Find a Mobius transformation that maps the right half-plane to the upper half-plane carrying the point 7+5i to 3i.

Can I get some help please?

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- March 1st 2010, 11:12 AMdori1123mobius transformation
Find a Mobius transformation that maps the right half-plane to the upper half-plane carrying the point 7+5i to 3i.

Can I get some help please? - March 2nd 2010, 07:44 AMOpalg
This method is messy, but at least it ought to work. The idea is that you should have a library of known Möbius transformations that you combine in order to get the desired result.

The transformation maps the right half-plane to the unit circle, and it takes 7+5i to (that's where the numbers start to get messy).

The transformation maps the upper half-plane to the unit circle, and it takes 3i to 1/2. Its inverse transformation, from the unit disk to the upper half-plane, is given by .

For any point*a*in the unit disk, the transformation maps the unit disk to itself, and takes*a*to the origin. Its inverse transformation is given by , and it obviously takes the origin to .

Thus the composition of mappings , where and , will be a Möbius transformation that maps the right half-plane to the upper half-plane and carries the point 7+5i to 3i.

Having written all that, I have a better idea. You can think of the right half-plane as consisting of all those points that are closer to 7+5i than they are to –7+5i. In other words, they are the points for which . It follows that the Möbius transformation maps the right half-plane to the unit disk, and it takes 7+5i to the origin. So you can use that map to replace the composite map in the above construction by the single map*k*, and that will avoid all the messy arithmetic.