Originally Posted by

**InvisibleMan** The problem is obviously in the last step.

When you rasing some number by a complex you should use the following definition:

$\displaystyle z^i = exp( i Log z)$

So in your case we have $\displaystyle z=1=exp (2\pi i n)$, raising to the power i we get:

$\displaystyle exp( i Log(exp(2 \pi i n)))$

Now remember the definitino of Log in complex plane:

$\displaystyle Log(z)= log |z| + i arg(z)$, and this function is multivalued so you need to pick a branch cut for the argument.

In your case obviosuly the equality is wrong cause one side of the equality you pick n=0 and on the other you picked another different number, you can't do this.

Edit: Does some know why the tex doesn't get compiled?

Thanks, yeah, you're completely correct about the log. I realized it had to be something like that but couldn't put 2 and 2 together!

Tex is broken so I suggest you use http://www.quicklatex.com/ to make images that you can imbed. Your (slightly modified tex) can be rendered as:

Which is generated by

Code:

\begin{align*}
z^i = \exp( i \log (z)) \\
z=1=\exp (2\pi i n) \\
\exp( i \log(\exp(2 \pi i n))) \\
\log(z)= \log |z| + i \arg(z)
\end{align*}