06-1350/Some Equations by Kurlin
Claim [Kurlin]. The function defined by
If this class is successful, at some point in the future I will tell you why I care. But meanwhile a few questions to expose my ignorance of simple algebra. Note that is is a rational function in its variables and their exponentials. Let's call such functions "tame". The equation satisfies involves rational expressions in the variables and their exponentials and linear substitutions applied to . Let us call such equations "tame". (The equation is also linear in . You may or may not include this in the definition of "tame"). So we see that a tame equation has a tame solution.
- Is that always the case?
- A prety example is the tame equation , whose solution is the "first" transcendental function .
- Are we done defining "new" special functions after we've defined or are there further ones that cannot be defined in terms of it?
- If there are further ones, why haven't we heard about them? Or else, where do we read about them?
- If there aren't further ones, this is a lovely "closure" property