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Questions tagged [rouches-theorem]

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29 views

Apply Rouché's theorem to count number of zeros inside disk

Attempt to find the number of zeros of $f(x) = x^5 + 3x - 1$ for $1 < |x| < 2$. I think I need to use Rouché's theorem to solve it. Rouché's theorem states that "suppose $f(z)$ and $g(z)$ are ...
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45 views

show $\frac{z}{(1+z)^2}$ is injective in the unit disk

So I need to show that $\frac{z}{(1+z)^2}$ is injective in the unit disk, $\mathbb{D}$. Here is what I have thought so far: If $\frac{z}{(1+z)^2}$ is injective, then $\frac{z}{(1+z)^2} - c$ always ...
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1answer
37 views

rouche's theorem without strict inequality

I've come across a problem involving Rouche's theorem. It asks whether we can say something about the roots of $f(z)=z$ if we know that on the boundary $ \mid z \mid = 1$ we have $\mid f(z) \mid \leq ...
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2answers
43 views

Applying Rouché's theorem in a square

I've got to determine the number of zeroes of the function $f: \mathbb{C} \rightarrow \mathbb{C}$, $f(z)=z^2+e^{z-1}$ inside the square with the corners $3 \pm 3i $ and $-3 \pm 3i $. Of course I ...
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63 views

Can you help on complex analysis problem

The question: Let $D = {z : |z| < 1}$, and let $f : D → D$ have a zero of order $n$ at zero. Show that $|f(z)| ≤ |z|^{n}$ on $D$. My attempt: I am not sure what theorem's are applied to this ...
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3answers
93 views

Find the number of roots of the polynomial $P(z)=z^5+2z^3+3$ in the closed unit disk $\{z\colon |z|\le 1\}$.

Find the number of roots of the polynomial $P(z)=z^5+2z^3+3$ in the closed unit disk $\{z\colon |z|\le 1\}$. My solution: Set $f(z)=3$. For every $0<\delta<1$ and $|z|=\delta$, we have $|P(z)-...
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1answer
38 views

Bound on constant in Polynomial so that zeros are bounded (Rouche)

Statement of the Problem: Let $a\in\mathbb{C}$ be a constant and consider the polynomial $$P(z)=z^{10}+a(z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1).$$ Given a radius $\rho>0$, use complex analysis to ...
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1answer
25 views

$Re(\frac{f}{g})\geq 0$ on the boundary of disk implies $f$ and $g$ have same number of zeros in the disk

I came across the following problem while solving some question paper. I have not been able to make any progress on it. “Let $f$ and $g$ be holomorphic in a neighbourhood of unit disk. Assume that $...
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68 views

Inverse statement to Rouché's theorem in complex analysis.

Let $f$ and $g$ be two analytic functions with the same finite number of zeroes. Then there exists an analytic function $h:\mathbb{C}\to\mathbb{C}\setminus\{0\}$ and a closed path $\gamma$ with ...
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127 views

Show that $z^5 - z +16$ has two roots in the right half plane

Show that the polynomial $$z^5 - z +16$$ has all of its roots in the region $$\{z\in \mathbb{C} \; | \; 1< |z| < 2\},$$ and show that two of its roots have positive real part. I have used ...
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1answer
74 views

Zeroes of $\sin(z)-z^2$

I am studying for my prelims exam. I stumbled upon the following question. Show that there are infinitely many zeroes of $\sin(z)-z^2$ in the complex plane. Had it just been $f(z)=\sin(z)-z$, one ...
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2answers
66 views

non-existence of a polynomial [duplicate]

This is a problem from a practice prelim exam I found online. Show that there cannot exist a polynomial of the form $p(x) = z^n +a_{n-1}z^{n-1} + \dots + a_1 z + a_0$ such that $|p(z)| < 1$ for ...
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3answers
45 views

Number of zeros of $p(z)=z^5+10z-3$ on annulus $A(0,1,2)$

How many zeros does the function $p(z)=z^5+10z-3$ have on the annulus $A(0,1,2)$? For $z\in C(0,2)$, we have $|f(z)|=|z^5|=32$ and $|g(z)|=|10z-3|\leqslant 10\cdot 2+3=23$, thus $|f|<|g|$ on $C(0,...
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28 views

