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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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Proving existence of a root in the unit disk using Rouché's Theorem

Let $a\in\mathbb{C}$, and let $n \ge 2$. Prove that the polynomial $2022+az+2023z^n$ has a root in the unit disk, $D(0,1)$. There's an algebraic way to solve this with Vieta's formulas, by observing ...
Ty Perkins's user avatar
1 vote
3 answers
71 views

Find the number of zeros of $f(z) = 6z+z^3+e^{z-2}$ in the annulus $1 <|z|<2$

I was looking at this post for advice on how to use Rouche's theorem to find the number of zeros a function has in a given annulus. An old qualifying exam problem says Find the number of zeros of $f(...
Grigor Hakobyan's user avatar
3 votes
2 answers
114 views

Finding the number of zeros on a half plane of $z^4+3z^2 + z + 1$

I found this problem in Berkeley problems in Mathematics: How many roots has the polynomial $z^4+3z^2 + z + 1$ in the right half $z$-plane? To this point, I have tried using the method prescribed by ...
sireesh's user avatar
  • 73
0 votes
0 answers
48 views

Rouché's theorem; $z^n(z−2)=1$

Statement: $p(z)=z^n(z−2)-1$ has $n$ zeroes on $D(0,1)$ Considering $f(z)=z^{n+1}-1$ for sufficiently small $\epsilon$ such that $z^{n+1}\neq-1$ for $z \in \overline{D(1,\epsilon)}$ ; $$|z^{n+1}-1|&...
J P's user avatar
  • 893
4 votes
1 answer
72 views

Finding $\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$ over a given contour [duplicate]

Let $f(z)=z^4-2z^3+2z^2-3z+60$ and let $\Gamma$ be the circle $|z|=5$. I want to find $$\int_\Gamma \frac{z f'(z)}{f(z)} \, dz$$ Supposing we had $f'(z)$ in the numerator instead of $z f'(z)$, this ...
Grigor Hakobyan's user avatar
4 votes
1 answer
140 views

Analytic $f: \mathbb{D} \to \mathbb{D}$, $f(0)=0$, and $f$ has five zeros in $\overline{\frac{1}{2}\mathbb{D}}$

Suppose $f: \mathbb{D} \to \mathbb{D}$ is a holomorphic function and $f(0)=0$. The function $f$ has a total of five zeros (counting multiplicities) in the closed half-disc $\overline{\frac{1}{2}\...
Grigor Hakobyan's user avatar
2 votes
2 answers
95 views

How to show that $\left\{z \in \mathbb{C} : \left|\frac{z - 1}{z + 1}\right| < \frac{1}{2}\right\}$ is a disk [duplicate]

I have to prove that the equation $$(z + 1)e^{-z} = 2(z - 1)$$ has only one solution in the open right half-plane. I know that if $z$ is a solution such that $\Re(z) > 0$ then $\left|\frac{z - 1}{z ...
Cyclotomic Manolo's user avatar
1 vote
1 answer
74 views

How many roots has the equation $z = \varphi(z)$ in $|z| < 1$ if for $|z| \leq 1$, $\varphi(z)$ is analytic and satisfies $|\varphi(z)| < 1$? [duplicate]

Problem How many roots has the equation $z = \varphi(z)$ in $|z| < 1$ if for $|z| \leq 1$, $\varphi(z)$ is analytic and satisfies $|\varphi(z)| < 1$? I'm suspecting this is an application of ...
Grigor Hakobyan's user avatar
0 votes
1 answer
40 views

Need help in Understanding the proof of Rouche's theorem

I have found proof of Rouche's theorem in various text and on different post on this site too. But I am only interested in the proof given here https://proofwiki.org/wiki/Rouch%C3%A9%27s_Theorem ...
General Mathematics's user avatar
0 votes
0 answers
32 views

Order of zeros of $2(z-1)^n-e^{-z}$ in $B_1(1)$.

I know there are $n$ zeros in the open ball $B_1(1)$ from Rouché's Theorem. However, how do I show that the order of these zeros are all one? I don't know where the zeros are, and this should be true ...
Aadi Rane's user avatar
  • 337
2 votes
2 answers
137 views

Tricky Application of Rouche's Theorem

I'm supposed to use Rouche's theorem to solve this problem, but I'm pretty sure it's not possible. Can anyone confirm this? I want to determine how many zeros $e^z-z$ has on $B_1(0)$. The obvious set ...
Ty Perkins's user avatar
1 vote
1 answer
66 views

Rouché's Theorem: How many roots does $\lambda-z=\frac{1}{3}e^{z^2}$ have in the strip Re$(z)\in [-1,1]$?

