Help with complex analysis

Metric_Space

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.

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).

Homework Helper
Ok, so what's that formula for log(z)? You only have to worry about defining a continuous angle function in the half planes, right?

Metric_Space
I think it is:

Log z: = ln r + iθ = ln | z | + iArg z.

Staff Emeritus
Homework Helper
I think it is:

Log z: = ln r + iθ = ln | z | + iArg z.

So, this is well-defined in the upper-half plane right? Can you show it's also holomorphic there?

Metric_Space
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?

Homework Helper
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.

Staff Emeritus
Homework Helper
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?

Staff Emeritus
Homework Helper
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...

Homework Helper
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.

Metric_Space
how would I use the C-R equations in this case?

Staff Emeritus