# Complex Analysis: Is my proof of an easy theorem correct?

• Nikitin
In summary: Therefore, when you divide h(z)/f(z) by g(z), you will get a laurent series of order n, and thus a pole of order n at z0. In summary, the theorem states that if a function f(z) has a zero of nth order at z0, then the function h(z)/f(z) has a pole of order n at z0, where h(z) is analytic and nonzero at z0. This can be proven by expanding f(z) as a taylor series and using some algebra to split up the expansion of 1/f(z) to obtain a laurent series of order n, making it clear that h(z)/f(z) has a pole of order n at z0
Nikitin
Theorem: If a function f(z) has a zero of nth order at z0, then the function h(z)/f(z) has a pole of order n at z0 (where h(z) is analytic at ##z_0##).

Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof something like this?

f(z) has a taylor expansion around z0 which begins on the nth term (since ##f^{(1)}(z_0),f^{(2)}(z_0),...,f^{(n-1)}(z_0) = 0##, and thus when you find the taylor expansion of f(z), you divide 1 by said taylor series to gain the expansion of 1/f(z). After splitting up the expansion of 1/f(z) into several different fractions (with the help of some fancy algebra), you will gain a laurent series of order n.

But how do I get to the last part? I am completely confused right now..

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Nikitin said:
Theorem: If a function f(z) has a zero of nth order at z0, then the function h(z)/f(z) has a pole of order n at z0 (where h(z) is analytic at ##z_0##).

Can somebody explain this theorem for me? It isn't proved in my book because it's so "easy", but I don't get it? Is the sketch of the proof something like this?

f(z) has a taylor expansion around z0 which begins on the nth term (since ##f^{(1)}(z_0),f^{(2)}(z_0),...,f^{(n-1)}(z_0) = 0##, and thus when you find the taylor expansion of f(z), you divide 1 by said taylor series to gain the expansion of 1/f(z). After splitting up the expansion of 1/f(z) into several different fractions (with the help of some fancy algebra), you will gain a laurent series of order n.

But how do I get to the last part? I am completely confused right now..

You should also specify that h(z) is nonzero at z0. If f(z) has a zero of nth order at z0, then you can write it as f(z)=(z-z0)^n*g(z) where g(z) is analytic and nonzero at z0. Does that make it clearer?

1 person
Damn, of course! Thanks now I understand the theorem.

Since g(z) is analytic and nonzero, it can be expanded as a taylor series, and thus the largest order negative polynomial possible in the total series is n!

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## 1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that deals with the study of complex numbers and their functions. It involves the use of techniques from calculus, algebra, and geometry to understand the properties and behavior of complex functions.

## 2. What is a proof in Complex Analysis?

A proof in Complex Analysis is a logical and rigorous argument that demonstrates the validity of a statement or theorem. It typically involves the use of mathematical definitions, axioms, and previously proven theorems to arrive at a conclusion.

## 3. Why is it important to check the correctness of a proof in Complex Analysis?

Checking the correctness of a proof is crucial in Complex Analysis because even a small error or oversight can lead to incorrect conclusions and potentially invalidate the entire proof. It ensures the accuracy and validity of the results obtained and helps to avoid any misunderstandings or false claims.

## 4. What are some common mistakes made in proofs of easy theorems in Complex Analysis?

Some common mistakes made in proofs of easy theorems in Complex Analysis include incorrect use of mathematical notation, misinterpretation of definitions or axioms, and faulty logic or reasoning. It is important to carefully review each step of the proof to identify and correct any errors.

## 5. How can I improve my proof-writing skills in Complex Analysis?

To improve your proof-writing skills in Complex Analysis, it is important to practice regularly and seek feedback from peers or instructors. Reading and studying well-written proofs can also help to improve your understanding of the subject and strengthen your proof-writing abilities.

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