Confusion in the proof of Rouché's theorem (joint continuity part)

Let, $f,g$ be two holomorphic functions in a region $\Omega$. $C$ be a circle in $\Omega$ containing interior such that $|f(z)|>|g(z)|\ \forall z\in C$. g vanishes nowhere on $C$. Then $f$ and $...
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Solving non-homogeneous system of equations with Rouche Capelli theorem

I have the following system of equations: x + y + z = 2 x - y + z + t = 1 I have converted this into a following matrix: \begin{bmatrix}1&1&1&0|2\\...
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1answer
99 views

Roots of $z \sin(z) =1$

this question has been answered here ,but I still have many questions. So I applied Rouche to a circle of radius $(n+\frac{1}{2}) \pi$, and set $f(z)=z \sin(z) -1$ And $g(z)=z \sin(z)$. Then I ...
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77 views

Prove that $ze^{\lambda - z} - 1$ has a real root in the unit disk

I am trying to show that $ f(z) = ze^{\lambda - z} - 1$, $\lambda > 1$ has a real root inside the disk. I have already showed, using Rouche's Theorem, that there is exactly one root inside the disk....
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1answer
78 views

Using Rouché's theorem to bound zeroes of partial sums of $e^z$

I'm having trouble with a question from an exam I'm studying for: " Prove that all roots of the polynomial $\sum\limits_{k=0}^n \frac{z^k}{k!}$ (for $n\geq 1$) are in the annulus $\{z: \frac{n}{e}<...
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2answers
63 views

Showing that, for $|a|>3$ and $n\geq1$, the function $f(z)=e^z-az^n$ has exactly $n$ roots (all simple and different) in the open unit disc [duplicate]

Let $|a|>3$, $n\geq1$, $n\in\mathbb{N}$, then the function $$f(z)=e^z-az^n$$ has exactly $n$ roots in the disc $\{z\mid|z|<1\}$, and that they are all simple. Hint: look at $f(z)-f'(z)...
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1answer
55 views

Finding number of roots to an equation

Let a be a complex number with $Re(a)>1$. How many solutions exist for the equation $e^z-z=a$ at $Re(z)<0$. I assume this is an exercise on Rouché's theorem, but this is the first time I see ...
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1answer
68 views

Rouché's Theorem/Argument Principle Application

Preparing for some prelim. exams, I encountered this problem: Show that $p(z)=z^6+3z^4+1=0$ has precisely two zeros in the upper half of the unit disk. There are two good ways to solve this. The ...
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1answer
90 views

finding the number of solutions $9z^4=\sin^2(z)$ in a complex sector

I want to find the number of solution of the equation: $9z^4 = \sin^2(z)$ in the complex sector: $S=(z\in \Bbb C|-1\le\Im(z)\le 1 )$ I tried to wrote down: $f(z) = \sin^2(z), g(z)=-9z^4$ so we need ...
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1answer
53 views

How many roots does $g(z)=z^7-2z^5+6z^3-z+1$ have inside the unit disk - Rouche's Theorem Application Verification

$g(z)=z^7-2z^5+6z^3-z+1$ and choose $f(z)=2z^5-6z^3$. On $\mid z\mid =1$, we have $\mid f(z)-g(z) \mid=1$ and $\mid f(z) \mid=4$ so $$\mid f(z)-g(z)\mid \leq \mid f(z)\mid$$ and by Rouche's ...
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67 views

Find the number of zeros of $f(z)=e^{z-1}-az$ inside unit disk, assuming $\mid a \mid >1$

This is an application of Rouche's theorem, I want to make sure I am doing it correctly: Let $f(z)=e^{z-1}-az$, where $\mid a \mid>1$ and $g(z)=-az$ Now, on the unit circle we have: $$\mid g(z) \...
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1answer
131 views

Rouche's theorem in the right half plane

I need to determine the number of zeros in the right half plane Re z>0 of the polynomial: $$ f(z)=z^3-z+1 $$ My attempt to solve the problem: I'm using Rouché's theorem and consider $$ g(z)=z^3+1 ~~ ...
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25 views

