In mathematics, Lambert's W function, named after Johann Heinrich Lambert, also called the Omega function, is the inverse function of f(w) = w.ew for complex numbers w; where ew is the exponential function.
This means that for every complex number z, we have
- W(z) eW(z) = z
By implicit differentiation, one can show that W satisfies the differential equation
- z (1 + W) dW/dz = W for z ≠ −1/e.
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 ex, at which point the W function provides the solution. For instance, to solve the equation 2t = 5t, we divide by 2t 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).
Similar techniques show that has solution .
The function W(x), and many expressions involving W(x), can be integrated using the substitution w = W(x), i.e. x = w ew:
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)