I'm studying for my complex analysis qual, and I've found a problem that I haven't seen solved here yet: Take $\lambda$ to be purely imaginary. Prove that $z-\lambda=\frac{1}{3}e^{z^2}$ has exactly ...
qualsqualsquals's user avatar
6 votes
0 answers
202 views

Using Rouche's theorem to prove injectivity

$\textbf{Problem setup}$: Let $F : R_{\tau} \rightarrow R_{\tau}$ be a biholomorphism onto its image (i.e. $F : R_{\tau} \rightarrow V$ is a holomorphic function with holomorphic inverse, here $V = F(...
porridgemathematics's user avatar
3 votes
0 answers
124 views

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 ...
Tropax's user avatar
  • 373
0 votes
2 answers
243 views

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 ...
Lucius Aelius Seianus's user avatar
2 votes
1 answer
103 views

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 ...
James S. Cook's user avatar
0 votes
0 answers
43 views

Sequence of functions uniform converges to f, then f is either zero or has no zeros, Rouche.

Question: Let $\Omega \subset \mathbb{C}$ be open and connected. Suppose $f_n: \Omega \to \mathbb{C}$, $n \ge 1$, $f: \Omega \to \mathbb{C}$ holomorphic s.t. $f_n \to f $uniformly on each ball $B \to \...
i_like_codes_and_girls's user avatar
2 votes
1 answer
107 views

Conjecture, a monic polynomial with integer coefficients cannot have exactly one non-real root outside the open unit disk

For context, I was reviewing some old notes of mine about polynomials and got stuck in a proof (which I can only suppose it was mine and without any revision). The problem was, let $$h(t) = t^n + a_{n-...
amrsa's user avatar
  • 13.1k
1 vote
1 answer
146 views

An Alternate Formulation of Rouche's Theorem?

The general form of Rouche's Theorem states: For any two complex-valued functions $f$ and $g$ holomorphic inside some region $K$ with closed contour $\partial K$ if $|g(z)| < |f(z)|$ on $\...
Josh's user avatar
  • 1,106
0 votes
0 answers
27 views

Missing detail FTA proof

In the answer: https://math.stackexchange.com/a/3741902/987127, I don't see at which point there is a strict inequality, to satisfy Rouche's theorem.
Shean's user avatar
  • 893
3 votes
2 answers
70 views

Find parameters $a,b,c \in\mathbb{R}$, such that following system is compatible

Given $$\begin{cases}2x-3y+4z-5t = 1 \\ x+9y+az+t=-3 \\ 5x-6y+10z-bt=c\end{cases}$$ how to find $a,b,c\in\mathbb{R}$ so that the system is compatible? How should one approach such a problem? Where to ...
J__n's user avatar
  • 1,123
0 votes
1 answer
42 views

Rouches proof question regarding the curve $\gamma$

https://en.wikipedia.org/wiki/Rouché%27s_theorem I got a question regarding the proof of Rouches theorem. The proof our prof gave us tells us that the contour $\partial K$ needs to be partwise smooth, ...
uoiu's user avatar
  • 593
2 votes
1 answer
94 views

Rouché's theorem proof question

I'm currently studying for a re-exam in Complex analysis, and have a question regarding the proof for Rouches theorem. Below is the exact proof, word by word, our professor have presented. I ...
uoiu's user avatar
  • 593
-1 votes
1 answer
69 views

Rouché Theorem Application

I have a function $n \geq 2$ $h(z) = f(z)+g(z) = z^{n+1}-1+z-1$ $f(z)=z^{n+1}-1$ $g(z) = z-1$ is my proof is correct here By Using Rouché for $|z|\leq 1$ $|z^{n+1}-1| \leq |z|^{n+1}-1 \leq 0 \leq |z-1|...
Will's user avatar
  • 145
0 votes
1 answer
111 views

How to obtain the relationship between a plane and a line in $\mathbb{R}^3$

I want to know if my reasoning can hold in general, of if there are caveats or if it's simply a load of nonsenses. Say I have to determine the position of a line with respect to a plane. Say I have ...
Heidegger's user avatar
  • 3,482
1 vote
1 answer
97 views

Show that all the roots of $p(z)= z^5 – z^3 + 1$ are in the ring $\frac{1}{2}\lt \vert z \vert \lt \frac{3}{2}$.

