Editing Lambert W function (section)

=== Bernoulli numbers and Todd genus ===
The equation (linked with the generating functions of [[Bernoulli number]]s and [[Genus of a multiplicative sequence|Todd genus]]):
: <math> Y = \frac{X}{1-e^X}</math>
can be solved by means of the two real branches {{math|''W''<sub>0</sub>}} and {{math|''W''<sub>−1</sub>}}:
: <math> X(Y) = \begin{cases}
W_{-1}\left( Y e^Y\right) - W_0\left( Y e^Y\right) = Y - W_0\left( Y e^Y\right) &\text{for }Y < -1,\\
W_0\left( Y e^Y\right) - W_{-1}\left( Y e^Y\right) = Y - W_{-1}\left(Y e^Y\right) &\text{for }-1 < Y < 0.
\end{cases}</math>

This application shows that the branch difference of the {{mvar|W}} function can be employed in order to solve other transcendental equations.<ref>[https://web.archive.org/web/20150212084155/http://www.apmaths.uwo.ca/~djeffrey/Offprints/SYNASC2014.pdf D. J. Jeffrey and J. E. Jankowski, "Branch differences and Lambert ''W''"]</ref>