Finding Holomorphic Logarithmic Formulas for Half-Planes in Complex Analysis

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Metric_Space
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Homework Statement




i) Find a suitable formula for log z when z lies in the half-plane K that lies above the x-axis, and
from that show log is holomorphic on K

ii) Find a suitable formula for log z when z lies in the half-plane L that lies below the x-axis, and
from that show log is holomorphic on L.

Homework Equations





The Attempt at a Solution



I've found a formula log Z on wikipedia but not sure how to relate it to the half plane(s).
 
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I think it is:

Log z: = ln r + iθ = ln | z | + iArg z.
 
I think it's well-defined on the upper-half plane.

Not sure how to show it's holomorphic -- should I just substitute into the definition of derivative?
 
micromass said:
So, this is well-defined in the upper-half plane right? Can you show it's also holomorphic there?

I wonder if they mean the closed half-plane? The problem would be a little more challenging if they do.
 
Dick said:
I wonder if they mean the closed half-plane? The problem would be a little more challenging if they do.

Good point. But is it even true in the closed half plane? You can easily pick your line of discontinuity to lie under the x-axis, but you cannot define the logarithm of 0 in any satisfying way, can you?
 
Metric_Space said:
I think it's well-defined on the upper-half plane.

Not sure how to show it's holomorphic -- should I just substitute into the definition of derivative?

You could do that. But if I were you, I would try the Cauchy-Riemann equations though. Maybe you can even use the inverse function theorem...
 
micromass said:
Good point. But is it even true in the closed half plane? You can easily pick your line of discontinuity to lie under the x-axis, but you cannot define the logarithm of 0 in any satisfying way, can you?

Good point also. Guess I was thinking closed half plane\{0}. So never mind.
 
how would I use the C-R equations in this case?