1
vote
0answers
35 views

Find the number of zeros of the polynomial in the first quadrant [closed]

$$p(z) = z^6+9z^4+z^3+2z+4$$ Please help! I have an exam coming up and I don't completely understand!!!
2
votes
3answers
66 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
61 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
28 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
30 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
23 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
35 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
159 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
60 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
121 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
0answers
21 views

Show that the full series have no zeros in $A$

Let us consider the convergent series $h:Ω→ℂ$ given by $$h(α,β)=∑_{n=2}^{∞}(-1)ⁿ⁻¹a_{n}(α)n^{iβ}$$ where $α,β$ are reals in the domain of convergence and $a_{n}(α)$ is a real increasing sequence with ...
0
votes
2answers
54 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
42 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
36 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
29 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
69 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
64 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
60 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
83 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
49 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
138 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
61 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
89 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
36 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
67 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
43 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
58 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
46 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
82 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 $\phi \in \mathbb{R}$, ...
1
vote
1answer
63 views

Find the root of the polynomial?

Consider the root of the polynomial $p(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dots+a_1x -1$. Suppose that $p(x)$ has no roots in the open unit disc in a complex plane and $p(-1)=0$. Show that ...
3
votes
2answers
60 views

Existence of holomorphic function with a sequence of zeros in the unit disc

The question is : Prove that there exists a holomorphic function $f$ on the open unit disc $\{z \in \mathbb{C} : |z| <1\}$ with the properties that $f(0) = 0$ and $f(1-1/n)=1$ for every integer $n$ ...
3
votes
1answer
114 views

zeros of a function holomorphic in the closed unit disc

Let $f$ be a holomorphic function in a neighborhood of the closed unit disc $\{z \in \mathbb{C} : |z| \leq 1\}$, and suppose that $\Re{(\bar{z}f(z))} > 0 $ when $|z| = 1$. Prove that $f$ has ...
3
votes
0answers
113 views

solution set in $\mathbb{C}$ of $ z^{\frac1{z}}=\left(\frac1{z}\right)^z$

if $z \in \mathbb{C}$ what can be said about the solution set of: $$ z^{\frac1{z}}=\left(\frac1{z}\right)^z $$ aside from the fact that it contains the fourth roots of unity? i will add as a footnote ...
4
votes
0answers
89 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
3
votes
2answers
109 views

Roots of $e^z=1+z$ on complex plane

What are the roots in the complex plane of $e^z=1+z$? Clearly $z=0$ is one root. On the real line, we can show that $e^x>1+x$ for all $x\neq 0$. But what about the rest of the complex plane?
0
votes
0answers
40 views

How to find the number of zeros in the left half plane?

Given a rational function $P(s)/Q(s)$ with $deg(Q(s))\geq deg(P(s))$. How to show that $ Q(s)$ and $P(s)-Q(s)$ have same number of roots in the left half plane using Rouche's theorem? Instead of ...
1
vote
3answers
119 views

Examples of complex functions with infinitely many complex zeros

What are some examples of complex functions with infinitely many complex zeros? There are no particular restrictions on the functions I am just curious and having a hard time finding examples. Also ...
4
votes
1answer
85 views

Real root of a complex equation.

I was working on a problem from Gamelin; where I was required to find out zeros of $2z^5+6z^1-1$ , in the unit disk (in $\mathbb C$). I applied Rouché's theorem and find out zeros in the unit ...
7
votes
2answers
120 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
0
votes
0answers
28 views

Ways to compute roots of complex numbers

I know how to use the De Moivre's Formula, but to caculate it one need to use caclulator. Is there any better way to take roots of complex numbers that is more "caclulator-free"? I am particulary ...
0
votes
1answer
80 views

If $|f(z)|>0$ then $f$ has no zeros.

I am trying to understand the proof of Rouche's Theorem. Rouche's Theorem Let $C$ denote a simple closed contour, and suppose that a) two functions f(z) and g(z) are analytic on and ...
0
votes
0answers
105 views

Zeros of a power series

Suppose we have a power series with (real or complex) coefficients $\sum_{n \geq 0} a_n x^n$ (that has nonzero radius of convergence). Can one say something about its zeros in terms of the ...
2
votes
1answer
42 views

Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
4
votes
0answers
141 views

Determine the number of zero points of $z^8-5z^3+z-2$ within the open unit circle (Rouché?)

How many zero points does the polynomial $z^8-5z^3+z-2$ have within the open unit circle? Hello, consider $$ \gamma\colon [0,2\pi]\to\mathbb{C}, \varphi\longmapsto\exp(i\varphi) $$ and ...
1
vote
6answers
364 views

How do I solve and plot the complex equation

I have the following complex equation: \begin{equation} z^6 + 1 = 0 \end{equation} I would like to be able to gain some intuition and understanding. I know from the fundamental theorem of algebra ...
3
votes
2answers
224 views

Determine the number of zeros of the polynomial $f(z)=z^{3}-2z-3$ in the region $A= \{ z : \Re(z) > 0, |\Im(z)| < \Re(z) \}$

Question: a). Determine the number of zeros of the polynomial $$f(z)=z^{3}-2z-3$$ in the region $$A= \{ z : Re(z) > 0, |Im(z)| < Re(z) \}$$. (b). Find the number of zeros of the function ...
0
votes
1answer
51 views

Is the case where the zeros of $f$ or $g$ are isolated possible? [closed]

Assume that $f,g:\mathbb{C}→\mathbb{R}$. Let us consider the following equation in $\mathbb{C}$ $$f(s)g(s)=0$$ My question is: What are the cases where the zeros of $f$ or $g$ are isolated?
12
votes
0answers
228 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
7
votes
4answers
352 views

How many zeros does $z^{4}+z^{3}+4z^{2}+2z+3$ have in the first quadrant?

Let $f(z) = z^{4}+z^{3}+4z^{2}+2z+3$. I know that $f$ has no real roots and no purely imaginary roots. The number of zeros of $f(z)$ in the first quadrant is $\frac{1}{2\pi ...