Let U be a simply connected domain and let f be a meromorphic function on U with only finitely

prensistath

prensistath

Answered question

2022-05-30

Let U be a simply connected domain and let f be a meromorphic function on U with only finitely many zeroes and poles. Prove that there is a holomorphic function g : U C and a rational function q, such that
f ( z ) = e g ( z ) q ( z )

Answer & Explanation

Danube2w

Danube2w

Beginner2022-05-31Added 7 answers

We at first prove the case were f has neither poles nor zeros: We want to prove that if f has no pole nor zeroes, it holds that g : U C such that z U : f ( z ) = e g ( z )
Proof:
We know that since f has no zeroes in U the function h ( z ) = f ( z ) f ( z ) is well defined in U.Futhermore U is simply connected therefore F : U C : F = h = f ( z ) f ( z )
We now define ϕ : U C and ϕ ( z ) := e F ( z ) f ( z )
We see that ϕ is constant in U since
ϕ ( z ) = e F ( z ) ( f ( z ) F ( z ) f ( z ) ) = e F ( z ) ( f ( z ) h ( z ) f ( z ) ) = e F ( z ) ( f ( z ) f ( z ) f ( z ) f ( z ) ) = 0
The last little definition, which will define our searched g, needs that we pick a point z 0 U and another w 0 (depending on z 0 ) such that e w 0 = f ( z 0 )
Now
g : U C         g ( z ) := F ( z ) F ( z 0 ) + w 0
is holomorphic since F is and it holds for all z U ::
e g ( z ) = e F ( z ) e F ( z 0 ) e w 0 = e F ( z ) ϕ ( z 0 ) = e F ( z ) ϕ ( z ) = f ( z )
If f has poles and zeroes:
Let z 1 , . . . , z k be the poles (multiplicity m z 1 , . . . , m z k ) and w 1 , . . . , w l (multiplicity m w 1 , . . . , m w l ) the zeroes of f then we just have to write f as
f ( z ) = e g ( z ) j = 1 l ( w w j ) m w j m = 1 k ( z z m ) m z m

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