Complex analysis zero/branch problem

[Scousergirl]

Homework Statement

Let f be a holomorphic function in the open subset G or C. Let the point Z of G be a zero of f of order m. Prove that there is a branch of f^(1/m) in some open disk centered at Z

Homework Equations

Branch- a continuous function g in G such that, for each x in G, the value of g(x) is a value of f^(1/m)

The Attempt at a Solution

My biggest problem is formulating a proof that a branch exists. If somebody could give me a push start on how to start and formulate the proof that the branch exists I will be able to figure out the rest.

 

[mighty2000]

Hello!

To prove the existence of a branch of f^(1/m) at point Z, we can use the Cauchy Integral Formula. This formula states that for a holomorphic function f defined on an open disk D centered at a point Z, we can write f as a power series centered at Z. In other words, we can write f(z) = ∑(n=0 to ∞) c_n (z-Z)^n.

Since Z is a zero of f of order m, we know that c_0 = c_1 = ... = c_(m-1) = 0 and c_m ≠ 0. Therefore, we can write f(z) = c_m (z-Z)^m + c_(m+1) (z-Z)^(m+1) + ...

Now, we can define a branch of f^(1/m) as g(z) = (c_m)^(1/m) (z-Z)^(1/m) + (c_(m+1))^(1/m) (z-Z)^(1/m+1) + ..., which is a continuous function in D.

To show that g is a branch of f^(1/m), we need to show that for any x in D, the value of g(x) is a value of f^(1/m). We can do this by substituting x for z in the power series representation of f and simplifying.

Therefore, we have shown that a branch of f^(1/m) exists in the open disk D centered at Z.