This value $0^i$ is usually left undefined. But if to define it should be seemingly the most naturally assigned the value 1. Exponential function $a^z$ has the period $i\left|\frac{2 \pi }{\log (a)}\right|$. When $a$ comes to $0$, the period also comes to $0$. This means the value in $i$ should b...

Suppose, the following holds for all $z$. $$\frac z\pi \log \left(\frac{\tau -\frac{z}{\pi }+\frac{1}{2}}{\tau +\frac{z}{\pi }-\frac{1}{2}}\right)=\cos \left(2 \tau z \pm z\right)$$ What would be the algebraic properties of $\tau$ if it can exist?