# Help with complex analysis

## 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).

Dick
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?

I think it is:

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

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?

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
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.

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?

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
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.

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

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

great

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