Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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1answer
20 views

Sum of $n^{\text{th}}$ powers of roots of quadratic

How would I go about finding an expression (preferably closed form) for the sum of $\alpha^n+\beta^n$ in terms of $\alpha + \beta$ and $\alpha\beta$ (where $\alpha$ and $\beta$ are the roots of a ...
3
votes
1answer
62 views

Why do we say that $\sqrt{-0} = -0$?

According to wikipedia's page on signed zeroes, we agree that $\sqrt{-0} = -0$. I would always have guessed that it would be $0i$, as $(0i)^2 = 0^2*i^2 = 0 * (-1) = -0$. I know that my own ...
1
vote
1answer
28 views

The root of $x^2+[1]=[0]$ in $\mathbb{Z}_p$

In $\mathbb{Z}_p$, where $p$ is a prime, how many roots of $x^2+[1]=[0]$? It is equivalent to show $[x^2]=[p-1]$,when p=3,there is non. When p=5, $x=2$,does there exist any rule of it
0
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1answer
23 views

Solution of $A = e^{\alpha t}\cos(\omega t + \phi)$

I would like to find the real roots of the function $$i(t) = \frac{\hat{V}}{R}\left(\frac{\omega^2}{(\alpha^2 + \omega^2)} \cos\left(\omega t + \tan^{-1}\left(\frac{\alpha}{\omega}\right)\right) + ...
1
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2answers
48 views

Roots of $f(x)=a_0+a_1\cos x+a_2\cos 2x+\dots+a_n\cos nx$

If $a_i$'s are nonzero real numbers such that $a_n > {\sum^{n-1}_{i=0}}|a_i|$ prove that the number of roots of $f(x)=a_0+a_1\cos x + a_2\cos 2x+\dots+a_n\cos nx$ is at least 2n.
2
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2answers
15 views

Roots of $i(t) = Ae^{\alpha t}cos(\omega t + \phi)$

I would like to find the roots of the function $i(t) = Ae^{\alpha t}\cos(\omega t + \phi)$ in the form $t = f(A, \alpha, \omega, \phi)$.
2
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1answer
19 views

Roots of a complex polynomial with leading coefficient larger than absolute sum of rest

Suppose I have an $N^{\text{th}}$ degree polynomial $P_N(z)=\sum_{i=0}^N a_i z^i$ where $\{a_i\}_{i=0}^N$ are complex numbers such that $|a_N|> \sum_{i=0}^{N-1}|a_i|$, can I claim that all its ...
0
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0answers
43 views

Real roots of a quintic polynomial with constraints

This is a question on the edge of math and programming. I pondered about the best way to state the problem: should I provide context, or get straight to the point of the question? Given various ...
2
votes
1answer
76 views

About the identity $e^{i\pi}=-1$

I have a question about the famous identity of Euler $e^{i\pi}=-1$. I opened the other day this question about the number of roots of a complex number with irrational exponent. Under this light and ...
0
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1answer
17 views

Lebesgue Measure of the set of roots of a complex exponential equation

In the following equation $\{\beta_i\}_{i=1}^N$ and $\{\alpha_i\}_{i=1}^N$ are non-zero complex numbers: $\sum_{i=1}^N \beta_i e^{\alpha_i t} = 0$. I would like to know if the Lebesgue measure of the ...
0
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1answer
44 views
+50

Sign of Laguerre root finding iteration

I'm trying to understand the method by Laguerre for polynomial root finding. However, I have some difficulties to understand one sentence of the book Applied Computational Complex Analysis (vol. 1) by ...
4
votes
1answer
36 views

Prove that this Newton sum value is unique

$$\begin{align}a+b+c+d&=1\\ a^2+b^2+c^2+d^2&=2\\ a^3+b^3+c^3+d^3&=3\\ a^4+b^4+c^4+d^4&=4\\ a^5+b^5+c^5+d^5&- ?\end{align}$$ The usual method I see for solving this kind of ...
0
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2answers
25 views

Finding a cubic equation from the relation between the roots

I'm trying to solve this problem: $ x^3 - x^2 - 3x -10 = 0$ has roots α,β,γ. Let u = −α+β+γ. Show that u+2α=1, and hence find a cubic equation having roots −α+β+γ, α−β+γ, α+β−γ. I was able to ...
0
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0answers
25 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
1answer
41 views

Writing roots of f(x) as f(a) for some a

I was solving a problem when this random thought popped into my head. Suppose you have a function, say $f(x)=x^2-1=(x-1)(x+1)$. The roots of this function are $-1$ and $-1$. We can write these roots ...
0
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0answers
29 views

Rayleigh quotient iteration and root finding

I'm trying to find the roots of a polynomial by finding the eigenvalues of its companion matrix. I understand that it is possible to use QR algorithm as the matrix happens to be in Hessenberg form ...
1
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2answers
37 views

Relation between the roots and the coefficients of a polynomial

I have studied that: For the polynomial $ax^3+bx^2+cx+d=0$, with roots $\alpha, \beta, \gamma$: We have: $$\begin{align} & \alpha + \beta + \gamma = -\frac ba \\ & \alpha\beta + \beta\gamma ...
2
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2answers
80 views

Method of dominant balance and perturbation

Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$ I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
1
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0answers
40 views

Solve $z+\sin{z}=i$

How can I find how many solutions following equation have? $$z+\sin{z}=i$$ I can make substitution $z=it$ and get $$t+\sinh{t}=1$$ which has one real solution $t\approx0.4900730685$ thus ...
2
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2answers
34 views

Finding the Roots of Cubic (Boas 2.14.25)

I'm working my way through Mathematical Methods in the Physical Sciences and came across the following problem: Use a computer to find the three solutions of the equation $x^3−3x−1=0$. Find away ...
1
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1answer
37 views

