I have been reading all the notes I could find about Riemann surfaces but still have no clue how to start these 2 proofs... I know they should be easy proofs...
1 first one being, prove/show that Mobius Transformation az+b/cz+d can be extended to meromorphic function in the extended complex plane.
I guess it has something to do with the pole and how to define at the infinite point... but don't know how to continue from that...
2 Second one is to show that Mobius Transformation is a holomorphic map from Cˆ to Cˆ. C^ is the extended complex plane again
Define $\frac {az+b} {cz+d}$ to be $\frac a c$ if $c \neq 0$ and $\infty$ if $c=0$ . (By definition of MT $ad-bc \neq0$ so $c=0$ implies $a \neq 0$). Rational functions are analytic except for poles so the extended function is meromorhic. (By definition analyticity of $f(z)$ at $\infty$ is same as analyticity of $f(\frac 1 z )$ at 0. In 2) differntiability of f at a point $z$ where $f(z)=\infty$ is defined as differentiability of $\frac 1 f$ at that point. With is definition it is fairly straighforward to verify that the extended MT is holomorphic on the extended palne.