Synopsis:

The expression: \[ f(w) = w \cdot e^{w}, \, f(w) \in \mathbb{C}. \] Contains the following observations:

Illustration One:

\[ W(2) = W_{0}(2), \, W_{0}(2) < 1, \, W_{0}(2) \approx 0.852. \]

Details for Illustration One:

\[ W(2) = 2 \cdot e^{2}. \]

Illustration Two:

\[ W(2i) = -1.818594853651363390792039731823489685404509942895780536757946`61. - 0.83229367309428477399513645900152437953200154215108978151029994`61.9 I \]

Details for Illustration Two:

\[ W(2i) = 2i \cdot e^{2i}. \]

Notes for Illustration Two:

Unlike \( W(2) \), \( W(2i) \) is a transcendental number.