0
votes
1answer
21 views

complex conjugate pairs of a quartic

I tried my hand at this question, which included finding the partial fractions of $\frac{x^2}{1-x^5}$. I found a factor of $1-x$ for the denominator, but I do not know how to work out the complex ...
1
vote
4answers
65 views

Finding all roots of $z^4-4z^3+9z^2-4z+8$

I need to know all the roots of $z^4-4z^3+9z^2-4z+8$. I know only one root: z=i. Is there an easy way to find the 3 roots that are unknown? thanks.
0
votes
2answers
43 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
1
vote
1answer
69 views

Tangent at average of two roots of cubic with one real and two complex roots

I was able to easily prove that the tangent at the average of two roots of a real cubic polynomial passed through the third root of the function. But I have only done this for functions with three ...
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 ...
1
vote
2answers
89 views

Evaluation of complex real numbers

The much anticipated math.SE community blog will $\tiny\mathrm{hopefully}$ contain a contribution from Alex Becker with the topic The Complex Real Roots of $x^3-3x+1$, which I'm really looking forward ...
1
vote
2answers
41 views

Finding matching roots

If ${4 + \sqrt{2}}$ is one root of a quadratic equation given by ${x^2 - Px + Q =0}$ where P and Q are rational numbers then find the missing root. The answer is ${4 - \sqrt{2}}$. And I'm a bit ...
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 ...
1
vote
1answer
53 views

Weird doubt about complex roots of second grade equation

I'm at the beginning of complex numbers study. I have the following equation: $$ x^2-6i=0$$ It's a second grade eq. so I expect to get two solutions. But: $$x=\sqrt{6}\sqrt[4]{-1}$$So I get 4 ...
1
vote
2answers
23 views

ODE with complex char roots gives strange solutions

$y''-4y'+5y=0$ has char roots - $\{e^{(2+i)x},e^{(2-i)x}\}.$ So its solutions is $e^{2x}\cos(x), e^{2x}\sin(x).$ But when i plug, e.g., first of them into original eq. i get: $-4 e^{2x} cos(x) + 8 ...
2
votes
2answers
69 views

Prove that 1 has n distinct roots of order n

I am trying to show that 1 has n distinct roots of degree n, or in other word that the equations $$z^n=1$$ has n different roots over the complex field. I know that the fundamental theorem of Algebra ...
2
votes
2answers
119 views

Find all solutions of $z^5+a^5=0$

The task is as follows: Find all solutions of $z^5+a^5=0$, where $a$ is a positive real number. My initial attempt (which leads nowhere) My guess is that i'll have to find the 5 5th roots of ...
3
votes
2answers
85 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 $n$ ...
2
votes
2answers
33 views

Polynomial With Imaginary Roots

Working on question 1 here http://www.sosmath.com/cyberexam/precalc/EA2002/EA2002.html Find a polynomial with integer coefficients that has the following zeros: ...
4
votes
2answers
140 views

Use $\alpha, \beta, \gamma $ roots of a polynomial to construct another polynomial [duplicate]

Let $\alpha, \beta, \gamma $ be roots $\in \mathbb{C}$ of $x^3-3x+1$. Determinate a monic polynomial, degree $3$, witch roots are $1- \alpha^{-1},1-\beta^{-1},1-\gamma^{-1}$ The catch is that i can't ...
0
votes
2answers
125 views

Polynomial divisibility

Given $p(x) \in \mathbb Q[x] $ an irreducible polynomial, and $\alpha \in\mathbb C $ root of $p(x)$. Prove that if $q(x) \in \mathbb Q[x]$ it's a polynomial, such $q(\alpha) = 0$ then $p(x) \mid ...
3
votes
2answers
125 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
2answers
62 views

Find the $8^{\text{th}}$ root of $1$ in the form $x+iy$.

I have squared each side $3$ times (not sure on the correct word but made it so it's $1=(x+iy)^8$ and expanded, is this the answer or is there a step to simplify everything?
1
vote
0answers
58 views

Zeta zeros by recurrence of zeta function, but this is useless isn't it?

One more useless question of mine can't do this site any harm. So here we go. The following Mathematica program converges to most of the riemann zeta zeros, by using an approximation as a starting ...
0
votes
2answers
58 views

Finding the roots of 4096x^3-10496x^2+152576x - 961=0 (1 root and 2 complex)?

I don't know how to find the roots of 4096x^3-10496x^2+152576x - 961=0 I try using wolfram and http://en.wikipedia.org/wiki/Cubic_function. I don't really understand it can someone please explain how ...
1
vote
3answers
53 views

Complex number: Roots

Solve all the roots of the following equation: $$(z-i)^2(z+i)^2=\frac{1}{4}.$$ Find the set of complex numbers $z$ such that $$\left|\frac{z-3}{z+3}\right|=2.$$ Would anyone mind telling me how ...
2
votes
1answer
94 views

Complex Number - Find all roots of the equation

$$e^{i \frac{{\pi}}{3}}z^5+4e^{i\frac{(2+3){\pi}}{6}}z^3 + z^2 + 4i = 0.$$ By using Euler's formula, I got: $$e^{i \frac{{\pi}}{3}} = \cos{\frac {\pi}{3}} + i\sin{\frac {\pi}{3}} = (\frac{1}{2} + ...
0
votes
0answers
30 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 ...
4
votes
3answers
127 views

Show that the real part of the root of an equation is constant

I've been stuck for a while on the following question: Let $z$ be a root of the following equation: $$z^n + (z+1)^n = 0$$ where $n$ is any positive integer. Show that $$Re(z) = -\frac12$$ ...
1
vote
6answers
452 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 ...
1
vote
5answers
241 views

