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Knissp
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Homework Statement
Justify for which complex values of a does the principal value of [tex]z^a[/tex] have a limit as z tends to 0?
Homework Equations
[tex]z^a = e^{a log(z)} [/tex]
[tex]log(z) = log|z| + (i) (arg(z)) [/tex]
The Attempt at a Solution
[tex]Lim_{z \rightarrow 0} z^a = Lim_{z \rightarrow 0} e^{(a) (log(z))} [/tex]
[tex]=Lim_{|z| \rightarrow 0} e^{(a) (log|z|) + (i) (a) (arg(z))}[/tex]
Let [tex]a = u + i v [/tex].
[tex]=Lim_{|z| \rightarrow 0} e^{(u+iv) (log|z| + (i) (u+iv) (arg(z)))}[/tex]
[tex]=Lim_{|z| \rightarrow 0} e^{(u) (log|z|) + (i) (v) (log|z|) + (i) (u) (arg(z)) - (v) (arg(z))}[/tex]
[tex]=Lim_{|z| \rightarrow 0} e^{(u) (log|z|)} e^{(i) (v) (log|z|)} e^{(i) (u) (arg(z))} e^{-v (arg(z))}[/tex]
[tex]=Lim_{|z| \rightarrow 0} |z|^u e^{(i) (v) (log|z|)} e^{((i) (u) - (v)) (arg(z))}[/tex]
I just noticed a big mistake here, so I'm erasing this part. Any ideas?
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