Number of zeros of $h(z)=3-z+2e^{-z}$ in the right half-plane $\Re(z)>0$ [duplicate]

I am supposed to show that $h(z)=3-z+2e^{-z}$ has exactly one zero in the right half-plane $\Re(z)>0$. I want to use Rouchés Theorem, which says that if $f$ and $g$ are analytic inside and on a ...
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1answer
197 views

If $11z^8+20iz^7+10iz-22=0$,then show that $1<|z|<2$

If $$11z^8+20iz^7+10iz-22=0$$then show that $$1<|z|<2$$ My Attempt: If $z=x+iy$; then$$z^7=\frac{22z-10iz}{11z+20i}$$ $$|z|^7=\sqrt{\frac{400+440y+100(x^2+y^2)+84}{400+440y+100(x^2+y^2)+21(x^...
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1answer
73 views

Question on Rouche's Theorem - Boundary Included?

Consider the region $1<|z|<3$. Using Rouche's theorem, it is possible to show that zero roots lie inside $|z|=1$ and three roots lie inside $|z|=3$. My question is, does Rouche's theorem ...
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43 views

Number of solutions of $z^5+4z^3=e^{iz}$ in $A=\{z:1 < |z| < 3\}.$

So we need the number of solutions in an this annulus and we will use Rouche's theorem. Let $p(z)=z^5+4z^3-e^{iz}$. We will first find the number of zeros in $D[0,3]$ and then subtract from this the ...
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1answer
100 views

Show that $f_a(z)=z+a-e^z$ has only 1 zero in $Re(z)<0$ and this zero is $<0$. $(a>1)$

I am trying to use Rouche's Theorem somehow but I can't seem to be able to find a proper function to compare $f_a(z)$ with. I tried $g(z)=z+a$ but then I can't deal with the $e^z$ term. Any ...
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1answer
33 views

Find number of elements $z$, $1<|z|<2$ satisfying $f(z)=0$ where $f(z)=z^5+z^3+5z^2+2$.

Let $g(z)=z^5$ then $\forall z$ s.t $|z|=2$ $|g(z)-f(z)|\leq |2|^3+5|2|^2+2< |2|^5=|g(z)|$ Therefore, By Rouché's $f(z) $ has 5 zeroes in $D(0,2)$. I do not know how to proceed any further. ...
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1answer
66 views

Application of Rouché's theorem outside the unit circle; Confusion about argument principle

I am preparing for my function theory exam and came across this problem. Let $f(z) := z^8 - 8z^5+2$. I want to count the number of roots in B:= $\big\{z \ \big| \left| z-2 \right| < \frac{1}{2} \...
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45 views

Finding the number of zeros of a polynomial in the closed disk [duplicate]

Find the number of zeros of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq1$. My attempts have not gotten far. I know we can examine the related equation $f(w)=w^3-5w^2+3w-1$ in $|w|\leq1$, letting $w=z^2$. ...
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1answer
41 views

Using Rouché's theorem to prove the degree of a polynomial over $\mathbb{C}$

Let $f$ be analytic over $\mathbb{C}$ and for $z\in\mathbb{C}, |f(z)|\le 7|z|^5$. Prove that $f$ is a polynomial with degree $\le 5$. Can Rouché's theorem be applied here? can I say that $7z^5$ has ...
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2answers
348 views

How many zeros does the polynomial have in the right half plane?

The polynomial is $f(z) = z^4+\sqrt{2}z^3+2z^2-5z+2$ If you check the image of the imaginary axis, you see that there are no zeros, so we can use the right semicircle from $iR$ to $-iR$,and make $R$ ...
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1answer
79 views

how many roots does $p(z) = z^{10} + 100z + 1$ in $\{z:|z|<1\}$

how many roots does $p(z) = z^{10} + 100z + 1$ has in $\{z:|z|<1\}$ Can I use Rouche theorem and say that on that region $|100z|=100>2=1+1=|z^{10}|+1$ and thus the number of roots for p(z) is ...
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2answers
69 views

Prove that sufficiently large partial sums of the Taylor series expansion of $e^z$ have all roots outside of an arbitrarily large radius.

