# exponential analogues via replacing n! with n^n in the definition of exp(z)

Here’s some facts about what I’ve been calling Wāhi functions (after a Polynesian particle meaning “almost”) and Sivaram Ambikasaran has been calling Upsilon functions. (I have a copy of Erdelyi’s Bateman manuscript project to stare at, so I’m really hesitant to want to create more special functions named after greek letters — there are already two eta functions around!)

ʍ(z) = $1 + \sum_{n=1}^{\infty} \frac{z^{n}}{n^{n}}$

(one can make analogies for the ordinary trigonometric and the hyperbolic functions pretty easily, and I expect to get them domain colored and locate all their roots fairly close to the origin pretty soon.)

Here are two domain colored plots of ʍ(z):

To 100 decimal places, the real root of this function is
-1.4037610512177590552812326104857766034054904247513964159403731011554059273640619168745389681867183

The first root above the real axis, to as much precision as I could determine using Newton’s method is:
-8.442450690545156496022687549628700429619597265139327 + 15.810919833506912457867624737264102106899387937553i