I am going to try check that $\vert z^5 – z^3\vert \lt 1$ for $\vert z \vert \le \frac{1}{2}.$ $1=|-1|=|z^5+z^3|=|z|^3\cdot |z^2+1|\le |z^2+1|\le |z|^2+|1|=|z|^2+1 \le 1.$ Now I'm stuck. What do I do ...
Bruno's user avatar
  • 37
1 vote
2 answers
2k views

Suppose $f(z)$ is analytic on $|z|<1$ such that $|f(z)|<1$ for all $|z|<1$ and $f(0) = \alpha \neq 0$. Show $f(z) \neq 0$ for all $|z|<|\alpha|$. [duplicate]

This is a problem from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming exam. I've been playing with it for a while and am stuck. Problem: Suppose $f(z)...
Serafina's user avatar
  • 470
-1 votes
1 answer
103 views

What is the number of zeros of the this polynomial? [closed]

I wish to know the number of zeros of the polynomial $z^{10}-6z^7+3z^3+1$ in $|z|<1$. Does it have something to do with Rouche's theorem?
Firdous Mala's user avatar
0 votes
2 answers
163 views

Find how many solutions (counting multiplicity) the equation $\sin z = ez^4$ has on the unit disk $|z|<1$.

This is a problem from a past qualifying exam in complex analysis. I'm working through these to study for my own upcoming qual. For this question, I think my proof is fairly straightforward, but I'd ...
Serafina's user avatar
  • 470
1 vote
1 answer
81 views

Find number of zeros of $z^{113}-180z^{23}+115z^{7}-32z^2-3z-10$ in annulus $1≤|z|≤2$ [closed]

I know I have to use Rouché's theorem but I can't figure out how to choose $f$ and $g$
crackedPeanut's user avatar
2 votes
2 answers
87 views

Show $2z^5-6z^2+z+1$ has three zeroes in $\{z : 1<\vert z \vert < 2\}$

I need to show $$2z^5-6z^2+z+1$$ has three zeroes on $\{z : 1< \vert z \vert < 2\}.$ So do I split in into cases for when $\vert z \vert= 1$ and when $\vert z \vert = 2$ by Rouches theorem I can ...
homosapien's user avatar
  • 4,225
0 votes
2 answers
145 views

Number of zeros inside unit disk

What is the number of zeros of the equation $ze^{3-z}-5^{2}=0$ inside the unit disk? I believe that that the answer is zero, by some kind of Rouche Theorem. But, I am not %100 sure about that. A ...
Bob Dobbs's user avatar
  • 11.8k
2 votes
1 answer
173 views

Rouche’s Theorem?

This is from an exam meant for high-school students: Suppose $z$ is a complex number satisfying $z^5+z^3+z+3=0$. Then |z| satisfies: A. $|z|\lt 1$ B. $|z|\geq 1$ C. $|z|\leq \frac12$ D. $|z|\leq \...
insipidintegrator's user avatar
0 votes
0 answers
56 views

number of zeros/rouche's theorem

let $h(z)=z^5+\frac{1}{3}z^3+\frac{1}{4}z^2+\frac{1}{3}$ the numbers of zeros on $M$ should be determined with $M=\{ z \in \mathbb{C}: |z| < \frac{1}{2}\}$. So I tried to apply Rouche's theorem: ...
Inocenciaa's user avatar
0 votes
0 answers
42 views

Prove that an equation has only one complex solution

I need to prove that the equation $z - \varepsilon \sin z = a$ has only one solution for small $\varepsilon$. The problem doesn't state anything specifically but I assume that $a$ is an arbitrary ...
Hinko Pih Pih's user avatar
-1 votes
1 answer
99 views

Number of zeros of the complex function in the unit disk

$ \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\VP}{V.P.} \DeclareMathOperator{\e}{e} \DeclareMathOperator{\AC}{AC} \DeclareMathOperator{\BB}{B} \...
Dmitry's user avatar
  • 165
1 vote
0 answers
54 views

Computing the Number of Zeroes using Rouché + Argument Principle

I want to show that the function $f(z) = 1 + 2z + 7z^2 + 3z^5$ has exactly two zeroes in the unit disk counted with multiplicty. My approach: Consider $g(z) = -7z^2-3z^5$. Then, $h(z) = f(z) + g(z) = ...
John's user avatar
  • 463
2 votes
2 answers
176 views

$|f(z) - g(z)| < |f(z)|$ vs $|f(z) - g(z)| < |g(z)|$ in Rouche's theorem?

