytic function. From the defining equation one can deduce the asymptotic approximation

$W\left(z\right)\approx lnz-lnlnz$

which in practice is used as the starting point for numerical inversion of the equation. The appearance of the logarithm, however, means that one has a lovely singularity at the origin. This asymptotic expression is thus not well suited to analytic treatment there.

The derivative of the function is considerably more complicated than many other functions,

${W}^{\prime}\left(z\right)=\frac{1}{{e}^{W}(1+W)}=\frac{1}{{e}^{W}+z}$

which means that manipulation of expressions containing the Lambert function will be considerably more involved than other functions. One can of course evaluate a power series for the function,

$W\left(z\right)=\sum _{k=1}^{\infty}\frac{(-k{)}^{k-1}}{k!}\phantom{\rule{.2em}{0ex}}{z}^{k}$

the derivation of which generally presumes a knowledge of Lagrange inversion. Again, not what one expects in the context of most analytic functions.

It would perhaps be interesting to identify approximations to the Lambert function in terms of more common analytic functions. One way to do this is through delay differential equations. The simple equation

${f}^{\prime}\left(t\right)=f(t-T)$

has the exact solution

$f\left(t\right)=exp\left[\frac{W\left(T\right)}{T}t\right]$

which is found by substituting an exponential form into the equation and comparing the result to the defining equation of the Lambert function. Given a numerical implementation of the the Lambert function, such as that in Math, one is set to work with the exact solution at least numerically.

Another way to solve the delay differential equation is by successive approximations. For an initial approximation, expand the right-hand side of the equation to first order in the delay:

$\begin{array}{l}{f}^{\prime}\left(t\right)\approx f\left(t\right)-T{f}^{\prime}\left(t\right)\\ \frac{{f}^{\prime}\left(t\right)}{f\left(t\right)}\approx \frac{1}{1+T}\\ f\left(t\right)\approx exp\left(\frac{t}{1+T}\right)\end{array}$

The constant of integration is chosen here so that one can make the identification

$W\left(T\right)\approx \frac{T}{1+T}\approx T-{T}^{2}+{T}^{3}-\cdots $

upon comparison to the exact solution. This initial approximation only captures the first two terms of the exact power series, which can be seen from the expression above or the first line of this Mathematica notebook.

The expansion of the right-hand side need not stop at the first order. For a second-order approximation, and assuming a solution of exponential form $f={e}^{ct}$, one has

$\begin{array}{l}{f}^{\prime}\left(t\right)\approx f\left(t\right)-T{f}^{\prime}\left(t\right)+\frac{{T}^{2}}{2}{f}^{\u2033}\left(t\right)\\ \phantom{\rule{1.5em}{0ex}}c=1-cT+\frac{{c}^{2}{T}^{2}}{2}\\ \phantom{\rule{1.5em}{0ex}}c=\frac{1+T\pm \sqrt{1+2T-{T}^{2}}}{{T}^{2}}\end{array}$

Choosing the negative sign to keep the solution finite as the delay goes to zero, one has

$W\left(T\right)\approx \frac{1+T-\sqrt{1+2T-{T}^{2}}}{T}\approx T-{T}^{2}+\frac{3}{2}{T}^{3}-\frac{5}{2}{T}^{4}+\cdots $

which captures an additional term in the exact power series. Continuing this to third and fourth order adds an additional term at each order, as can be seen in the linked notebook.

This polynomial approximation clearly does not approach the Lambert function quickly enough considering the increasing complexity of exact solutions, and ends in practice anyway with the fourth order. Consider another way to extend an initial exponential estimate to higher order by writing the full solution in the form

