elementary superfunctions

[bo198214]

And also as we had seen, for 1/x there are also two different solutions. It is an open question thus how much independent superfunctions has a given function.

Its not an open question, there are infinitely many (even for real-analytic solutions which \( 1/x \) does not have).
If you have one solution \( F \) just take any 1-periodic function \( \theta \) and then \( F(x+\theta(x)) \) is another solution. Even elementary if \( \theta \) is elementary (say linear combination of some \( \sin(2\pi k x) \)).

Thatswhy I always try to find for elementary solutions whether they are regular at some fixed point because this reduces the number of real analytic solutions to two at one fixed point (analogously to \( \exp_{\sqrt{2}} \)) up to x-translation.