I was reading Marsden's Basic Complex Analysis and noticed that Rouche's Theorem was formulated as requiring $|f(z) - g(z)| < |f(z)|$ (for all $z$ on closed $\gamma$, etc.), and the statement was ...
dpbrewer's user avatar
1 vote
1 answer
305 views

Find out the number of zeroes of $f(z) = e^z -3z -2 =0$ , inside $|z|=1.$

In the first view I thought it to be a question of Rouche's theorem but when proceeded it was quiet tricky As $f(z) = e^z- 3z- 2$ If we take $g(z)= - 3z-1$ Then $f(z) - g(z) = e^z - 2 = h(z)$ (say) ...
Aryan Singh's user avatar
2 votes
2 answers
150 views

How can I show that $z^4-2z+3$ has no zeros within the unit circle in the complex plane?

How can I show that $z^4-2z+3$, $z \in \mathbb{C}$, has no zeros within the unit circle in the complex plane? It looks like the Rouche theorem, but i still cannot do it. Please help. Thanks in advance....
Logic_Problem_42's user avatar
0 votes
0 answers
22 views

Rouché for positive $n$, but what for negative $n$?

Suppose $\Omega$ is open that contains the closed unit disc. If $f:\Omega \rightarrow \mathbb{C}$ is holomorphoc with $|f(z)| < 1$ for all $z$ with $|z|=1$. How many solutions are there of $f(z)=z^...
Geigercounter's user avatar
0 votes
0 answers
49 views

Show that $P(z)=a_kZ^k$ has $n$ zeros in $B(0,1)$.

Let $P(z)=\sum_{k=0}^n a_kz^k$ where $0<a_1<\cdots<a_n$. Show that $P(z)$ has n zeros in unit disk $\Bbb D$. Moreover, show that $\sum_{k=0}^n a_kcoskt$ has 2n zeros in open interval $(0,2π)$. I want ...
Hanyu.Chern's user avatar
0 votes
0 answers
108 views

Why is only the boundary considered when using Rouche's Theorem?

A typical setup for a Rouche's Theorem problem is one where you have a polynomial, say $f(z)$, defined on some domain $U$. For simplicity let's take $U$ to be the unit disk. What often happens is one ...
CBBAM's user avatar
  • 6,265
1 vote
1 answer
146 views

Proving the polynomial $P_t(z) = \sum_{j=0}^N a_j(t) z^j$ has exactly one zero if $P_0$ has a simple zero with $a_j(t)$ and $P_t(z)$ continuous in $t$

Let $P_t(z)$ be a polynomial in $z$ of degree $\leq N$ for each value of $t \in [0,1]$. Suppose that $P_t(z)$ depends continuously on $t$ in the sense that $$P_t(z) = \sum_{j=0}^N a_j(t) z^j$$ and ...
fieke_2000's user avatar
1 vote
1 answer
61 views

Counting the zeros when a polynomial is perturbed by an exponential

I'm working on the following problem. Consider the function $f(z) = z^5 - 8 + e^{\gamma z}$, where $\gamma > 0$. How many zeros (counted w/ multiplicity) does $f$ have on the left half-plane $\{ z ...
AJY's user avatar
  • 8,769
6 votes
0 answers
191 views

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 ...
Espace' etale's user avatar
3 votes
0 answers
335 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 ...
john's user avatar
  • 1,288
0 votes
0 answers
194 views

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 ...
math's user avatar
  • 93
0 votes
0 answers
34 views

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 ...
math's user avatar
  • 93
1 vote
1 answer
360 views

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 $f: U\setminus S\rightarrow \mathbb{C}$ ...
Vinay Deshpande's user avatar