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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 *W*_{0} 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.

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