W_{k} (z e^{z}) = z, z \in S_{k} \subset C

where $S_{k}$ is the range of the $k^{th}$ branch of $W$ , i.e. the branch of $W_{k}$

Example

A^{x} + x = B, A^{x} = B - x 1 = (B - x) A^{- x} A^{B} = (B - x) A^{B - x} A^{B} \cdot ln A = ln A (B - x) e^{l n A (B - x)} W (A^{B} \cdot ln A) = W (ln A (B - x) e^{l n A (B - x)}) W (A^{B} \cdot ln A) = ln A (B - x) x = B - \frac{W ( A ^{B} \cdot ln A )}{ln A}

Notes