# Meromorphic functions

• Ant farm
In summary, the conversation involved discussing basic questions about the Riemann Sphere, particularly involving meromorphic functions. The first question was to find all meromorphic functions that satisfy f(f)=f, leading to the possibility of only the identity map or a constant map. However, the existence of f^-1 rules out the latter. The second question involved determining the possible values of n for a meromorphic function with a preimage of c containing n elements. It was suggested to use Laurent expansions to prove that f(f)=f only when degf=1 or 0, leading to the conclusion that f must be either the identity map or a constant map.

#### Ant farm

Hi there,
working on some basic questions involving the Riemann Sphere(sigma): C union infinity

firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...
was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...

secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??
stuck here, any hints would be great!
thank you.

Ant farm said:
Hi there,
working on some basic questions involving the Riemann Sphere(sigma): C union infinity

firstly, i was asked to find all meromorphic f: sigma -> sigma such that f(f)=f.

my thoughts are: since the degree of a composition f(g) is deg(f)deg(g), our only possibilities are f=identity map (whose degree is 1) or f=the constant map...but then the map f(z)= infinity is not meromorphic...
was also thinking that f(f)=f only when f^2=f which implies that f=f^-1...which only occurs with the identity map...

f^2=f does not imply f=f^-1. Firstly, f^-1 need not exist, indeed cannot exist, unless f=Id. There are also more maps than just Id that satisfy f=f^-1 (or f^2=Id).

secondly, let f: sigma->sigma be meromorphic and such that for each c belonging to sigma the preimage f^-1(c) contains precisely n elements(not counting multiplicities). what are the possible values for n??
stuck here, any hints would be great!
thank you.

My first thoughts are that meromorphic functions have Laurent expansions.

it seems you have proved that f(f) = f implies degf = 1 or 0.

that does sound as if f is id or constant, can you prove that?