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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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 \...
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Can we find a holomorphic function $f$ on an annulus such that $e^{f(z)}=z$
Can we find a holomorphic function $f$ on an annulus $A$ such that $\exp(f(z)) = z$ for all $z$ in $A$, where $A = {z:1/2<|z|<1}$.
I guess that we should apply the argument principle or Rouche ...
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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
$$h(t) = t^n + a_{n-...
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Rouche's theorem for finding real zeroes of holomorphic functions
I was wondering if we can apply Rouche's theorem for finding the real solutions/zeroes of a holomorphic function.
Rouche's theorem:
Let $\Omega$ be a simply connected set and $\gamma$ be a simple ...
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An Alternate Formulation of Rouche's Theorem?
The general form of Rouche's Theorem states:
For any two complex-valued functions $f$ and $g$ holomorphic inside
some region $K$ with closed contour $\partial K$ if $|g(z)| < |f(z)|$
on $\...
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Missing detail FTA proof
In the answer: https://math.stackexchange.com/a/3741902/987127, I don't see at which point there is a strict inequality, to satisfy Rouche's theorem.
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Find parameters $a,b,c \in\mathbb{R}$, such that following system is compatible
Given $$\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 ...
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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, ...
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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 ...
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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|...
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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 ...
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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 ...
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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)...
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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?
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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 ...
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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$
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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 ...
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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 ...
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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 \...
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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:
...
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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 ...
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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}
\...
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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) = ...
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$|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 ...
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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)
...
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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....
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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^...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $f: U\setminus S\rightarrow \mathbb{C}$ ...
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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 ...
user889034
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Using Rouché's theorem to find the number of zeros of $Z^8+5Z^7-20$
I am new to complex analysis and Rouché's theorem and would like to know if my choice of $f$ and $g$ are appropriate.
The version of Rouché's theorem I am most familiar with has the condition $|f(z)+g(...
user889034
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Is there any relationship between Rouché's Theorem and Newton's method
In trying to find complex zeros of equation $3^z+4^z=5^z$, we could divide both sides by $(\sqrt{15})^z$ to transform it into $(\frac{4}{\sqrt{15}})^z=(\sqrt{\frac53})^z-(\sqrt{\frac35})^z$.
Let's ...
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2
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Rouché's Theorem for a polynomial
Show that $P(z) = z^{47} − z^{23}+ 2z^{11} − z^5 + 4z^2 + 1$ has at least one zero in the unit disk $D(0,1)$.
Here's my attempt:
Let $f(z)=z^{47} − z^{23}+ 2z^{11} − z^5 + 4z^2 + 1$ and $g(z)=− z^5 + ...
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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 ...
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Calculate number of zeros of a polynomial (Rouché Theorem)
Let $m,n\in\mathbb{N}$, I have to prove that the polynomial
$P(z)=1+z+\dfrac{z^2}{2!}+...+\dfrac{z^m}{m!}+3z^n$ has exactly $n$ roots in the $D(0,1)$.
My attempt:
Let's call $Q(z)=3z^n$, $R(z)=1+z+\...
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Prove a function has only simple roots
I am working on old qualifying exams, and I have stumbled on a problem that looks somewhat like this:
Prove that $f(z) = (z-1)^{50}e^z + .75(z+1)^{50}$ has exactly 50 simple zeroes with $\Re z > 0 ...
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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 ...
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Do I need to apply Rouche’s theorem twice in this proof?
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 ...
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