Manipulation of exponents in C.

Question

Show that if $ z \neq 0 $ and $a$ is a real number, then $|z^a | = e^{a \ln{|z|} } = |z|^{a} $ where the principal value of $ |z|^{a}$ is to be taken.

The far right seems to be the easiest to start with.

$ |z|^{a}= e^{a \log {|z|}} $ my best attempt at the second part was to write

$|z^a |= | z *z *z *z*....*z|$ ( say we have $a$ terms of z) $ = e^ {\log {|z*z*z*...*z|} }= \exp ( \log |z| +\log |z| + \log |z| + ... + \log |z| )= \exp( a \log |z|) = e^{a \log {|z|}} $

Which feels like it makes sense until one realizes $a$ is not an integers. though even if $a$ was rational i think you could expand this way but an irrational value doesn't make a lot of sense.

Any ideas how to show this is true for all $a$ that are not imaginary?

Answer 1

Write $z$ in polar. $z=|z|e^{i\theta}.$ Then $z^a=|z|^ae^{ia\theta}$ so that when we take the absolute value $|z^a| = |z|^a$ since $|e^{ia\theta}|=1$ when $a$ is real.

$|z|^a = e^{a\ln|z|}$ by the definition of natural log.