Finding Holomorphic Logarithmic Formulas for Half-Planes in 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.

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).

 

[Dick]

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.

 

[micromass]

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?

 

[Dick]

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.

 

[micromass]

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?

 

[micromass]

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...

 

[Dick]

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?

 

[micromass]

Just find the partial derivatives, show they are continuous and show that the CR-equations are satisfied. This would imply holomorphicness...

 

[Metric_Space]

great

 

[hunt_mat]

you have to change co-ordinates from /9x,y) to (r,theta), this is done via the chain rule.