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By implicit differentiation, one can show that W satisfies the differential equation
The Taylor series of W_{0} 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 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}:
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