Existence of (complex) limit z->0 (z^a)

[Knissp]

Homework Statement

Justify for which complex values of a does the principal value of z^a have a limit as z tends to 0?

Homework Equations

z^a = e^{a log(z)}

log(z) = log|z| + (i) (arg(z))

The Attempt at a Solution

Lim_{z \rightarrow 0} z^a = Lim_{z \rightarrow 0} e^{(a) (log(z))}

=Lim_{|z| \rightarrow 0} e^{(a) (log|z|) + (i) (a) (arg(z))}

Let a = u + i v.

=Lim_{|z| \rightarrow 0} e^{(u+iv) (log|z| + (i) (u+iv) (arg(z)))}

=Lim_{|z| \rightarrow 0} e^{(u) (log|z|) + (i) (v) (log|z|) + (i) (u) (arg(z)) - (v) (arg(z))}

=Lim_{|z| \rightarrow 0} e^{(u) (log|z|)} e^{(i) (v) (log|z|)} e^{(i) (u) (arg(z))} e^{-v (arg(z))}

=Lim_{|z| \rightarrow 0} |z|^u e^{(i) (v) (log|z|)} e^{((i) (u) - (v)) (arg(z))}

I just noticed a big mistake here, so I'm erasing this part. Any ideas?

 

[tiny-tim]

Hi Knissp!

Isn't it easier just to go polar, and put z = re^iθ ?