5
votes
1answer
71 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
3
votes
1answer
74 views

Find roots of $3z^{100} - e^z$ in the unit disc.

This question was given in an exam in complex analysis: Let $f \left( z \right) = 3z^{100} - e^z$. Find all of $f$'s roots in $D \left ( 0,1 \right)$ and show that they are simple roots. I've seen ...
1
vote
1answer
55 views

Rouches Theorem Applied to a family of Polynomials

I would like to prove that the family of polynomials $z^{2j+2} + \alpha z^{2j+1} - \alpha z - 1$ has only one root inside the open unit circle when $|\alpha|$ is greater than 1. This seems like an ...
0
votes
1answer
35 views

Proving that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to z_0}\frac{f^\prime(z)}{g^\prime(z)}$

Let $f,g$ both analythic in neighbourhood of $z_0$ and they both have zero of multiplicity $n$ in $z_0$. Prove that $\lim\limits_{z\to z_0}\frac{f(z)}{g(z)}=\lim\limits_{z\to ...
6
votes
1answer
69 views

Solution of $\exp(z)=z$ in $\Bbb{C}$.

I have posted a related question here. I thinkg this one is more interesting: What about the solution of $\exp(z)=z$ in $\Bbb{C}$? My try : $z \mapsto e^z - z$ is entire non-constant. Perhaps ...
4
votes
3answers
112 views

all complex solutions of $z\sin(z)=1$?

A possibly easy question, Can we find all complex solutions of $z\sin(z)=1$ ? My try: Let $$\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$$ so we have $$ z\frac{e^{iz} - e^{-iz}}{2i}=1 $$ Not sure how ...
1
vote
1answer
44 views

Using argument principle on $e^z + z$

I want to use the argument principle to estimate the number of zeros of $e^z + z$ inside the rectangle with sides $y=2\pi n i, 2 \pi (n+1)i$ and $x = R, -R$ for $R$ large. But $\int \frac {e^z + ...
6
votes
1answer
93 views

zeros of a polynomial

Given $P(z)=z^6+6z+10$, find how many roots are in each quadrant I have already seen that $P(z)$ has six different roots, and that none of them are real or of the form $ki$, $k\in \Bbb R$. Since ...
0
votes
1answer
32 views

Application of Rouché's (Rouche's) Theorem to a Polynomial

Here is my question: State Rouché's theorem. How many roots of the polynomial $p(z) = z^8 + 3 z^7 + 6 z^2 + 1$ are contained in the annulus {$1 < |z| < 2$}? The statement is fine. I then ...
0
votes
1answer
29 views

Simple Pole Search

How do I find poles of: $H(z) = \dfrac{z^3}{z^3+\alpha}$. I know I must find the z values that do $z^3 = -\alpha$. I know how to do it in Matlab (with "residuez" function) but, how can i solve this ...
0
votes
1answer
48 views

Set of Solutions of A Quadratic Equation with Coefficients in $\{0,1,\cdots , \ p-1\}$

I was just playing with quadratic equations and this interesting question came into my mind. Say I have a set of quadratic polynomials $S=\{f_{(b,c)}(x)=x^2+bx+c:b,c\in \{0,1,\cdots, p-1 \}\}$ where ...
2
votes
2answers
46 views

Prove that $p(z) = 2z^5 + 6z - 1 $ have roots (in two sets)

Prove that $p(z) = 2z^5 + 6z - 1 $ have one real root in $(0,1)$ and four root in $\left\{ z \in \mathbb{C} : 1<|z|<2 \right\}$. I suppose that we should use Rouché's theorem but I have no ...
1
vote
1answer
59 views

Complex Analysis: Isolated Singularities, Poles, and Residues

I was given the following question. Show that the isolated singularities of the function $f(z) = \frac{z}{z^4+4}$ are poles. Determine the order of each pole and find the corresponding ...
2
votes
1answer
38 views

How to show that it holds $|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$ for any root of $X^n+\sum_{k=0}^{n-1}a_kX^k$?

