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1. Compute all the values of [tex] e^ {\pi i} [/tex], indicating clearly whether there is just one or many of them.

Trivially, exp(pi * i) = -1. However, we can also consider e to be the complex number z, and pi * i to be the complex number alpha. Then we get:

[tex]e^{\pi i} = z^{\alpha} = e^{\alpha log(z)}

= e^{\alpha (Log |z| + i arg(z))}

= e^{\pi i (Log |e| + i arg(e))}

= e^{\pi i (1 + i2k\pi)}

= e^{\pi i}e^{-2\pi^{2}k}

= - e^{-2\pi^{2}k}

[/tex]

where k is an integer.

So what exactly is going on here? does exp(pi*i) = -1 or -exp(-2kpi^2)??

P.S. I hope all this tex doesn't mess up :(