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Power series for the Lambert W function

This development of the Lambert W function also applies to complex values of x

 


Taylor series

The Taylor series of W0 around 0 can be calculated with the inverse Lagrange theorem and is

The radius of convergence is −1/e, which you can determine in the ratio test. The function defined by this sequence can be extended to a holomorphic function defined for all complex numbers with a branching point along the interval (−∞, −1/e]. This holomorphic function is the main branch of the Lambert W function.

 


History

The German-Swiss mathematician Johann Heinrich Lambert (1728 - 1777) described this power series.


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