Consider the complex-valued family of functions $$ f_n(z) = \sum_{k=0}^n \frac{1}{k!}z^k. $$ Is it possible to use Rouché's Theorem to prove that for any $R \in \mathbb R$ there exists some $n$ ...
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3answers
60 views

Rouché for Polynomial $p(z)=z^7+z(z-3)^3+1$ around non-centered annulus.

Let $p(z)=z^7+z(z-3)^3+1$. Find the numbers of zeros (including multiplicities) in $B_{1}(3)$. I want to apply Rouché's Theorem for $p(z)$ and $g(z)=-z(z-3)^3$, but I don't know how to confine on $|z-...
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1answer
74 views

Find the number of zeroes of $6z^3 + e^z + 1$ in the unit disc $|z|<1$

I have studied Rouche's theorem and applied it to polynomial expressions but I don't seem to understand the problem in expressions with an exponential term. My approach to the above question is as ...
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1answer
189 views

Question about a proof (Rouches Theorem)

Context: I made a video some time ago, where I tried to prove Rouches Theorem with the help of Argument Principle and Cauchy Integral Theorem. But now it seems like I made a mistake in the video ...
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2answers
43 views

An Application of Rouché's Theorem for the estimating the position of zeros

I am working on the following exercise: Let $p$ be a polynomial function of the form $p(z) = z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0$. Show that all zeros of $p$ are in the open circular disc $\...
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2answers
93 views

Number of zeroes outside $|z|>2$ (Rouche's theorem)

So I have been given the following equation : $z^6-5z^3+1=0$. I have to calculate the number of zeros (given $|z|>2$). I already have the following: $|z^6| = 64$ and $|-5z^3+1| \leq 41$ for $|z|=2$...
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1answer
30 views

solutions to $cz^n=e^z$ in $\{z\in\mathbb{C}:\Re(z)<1\}$

Let $c\in\mathbb{C}$ such that $|c|>e$. How many solutions are there to the equation $cz^n=e^z$ in $G=\{z\in\mathbb{C}:\Re(z)<1\}$ ? Equivalently, we can look at zeroes of the function $f(z)=...
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36 views

Geometric explanation of Rouché theorem

A similar question can be found in this post. I apologize for any inconvenience. I can't understand the geometric interpretation of Rouché's theorem in Wikipedia. The standard formulation of Rouche'...
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42 views

Zeros of holomorphic function on Annulus

This is an example from a book. Let $f(z) = z^5 + z^4 + 6z +1.$ Then since for $|z| = 2,$ $$|z^4 + 6Z + 1| \leq 29 < 32 = |z|^5$$ and for $|z| = 1,$ $$|z^5 + z^4 + 1| \leq 3 < 6 = |6z|.$$ By ...
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1answer
87 views

Rouché's theorem $e^{-z}-2z^2=1$

Show that the equation $$e^{-z}-2z^2=1$$ has exactly two solutions in the disk $|z|<1$, and that they are both real. I'm trying to use Rouché's theorem to find the zeros for $f(z)=e^{-z}-2z^2-1=0$...
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1answer
81 views

Rouche's Theorem of functions other than polynomial

All problems of Rouche's theorem are about zeros of polynomials , Is there another functions can I apply Rouche's Thm. to find the number of zeros other than polynomials ? please give me an example ...
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1answer
41 views

Complex Analysis - Rouches Theorem

Using Rouches Theorem, Determine the number of roots of the equation $z^4-6z^3+9z^2-24z+20=0$ inside the circle $|z|=2$. My problem here is that i cannot find a single dominant term? Can i choose ...
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1answer
214 views

Counting Zeros with Rouche's Theorem

I'm attempting to answer the question, "Prove that for any positive number $\epsilon$, the function $f(z)=\frac{1}{z+i} + \sin(z)$ has infinitely many zeros on the strip $|Im(z)|<\epsilon$". My ...
3
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1answer
185 views

Complex roots of $z^{10} - 6z^9 + 6^9$ inside the disk of radius six

Show that there are exactly $8$ roots of the polynomial $z^{10} - 6z^9 + 6^9$ inside the disk of radius $6$ centered at the origin in the complex plane. I am trying to use Rouche's theorem but am ...