In mathematics, **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*; where *e*^{w} is the exponential function.

This means that for every complex number *z*, we have

*W*(*z*) e^{W(z)}=*z*

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

*z*(1 +*W*) d*W*/d*z*=*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* *e*^{x}, at which point the *W* function provides the solution. For instance, to solve the equation 2^{t} = 5*t*, we divide by 2^{t} to get 1 = 5*t* *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* e^{w}:

**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)