Don't know how to find all the roots

So i got this problem : Find all the roots of $r^{3}=(-1)$ i can only think to use : $\sqrt[n]{z} =\sqrt[n]{r}\left[\cos \left(\dfrac{\theta + 2\pi{k}}{n}\right) + i \sin\left(\dfrac{\theta + 2\pi ...
15
votes
0answers
324 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 ...
1
vote
1answer
96 views

Analog to bisection: Converging on complex roots of a polynomial

I am working on a Perl module that, among other features, will solve all the zeroes of a polynomial. Thus far, I am doing OK for $2$, $3$, $4$th degree using quadratic, Cardano's and Ferarri's ...
6
votes
2answers
159 views

Proving that $\sum\limits_{n = 0}^{2013} a_n z^n \neq 0$ if $a_0 > a_1 > \dots > a_{2013} > 0$ and $|z| \leq 1$

I'm going to teach a preparation course for the complex analysis qualifying exam from my university (which basically consists of me doing some problems from past exams) and I'm trying to solve some ...
2
votes
0answers
63 views

Overdetermined system - showing that there are no roots that satisfy the set of equations

We consider an overdetermined set of equations, consisting of two equations for one complex variable $x$. I want to show that there are no roots for $x$ in the complex unit disc but without the ...
13
votes
3answers
141 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
-4
votes
1answer
184 views

Root of a quadratic equation that has modulus $1$

Let us suppose $\alpha \in \mathbb C$ and $|\alpha|=1$ and $\alpha$ satisfies a monic quadratic equation. Then prove that $\alpha^{12} =1$. Show me the right way to solve this. Thanks in advance.
4
votes
3answers
474 views

Solve $\sin(z) = z$ in complex numbers

Show that $\sin(z) = z$ has infinitely many solutions in complex numbers. Little Picard theorem should help, but using big Picard theorem is undesirable. Thanks a lot!
0
votes
1answer
86 views

Complex numbers and absolute values

If i have equation: \begin{align} P = \left|\psi\right|^2 \end{align} where $P$ is a probability and we know there is no negative probability. This means $P$ must belong to $\mathbb{R}$. If i want ...
2
votes
2answers
129 views

Roots of cubic polynomial lying inside the circle

Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle $|z|=max{\{1,|a|+|b|+|c| \}}$ Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers. What might ...
0
votes
1answer
42 views

Complex solutions to $a = (z+b)^n$

I have tried the whole afternoon trying to figure out how to approach an equation of the form $a = (z+b)^n$, more specifically the equation: $1 = (z+1)^4$. Is there a general approach to equations of ...
1
vote
2answers
153 views

Comparing square roots of negative numbers

If we have for instance $\sqrt{-25}$, that is, a square root of $-25$, I know the answer can be $5i$ (Is $-5i$ also correct? Sorry not professional in mathematics). My main question here is how to ...
1
vote
1answer
168 views

Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials?

Is the Fujiwara bound the most precise bound on maximum absolute value of complex roots of real polynomials ? Or does it exist some improved version for this special case of real polynomials ?
3
votes
2answers
435 views

geometric interpretation of quadratic equation with complex coefficients

When an equation has real coefficients and non-negative discriminant, the geometric meaning of it's roots is intersection of the function with the x-axis. I know how to get roots of quadratic ...
18
votes
4answers
757 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
5
votes
4answers
166 views

Can a cubic that crosses the x axis at three points have imaginary roots?

I have a cubic polynomial, $x^3-12x+2$ and when I try to find it's roots by hand, I get two complex roots and one real one. Same, if I use Mathematica. But, when I plot the graph, it crosses the ...
0
votes
4answers
112 views

For $\sqrt[3]{-1+i}$, is $r$ (when put in polar form) $\sqrt[6]{2}$?

And when you put that into the nth root form... It becomes $2^{1/18}\cos\theta + 2^{1/18}\sin\theta$? $n$th root form given is: $\sqrt[n]r\cdot\cos(\theta+2\pi k)n$
0
votes
2answers
90 views

Multiple root in a polynomial

I'm doing some old multiple tests. It seems I'm pretty stuck around the topic off complex numbers, could someone elaborate how to: Show that 1 is a multiple root of 2nd degree in p$p(x)=x^3-x^2-x+1$
3
votes
3answers
193 views

How to find the roots of $x³-2$?

I'm trying to find the roots of $x^3 -2$, I know that one of the roots are $\sqrt[3] 2$ and $\sqrt[3] {2}e^{\frac{2\pi}{3}i}$ but I don't why. The first one is easy to find, but the another two roots? ...
0
votes
1answer
162 views

Complex n-th root question

Let $m$ and $n\neq0$ be any two integers.Show that $z^{m/n}=\left(z^{1/n}\right)^m$ has $n/(n,m)$ distinct values, where $(n,m)$ is the greatest common divisor of $n$ and $m$. Prove that the sets of ...
1
vote
1answer
66 views

Understanding a theorem of Marden's on the moduli of zeros of polynomials

My question is concerning Theorem 3.2 in this paper of Marden's. The gist of the theorem is stated below. Theorem 3.2. Every polynomial of the form $$ f(z) = \sum_{j=0}^{n} (b_j - ...
5
votes
2answers
225 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
10
votes
6answers
653 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
1
vote
6answers
824 views

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct.

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct. Also, show that $|3z - z^3 + 5z^4| < |z^6|$ when $|z| > 3$. I can see that 0 is a real ...
0
votes
1answer
7k views

Where to find information on shadow functions?

I happen to give some private lessons to an IB (International Baccalaureate) student. He asked me for help with writing some kind of a project on a set topic, given some materials (containing the ...