TITLE: Why is $e^{a\pi i}\neq (-1)^a$?
QUESTION [5 upvotes]: Why are the following statements incorrect? I have trouble understanding my mistake. $$e^{a\cdot \pi i} = e^{\pi i^a} = (-1)^a $$ $$e^{a\cdot 2\pi i} = e^{2\pi i^a} = (1)^a =1 $$
Any clues would be appreciated!

REPLY [2 votes]: I'm going to presume that by $e^{2\pi i^a}$ you meant $(e^{2\pi i})^a.$ You seem to suggest that since $e^{2\pi i} = 1$ you should have $e^{2\pi ia} = (e^{2\pi i})^a = 1^a = 1$ for every value of $a.$
The problem is that although exponential functions are one-to-one functions when their arguments are real, they are not one-to-one with complex arguments, and that upsets these usual identities. It is not only when $b=1$ that one can have $b^x=1.$
In conventional usage, $a^{b^{\,c}}$ means $a^{\left( b^{\,c}\right)},$ not $(a^b)^c.$