Derivative of an automorphic rational function

Suppose you have a finite subgroup of , viewed as the group of Mobius transformations acting on the Riemann sphere . An example of such a subgroup is given by the group of "harmonic" transformations, which take to . Take two points which have a trivial stabilizer in , and whose orbits are disjoint and do not contain the point . Form the rational function

.

This function has the property of being invariant under

, i.e.

.

**Problem 1** Show that we can write

, for some constant

.

**Problem 2 **Show that, for any

, the order of

at

is equal to the number of nontrivial elements in the stabilizer of

in

. (Remember that the order of

at a point

is the least integer

such that

is holomorphic at

.)