Let $z\in\mathbb{C}$ be a root of the complex polynomial $$f=X^n+\sum_{k=0}^{n-1}a_kX^k$$ I want to show that it holds $$|z|<2\max_{0\le k<n}|a_k|^{\frac{1}{n-k}}$$ Proof: For $s>1$, consider ...
12
votes
3answers
166 views

Why is this polynomial a monomial?

Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$. Why is it that $p(z) = az^n$ for some $|a| = 1$? I've noticed that we could easily prove this by induction if we ...
2
votes
1answer
31 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
2
votes
0answers
19 views

Extensions of the Hermite Bielher and Hermite-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
1
vote
1answer
32 views

Constant function with maximum modulus [duplicate]

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
6
votes
2answers
152 views

All roots of polynomial inside the open unit disc

I know from here that for a polynomial $p(z)=a_0+a_1z+...+a_nz^n$ with $0<a_0\leq a_1\leq...\leq a_n$ all roots are in the closed unit disk. What condition do we need to get that all roots are in ...
1
vote
1answer
67 views

Minimum Modulus Principle for a constant fuction in a simple closed curve

Suppose that $f$ is analytic on a domain $D$, which contains a simple closed curve $\gamma$ and the inside of $\gamma$. If $|f|$ is constant on $\gamma$, then I want to prove that either $f$ is ...
0
votes
1answer
49 views

Using Rouche's theorem to find number of roots.

I am still unsure how exactly one applies Rouche's Theorem to find the roots of polynomials. For example, to find how many roots $z^9+z^5-8z^3+2z+1$ has in between the circles $|z|=1$ and $|z|=2$. I ...
1
vote
0answers
48 views

How to explain this result due to Pôlya

How to explain this result due to Pôlya: An entire function is determined uniquely by the inverse images, counting multiplicities of three distinct non omited values. I cannot understand how this ...
2
votes
3answers
94 views

Determine the number of zeros in the first quadrant $f(z) = z^4- 3z^2 + 3$ [closed]

Determine the number of zeroes of the following function which are in the first quadrant: $$f(z) = z^4- 3z^2 + 3$$ Help please!!! I'm not that good at complex variables!
3
votes
2answers
84 views

Determine the number of zeros in the first quadrant

This is a homework question: $$f(z) = z^2 - z + 1$$ sorry for the poor code!
1
vote
1answer
32 views

How do I find zeros in D(0,2)

$p(z) = z^8 - 20z^4 + 7z^3 + 1$. I know there is 4 real roots, but how do i figure out how many zeroes are there in $D(0,2)$?
1
vote
1answer
36 views

Why does the Uniqueness Principle imply real identities are true in the complex analogue?

Uniqueness principle theorem :If $f$ and $g$ are analytic functions on a domain $D$, and if $f(z)=g(z)$ for $z$ belonging to a set that has a non isolated point, then $f(z)=g(z)$ for all $z\in D$. ...
2
votes
1answer
33 views

Open mapping principle complex?

Shows that if $f(z)$ is a non-constant analytic function on a domain D, then the image under f(z) of any open set is an open set. What I have so far: Since $f(z)$ is non-constant and and analytic, it ...
1
vote
0answers
39 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
4
votes
6answers
187 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
0
votes
1answer
98 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
1
vote
1answer
189 views

Show that $ z \sin(z) = 1 $ has only real solutions.

Here is my question - it is an example sheet question, completely non-examinable: Show that the equation $ z \sin(z) = 1 $ has only real solutions. [Hint: Find the number of real roots in the ...
0
votes
2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
1
vote
2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
1
vote
0answers
20 views

What is the possible structures (closed, discrete, etc…) of the set $A$

Let $f$ be a non identically zero holomorphic function on the set $B=(a,b)×ℝ$. Let $g$ be a non identically zero harmonic (not holomorphic) function on the set $B=(a,b)×ℝ$. Assume that there is a set ...
2
votes
1answer
46 views

Number of roots and Rouché's Theorem

Given a polynomial $p(z)=z^4 +6z+3$, I want to show that it has exactly one root $z_1$ with $|z_1|<1$. I am pretty sure it will be easy to show this using Rouché's Theorem. Using this I would have ...
2
votes
1answer
31 views

Extend rolle's theorem to complex functions?

