Solve for C

Homework Statement

The originial equation I have been working with is

P = ((alpha^C)/(C Factorial))/(summation of alpha^J/J factorial), J going from 0 to C)

After approximately the denominator to e^alpha, and applying stirling's approximation on C factorial, further simplifying, I have the follow:

ln P = C ln alpha - (C ln C) + C - alpha

Solve for C (C is a variable, not constant):

P is given, alpha is given, so the only unknown is C, but after spending hours on trying various things, I just could not isolate C as itself and come up with the solution. Please help

Thanks for the guidance! Actually I also tried looking at the Lambert W too, from the website I can understand how they come up with the W number, but how is W (value) calculated for example ..

Can you please enlighten me? Thanks

Borek
Mentor
To quote wikipedia:

The Lambert W relation cannot be expressed in terms of elementary functions.

Wiki also gives some ways to calculate value of W(x) using series.