E to the pi * i

1. Apr 15, 2010

endeavor



1. Compute all the values of $$e^ {\pi i}$$, 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:

$$e^{\pi i} = z^{\alpha} = e^{\alpha log(z)} = e^{\alpha (Log |z| + i arg(z))} = e^{\pi i (Log |e| + i arg(e))} = e^{\pi i (1 + i2k\pi)} = e^{\pi i}e^{-2\pi^{2}k} = - e^{-2\pi^{2}k}$$
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 :(

2. Apr 15, 2010

endeavor

something is wrong with LaTeX... it isn't displaying my tex right...

3. Apr 15, 2010

Staff: Mentor

4. Apr 15, 2010

NeoDevin

Is the complex exponential function invertible? (What is required for a function to have an inverse?)

5. Apr 16, 2010

endeavor

The function must be 1-1, right?

6. Apr 16, 2010

NeoDevin

Correct. Does the complex exponential satisfy this?

7. Apr 16, 2010

endeavor

Sorry, I misread your first question. So, no, the complex exponential is not an invertible function. Where does my initial post break down then?

8. Apr 16, 2010

NeoDevin

When you tried to invert it.