Understanding Complex Functions and Polynomials in the Complex Plane

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In summary, the conversation discusses the use of Rouché's Theorem to show that a function has exactly two distinct zeroes on a given complex set, and also how to prove that the composition of two non-constant holomorphic functions is a polynomial if and only if both functions are polynomials. One possible solution for the first problem is to use the substitution $u=z-2$ and then apply Rouché's Theorem to show that the function has two zeroes on $|u|<1$ and that these zeroes are distinct. For the second problem, one can prove the converse implication by assuming that one function is a polynomial and showing that the other function must also be a polynomial.
  • #1
Markov2
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1) Let $f(z)=e^{-z}+z^2-4z+4.$ Show that $f$ has exactly two zeroes on $\{z\in\mathbb C:|z-2|<1\}.$ Show that these zeroes are distinct, that is, it's not a zero of order two.

2) Let $f,g\in\mathcal H(\mathbb C)$ be no constant. Prove that $(f\circ g)(z)$ is a polynomial iff $f$ and $g$ are polynomials.

Attempts:

1) I know I have to use Rouché's Theorem, but I don't know how for this case, it annoys me the $|z-2|<1.$

2) The $\Longleftarrow$ implication is trivial, but not the another one. I don't know how to proceed though.
 
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  • #2
Markov said:
1) Let $f(z)=e^{-z}+z^2-4z+4.$ Show that $f$ has exactly two zeroes on $\{z\in\mathbb C:|z-2|<1\}.$ Show that these zeroes are distinct, that is, it's not a zero of order two.

Hint Using the substitution $u=z-2$ we obtain $f(z)=e^{-u-2}+u^2$ and $|\;e^{-u-2}\;|<|u^2|$ on $|u|=1$ .
 
  • #3

Very nice Fernando! Now let $|f(u)-u^2|=|e^{-u-2}|\le|u^2|=1\le|f(u)|+|u|^2,$ since $f(u)$ and $u^2$ have no zeroes for $|u|=1,$ then by Rouché's Theorem $f$ has two zeroes for $|u|<1,$ but how to justify that those zeroes are distincts? (Second part of the problem.)

Could you help me with second problem please?
 
  • #4
Markov said:
but how to justify that those zeroes are distincts? (Second part of the problem.)

Suppose $a$ is a double root of $g(u)=e^{-u-2}+u^2$ in $|u|<1$ then, $a$ is also a root of $g'(u)=-e^{-u-2}+2u$, but $g(a)=0$ and $g'(a)=0$ implies $a^2+2a=0$ that is, $a=0$ or $a=-2$ (contradiction) .
 
  • #5
Fernando last thing: is my solution correct by applying Rouché's Theorem?
 
  • #6
Can anydoby check if my solution by using Rouché is correct?
 
  • #7
Markov said:
Can anydoby check if my solution by using Rouché is correct?

I don't see any solution on your part. :)
 

FAQ: Understanding Complex Functions and Polynomials in the Complex Plane

What is complexity theory?

Complexity theory is a scientific framework that studies complex systems and how they behave. It looks at how simple rules and interactions between individual components can lead to complex and unpredictable behavior on a larger scale.

What are some real-world applications of complexity theory?

Complexity theory has been applied to various fields such as biology, economics, sociology, and computer science. Examples include studying the behavior of ecosystems, predicting stock market trends, understanding social networks, and developing algorithms for artificial intelligence.

How is complexity theory different from chaos theory?

Complexity theory and chaos theory are often used interchangeably, but they refer to different concepts. Chaos theory focuses on the behavior of systems that are highly sensitive to initial conditions, while complexity theory looks at how systems with simple rules can produce complex patterns and behavior.

Can complexity theory be used to solve complex problems?

While complexity theory does not provide direct solutions to complex problems, it can offer insights and tools for understanding and managing them. By studying the underlying dynamics and patterns of complex systems, complexity theory can help identify key factors and interactions that can inform problem-solving strategies.

What are some challenges of studying complex systems?

One of the main challenges of studying complex systems is that they are difficult to define and measure. They often involve a large number of components and interactions, making it challenging to isolate and study individual parts. Additionally, complex systems are dynamic and can change over time, making it challenging to predict their behavior.

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