$f\left(t\right)={e}^{{C}_{1}t}g\left(t\right)$

Inserting this in the delay differential equation gives

$\begin{array}{c}\left[{C}_{1}g\right(t)+{g}^{\prime}(t\left)\right]{e}^{{C}_{1}t}={e}^{{C}_{1}(t-T)}\left[g\right(t)-T{g}^{\prime}(t\left)\right]\\ {g}^{\prime}\left(t\right)[1+T{e}^{-{C}_{1}T}]=g\left(t\right)[{e}^{-{C}_{1}T}-{C}_{1}]\\ \frac{{g}^{\prime}}{g}=\frac{1-{C}_{1}{e}^{{C}_{1}T}}{T+{e}^{{C}_{1}T}}\\ g\left(t\right)=exp\left[\frac{1-{C}_{1}{e}^{{C}_{1}T}}{T+{e}^{{C}_{1}T}}t\right]\end{array}$

so that the next approximation to the solution is

$f\left(t\right)=exp\left[\right({C}_{1}+\frac{1-{C}_{1}{e}^{{C}_{1}T}}{T+{e}^{{C}_{1}T}}\left)t\right]=exp\left[\frac{1+{C}_{1}T}{T+{e}^{{C}_{1}T}}t\right]\equiv {e}^{{C}_{2}t}$

Thanks to the multiplicative separability of the exponential function, the process can be iterated any number of times with the rule

${C}_{n+1}=\frac{1+{C}_{n}T}{T+{e}^{{C}_{n}T}}$

so that one has successive approximations to the Lambert function of the form

$W\left(T\right)\approx T{C}_{n}\phantom{\rule{5em}{0ex}}n=1,2,3,\dots $

The first approximation, giving only two terms of the exact power series, has already been given above. The second functional approximation is relatively simple to write out

$W\left(T\right)\approx T\frac{1+\frac{T}{1+T}}{T+exp\left(\frac{T}{1+T}\right)}=\frac{T}{1+T}\frac{1+2T}{T+exp\left(\frac{T}{1+T}\right)}$

and as can be seen from the notebook gives the first six terms of the powers series, twice as many as the corresponding polynomial approximation. The third functional approximation reproduces fourteen terms of the exact power series, roughly doubling the number of correct terms. The fourth and fifth iterations continue the doubling, correctly reproducing thirty and sixty-two terms, respectively.

There are naturally caveats to the method. The successive functional approximations are not simple to write out in full, but as long as one has access to a CAS like Mathematica then one need not write them explicitly. The exact power series itself does not converge quickly and is inferior to numerical inversion for accurate evaluation of the function, but that is not the point of this presentation. What is desired here is a generally accurate analytic approximation of the function near the origin, and these approximations are surprisingly good for that purpose.

At least in the right-hand complex plane, that is. The Lambert function of course has a branch cut along the negative real axis, and this cannot be described in principle by single-valued functions. Here is a visualization of the exact numerical function along with a selectable one of the iterated set of approximations. The latter is displaced upward slightly to avoid z-fighting between the two surfaces:

In retrospect it should be obvious that there would be poles in the approximation in the vicinity of the negative real axis, since a series of poles can approximate the mathematical contribution of a branch cut, as is typically done in *S*-matrix theory. The functional approximation is thus only useful in the left-hand complex plane in the sense in which a full analytic *S*-matrix is approximated by simpler complex functions.

In light of the poles appearing in the previous approximation, then is another approach one can take: a Padé approximant. This is a rational function of two polynomials that reproduces a power series up to some given order. For the Lambert *W* function, the first few so-called diagonal approximants are

${P}_{1}^{1}\left(T\right)=\frac{T}{1+T}$

${P}_{2}^{2}\left(T\right)=\frac{T+\frac{4}{3}{T}^{2}}{1+\frac{7}{3}T+\frac{5}{6}{T}^{2}}$

${P}_{3}^{3}\left(T\right)=\frac{T+\frac{228}{85}{T}^{2}+\frac{451}{340}{T}^{3}}{1+\frac{313}{85}T+\frac{1193}{340}{T}^{2}+\frac{133}{204}{T}^{3}}$

as evaluated in the previously mentioned notebook.

Numerical Padé approximants are available in Math and Taylor coefficients around the origin are given at the outset. Together these can be used to visualize successive approximants compared to the exact Lambert function. The approximation is again displaced upward slightly to avoid z-fighting between the two surfaces:

The first Padé approximant is identical to the first iteration of the previous approximation. Successive approximants approach the exact function in the right-hand complex plane, but are noticeably worse in the left-hand plane than the previous approximation. Clearly the pole structure of the approximant is not complicated enough to account for the branch cut.

This presentation will be extended in the future as other analytic approximations are encountered.

*Uploaded 2021.12.02 — Updated 2022.12.06*
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