# Questions tagged [rouches-theorem]

For questions on Rouché's theorem, which relates the number of roots of two holomorphic functions $f,g$ in some bounded domain $D$ given that $|g|<|f|$ on $\partial D$.

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### Roots of the polynomial $z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$.

I came across the following problem: Find the number of zeros of $P(z)=z^4+4z^3+6z^2-4z+4$ in the circle $|z+1|<1$. I know that I can't apply Rouche's Theorem directly since we don't have a ...
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### the zeros of a polynomial depend in a continuous way on its coefficients. [closed]

We use $P_n$ to denote the vector space of complex polynomials of degree $\leq n$, and we write $$\mathcal{P}^{*}_n = \mathcal{P}_n \setminus \mathcal{P}_{n-1}$$ for the set of polynomials of exact ...
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### Rouche's Theorem Question, or not?

Consider $P(z) = z^4+4z^3+6z^2-4z+3$. How many zeros does $P(z)$ have within $|z-1|<1$. I've looked at wolfram alpha and it appears the answer is two. I've tried many things to apply Rouche's ...
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### Is Rouche's theorem valid with poles inside the contour

Can Rouche’s theorem be extended to the case when poles are inside the contour? In my textbook, the theorem is stated as Suppose $f(z)$ and $g(z)$ are analytic inside and on a simple closed contour C, ...
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### Given entire functions $f,g$ having equal number of zeros in $D_r(0)$. Then exists an entire, non-vanishing $h$ st $|f - hg| < |f|$ on a large circle [duplicate]

The problem: Given entire functions $f,g : \mathbb{C} \to \mathbb{C}$, such that all the zeroes of both functions are contained in a disk $D_r(0)$ and they have an equal number of them (counting ...
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### Roots of $1+x+x^{2}/2!+...+x^{n}/n!$ are larger than $\frac{n}{2e}$

I wish to prove that the absolute values of the roots of the polynomial $p(x)=1+x+x^{2}/2!+...+x^{n}/n!$ are between $n/2e$ and $2n$. I was able to prove the upper bound using the following: Claim: If ...
310 views

### Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$.

Let $\varepsilon >0$ and let $f:B(0,1+\varepsilon )\rightarrow B(0,1)$ be a holomorphic function. Show that there exists a unique $z_0\in B(0,1)$ s.t $f(z_0)=z_0$. For this problem, I don't want to ...
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### How the argument principle is used in proof of Rouche's Theorem.

The argument theorem states: If f is meromorphic in an open set containing a circle C and its interior and if f has no poles and never vanishes in C, then the number of zeros of f inside C minus the ...
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### Can I have any poles if $f(z) \neq 0$?

Can I have any poles if $f(z) \neq 0$? I'm trying to refresh my memory on poles... I was reading a proof for Rouche's Theorem an it, we needed to satisfy the assumptions of the argument principle ...
1 vote
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### usage of Rouche's theorem for winding number in proof of Residue theorem

A doubt from proof of residue theorem from Tao's notes. notes 4 Theorem 21 (Residue theorem) Let $U\subset \mathbb{C}$ be a simply connected open set, and let $f: U\setminus S\rightarrow \mathbb{C}$ ...
1 vote
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### Multiplicity of $z=0$ on $f(z)=z\cos(z)-\sin(z)$

I'm trying to find the multiplicity of $z=0$ on $f(z)=z\cos(z)-\sin(z)$ using complex analysis. I'm new to complex analysis and the argument principle/Rouché's theorem so I'm not quite sure where to ... 1 vote
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### Using Rouché's theorem to infer the amount of zeroes inside the given domain

Given $p(z)=i z^{5}-8 z^{4}-\pi$, How many zeros there is for $p(z)$ inside $D_{1}(0) \cap\{z \mid \operatorname{Im}(z)>0\}$? I can use Rouché theorem to infer how many zeros there are in the ...
1 vote
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### Number of zeros outside the disk $\{ z : |z| \leq 2 \}$
I need to count (including the multiplicities of the zeros) number of the zeros outside the disk $\{ z : |z| \leq 2 \}$ for the polynomial $f(z) = z^7 +9z^4 -7z +3$. I know this should be direct ...
Here's the problem I'm trying to proof: Let functions $f(z)$ and $g(z)$ be both analytic in the disc $|z| ≤ 2$, let $|f(z)| > |g(z)|$ on the circle $|z| = 2$and let $|g(z)| > |f(z)|$ on the ...