Lambert's W function

Lambert's W function, named after Johann Heinrich Lambert, also called the Omega function, is the inverse function of f(w) = w e^w for complex numbers w. This means that for every complex number z, we have

W(z) e^W(z) = z

Since the function f is not injective, the function W is multivalued. If we restrict to real arguments x ≥ -1/e and demand w≥-1, then a single valued function W₀(x) is defined, whose graph is shown. We have W₀(0) = 0 and W₀(-1/e) = -1.

The Lambert W function cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials and also occurs in the solution of time-delayed differential equations, such as y'(t) = a y(t - 1).

By implicit differentiation, one can show that W satisfies the differential equation

z (1 + W) dW/dz   =   W      for z ≠ -1/e.

The Taylor series of W₀ around 0 can be found using the Lagrange inversion theorem and is given by

<math>

W_0 (x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}\ x^n </math>

The radius of convergence is 1/e. This can be extended to a holomorphic function defined on all complex numbers except the real interval (-∞, -1/e]; this holomorphic function is also called the prinicipal branch of the Lambert W function.

Many equations involving exponentials can be solved using the W function. The general strategy is to move all instances of the unknown to one side of the equation and make it look like x e^x, at which point the W function provides the solution. For instance, to solve the equation 2^t = 5t, we divide by 2^t to get 1 = 5t e^-ln(2)t, then divide by 5 and multiply by -ln(2) to get -ln(2)/5 = -ln(2)t e^-ln(2)t. Now application of the W function yields -ln(2)t = W(-ln(2)/5), i.e. t = -W(-ln(2)/5) / ln(2).

The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = w e^w:

<math>

\int W(x)\, dx = x \left( W(x) - 1 + \frac{1}{W(x)} \right) + C </math>

References:

• Corless et.al. "On the Lambert W function" Adv. Computational Maths. 5, 329 - 359 (1996). http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.ps (PostScript)