By implicit differentiation, one can show that W satisfies the differential equation
The Taylor series of W0 around 0 can be found using the Lagrange inversion theorem and is given by
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 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).
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:
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