If $f(z)$ is a polynomial of degree n with n distinct real roots $r _1$<,..., <$r_n$, then there exists exactly one root of $f '(z)$ in between any consecutive root of $f(z)$. The context of ...
0
votes
2answers
75 views

Relationship between f and f' in terms of number of real/non-real roots

$f$ is a polynomial of degree $n\ge1$ and $\forall x,x\in \Bbb R \rightarrow f(x)\in\Bbb R$. Prove that: (a)$f$ has at most one more real root than $f'$ (b)$f'$ has no more non-real roots than $f$ ...
1
vote
2answers
68 views

Zeros of $e^{z}-z$, Stein-Shakarchi

This is an exercise form Stein-Shakarchi. Prove that $$f(z) = e^{z}-z$$ has infinite many zeros. What I have done : if not, by Hadamard theorem we obtain $$e^{z}-z = ...
1
vote
1answer
65 views

$h^{n} = f$, $h$ and $f$ entire functions

I found this exercise. Let $f$ be an entire function and $n$ a positive integer. Show that there exists an entire function $h$ such that $h^{n} = f$ if and only if the orders of the zeros of $f$ are ...
0
votes
1answer
96 views

Zeros of $f_{\epsilon}(z) = f(z) + \epsilon g(z)$ with $f$ and $g$ holomorphic

I'm stuck with this problem from Stein-Shakarchi: Suppose $f$ and $g$ are holomorphic in a region containing the disc $|z| \leq 1 $. Suppose that $f$ has a simple zero at $z = 0$ and vanishes nowhere ...
1
vote
4answers
73 views

Multiplicity of zeros

Can you explain me how to get the multiplicity of a zero? In particular, I would ask you how to determine the zeros' multiplicity of $$\cos(\frac{\pi}{2}z)$$ I suppose they are $z = 2k+1, k \in ...
3
votes
1answer
141 views

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \pmod 6$

Proof Verification: The polynomial $f(x) = (x+1)^n-x^n-1$ has a root of multiplicity 2 if and only if $n \equiv 1 \mod 6$ Let $r$ be a root, real or complex, of multiplicity 2 of $f(x)$. Then, by the ...
2
votes
1answer
69 views

If $f$ analytic in $|z|>1$ and $|f(z)|<|z|^n$, then $f$ has finitely many zeros in $|z|>2$.

Let $f(z)$ be analytic in $\Omega = \{|z|>1\}$. Suppose that $f$ satisfies $|f(z)| < |z|^n$ for all $z \in \Omega$ and for some n> 0. Prove that either $f$ has finitely many zeros in ...
5
votes
1answer
128 views

How to imagine zeros of an analytic function of several variables

Let $f(z_1,\cdots, z_n)$ be a holomorphic function of several variables in an open subset of $\mathcal C^n$. Let $Z(f)=\{ (z_1,\cdots, z_n) \: | \: f=0\}$ be the zero set of $f$. If $n=1$, the zeros ...
1
vote
2answers
39 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
2
votes
1answer
79 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
1
vote
1answer
44 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
2
votes
2answers
63 views

rectangle where $\cos{z} =iz$ has exactly one solution

Determine a rectangle inside which there is exactly one solution of the equation $\cos{z} = iz$. I know the following result: Let $f$ be holomorphic in $\Omega$ with $a \in \Omega$. Let $f(a)= b$ is ...
2
votes
1answer
50 views

uniformly bounded sequence of non constant holomorphic functions

Let $\{f_n\}_{n=1}^{\infty}$ be a uniformly bounded sequence of nonconstant holomorphic functions in a connected open set $\Omega$. Let $f \not \equiv 0 $ be a holomorphic function in $\Omega$. ...
4
votes
2answers
100 views

meromorphic function in the unit disc with only one pole of order n

Let $f$ be meromorphic in a neighborhood of $\{|z| \leq 1\}\setminus \{1/2\}$ and have a pole or order $n$ at $1/2$. Suppose that $|f| < 3$ on $\{|z|=1\}$. Show that for any $\phi \in \mathbb{R}$, ...