Thread: Gelfond's constant and complexity

1. Gelfond's constant and complexity

Gelfond's constant contains really odd complex math to me.

Consider that $\displaystyle e^{-i\pi} = -1$ is also proven by Taylor series.

So we can show that $\displaystyle e^{\pi} = (e^{-i\pi})^{i}} = (-1)^{i}$ and $\displaystyle e^{\pi} = (e^{i\pi})^{-i}} = (-1)^{-i}$ which goes against my normal logic.

Anyone know why it's shown like it is?

Thanks!

2. $\displaystyle (-1)^i=e^{i\log (-1)}=e^{i(0+\pi i)}=e^{-\pi}\neq e^{\pi}$

3. Thank you!!