Conformal Mapping

[Steven G]

Hi,
I was wondering how I would find a mobius transformation from the unit disk to the upper unit disk.
I suspect that I should just use the inverse transformation from the upper unit disk to the unit disk. However I am not seeing this one.
A small hint would probably suffice

 

[blamocur]

I suspect that I should just use the inverse transformation from the upper unit disk to the unit disk.

Do you know the transformation from the upper unit disk to the unit disk? Can you post it?

 

[blamocur]

Do you know the transformation from the upper unit disk to the unit disk? Can you post it?

Silly me: just realized that it is [imath]z^2[/imath]

 

[Steven G]

Silly me: just realized that it is [imath]z^2[/imath]

No, I do not know the mobius transformation from the upper unit disk to the unit disk.
But z^2 isn't a mobius transformation.

 

[blamocur]

Silly me: just realized that it is [imath]z^2[/imath]

... which isn't Mobius.

 

[blamocur]

No, I do not know the mobius transformation from the upper unit disk to the unit disk.
But z^2 isn't a mobius transformation.

Are you sure that the solution exists?

Mobius transformations are:
a) Bijective, a.k.a. one-to-one (unlike [imath]z^2[/imath]), and
b) differentiable, which means they preserve angles, unless the derivative is 0.

The above means that at [imath]z=\pm 1[/imath] the derivatives must be 0 because the image of the half-disk border does not have any straight angles. I am not sure any Mobius transform has zero derivative anywhere.