Can there be a definite solution for positive real values of a?
My musings so far:
0^{ix} = e^{ix\ln0} = (\cos x + i\sin x)^{\ln0} = ( \cos x + i\sin x )^{-\infty} = \left( e^{-\infty} \right)^\ln( \cos x + i\sin x ) } = 0^{\ln( \cos x + i\sin x )}
Since 0^0 can be assigned to produce...
Thanks to everyone participating, I think I got sorted it out by inspecting a similar behaving sine and its infinite product.
\sin x = x \prod_{n = 1}^\infty\left(1 - \frac{x^2}{\pi^2 n^2}\right)
If we assume Asin(pi*A*x)=Bsin(pi*B*x) we may easily inspect the existence of trivial roots...