Numerical root finding for 5th degree polyomial

I have the equation $y^5 -ay -b=0$. I need to get a solution whether numerical or analytical. I heard $5$th order polynomials are not solvable analytically, so how can I get the root numerically. ...
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2answers
55 views

Cube root equations 1

$$E_{1} : \sqrt[3]{1+z}-\sqrt[3]{1-z}=\sqrt[6]{1-z^{2}} $$ Let $a=\sqrt[3]{1+z}$ and $b=\sqrt[3]{1-z}$ $E_1$ is equivalent to $E_2$ : $$ E_2:\ ...
0
votes
1answer
31 views

Polynomial with one rational root or one imaginary root

In my textbook there is an example where we have to find all the roots of $2x^3-5x^2+4x-1$. After applying the Rational Root Theorem we can conclude that $1$ and $1/2$ are two solutions to this ...
7
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0answers
122 views

Is it true that $\gamma_{\lfloor\log\Gamma(x)\rfloor}\sim 2\pi x$?

I realise that Gram points can approximate the imaginary part on the $x$th zeta zero $(\gamma_x)$ accurately, and indeed, Guilherme França, André LeClair give another formula, namely ...
1
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0answers
52 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
1
vote
1answer
33 views

Roots of a polynomial with one parameter

Let $P$ be the polynomial defined by $P (Z) = Z^{4}-aZ^{2} +1$ Calculate $P\left(\sqrt{\dfrac{a}{2}}\right)$ and deduce the number of real roots of $P$ as the case $a = 2$, $a >2$ or $a ...
1
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2answers
68 views

How many roots has the equation?

Let $f(x)=x^3-3x+1$. How many roots has the equation: $f(f(x))=0$? I tried to solve it graphically and found 7 roots. If there exists analytical solution?
1
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1answer
37 views

Asymptotic behavior of the solution to an equation

Let $c\in(0,1)$ be a constant and let $k$ be a positive odd integer, and let $a(k)$ denote the value of $a$ that satisfies the equation $$(1-a)^kk\sqrt{a}=c$$. As $k\rightarrow\infty$, what can we ...
0
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0answers
35 views

Find the condition on b so that 6th degree polynomial has at least three real roots

Most of the questions on this site ask: Given a polynomial, how to find the number of real roots. My question is: given a 6th degree polynomial $P_b(x)$: (b lies between 1 and 4) ...
1
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2answers
67 views

I want to get a formula for solving this

If $abc = n$ where $a,b,c,n\in \mathbb{N}$ then can you derive a formula to find the total number of triples of a,b,c as such? eg : $abc = 12$ has $4$ such triples, ...
0
votes
2answers
79 views

Finding roots with seemingly no algebraic way

I have a graph of: $$y = \frac{x^3 + 2x^2 - 4}{x^2}$$ and I have to find the x-intercept. So I have the equation $(x^2)(x+2)-4 = 0$ And then I don't know what to do. Not sure if we can use ...
1
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0answers
47 views

How many zero's does a general real entire function $f(z)$ have?

Let $f(z)$ be a real entire function. How do we find the number of solutions for $f(w)=0$ ? Can we express the number of zero's of $f$ in terms of its Taylor coëfficiënts ? Im not looking for the ...
2
votes
1answer
29 views

Average Number of Roots of a Polynomial modulo p

Let $f \in \mathbb{Z}[X]$ be an irreducible non-constant polynomial, and consider this polynomial modulo $p$ for each prime $p$. On average, how many roots does $f$ have modulo $p$? I.e., if $r(p)$ ...
4
votes
3answers
255 views

How can I show why this equation has no complex roots?

I've been asked to show why an equation has no complex roots but i'm at a complete loss. The equation is $F_{n+2}=F_n$ Where $F_n=(x-1)(x-2)...(x-n)$ and n is a positive integer. I'd really ...
-3
votes
1answer
26 views

Find two roots for $\cos 5x=a$ [closed]

For some $a>0$ equation $\cos 5x=a$ has two roots, difference between them is $\large\frac{7\pi}{4}$. Find all roots of this equation!
0
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0answers
50 views

Polynomial of Degree 3 Solutions [duplicate]

If $p(x) \in F[x]$ is of degree $3$, and $p(x)=a_0+a_1x+a_2x^2+a_3x^3$, show that $p(x)$ is irreducible over $F$ if there is no element $r\in F$ such that $a_0+a_1r+a_2r^2+a_3r^3 =0$. If $p(x)$ is ...
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votes
2answers
67 views

Existence of a root

Let $f:[a,b] \rightarrow \Bbb R$ continuous, such that for every $x$ there is a $y$ such as that $|f(y)|\leq|f(x)|/2$. Show there exists a $\xi$ such that $f(\xi)=0$
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1answer
38 views

Finding complex roots of integer polynomials

How would one find approximates for complex root of polynomial with integer coefficients,I know for example the Newton's method $$x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$$ Anyway is it possible to ...
2
votes
2answers
41 views

Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
3
votes
1answer
116 views

The number of solutions of $z^5+2z^3-z^2+z=a$ for $a\in \mathbb{R}$

How we can calculate the number of solutions of $$z^5+2z^3-z^2+z=a\;\;,\;\;a\in \mathbb{R}$$ in the half-plane $\mathfrak {Re}(z)\ge 0$. Any hint would be appreciated.
2
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0answers
34 views

What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
4
votes
2answers
223 views

Problem getting the real roots of this complex expression

I'm trying to get the real roots of this expression: $$\dfrac{1}{z-i}+\dfrac{2+i}{1+i} = \sqrt{2}$$ Where $i^2=-1$ and $z=x+iy$. I tried to simplify that with Algebra, and then separate the real ...
0
votes
4answers
49 views

One root of the equation $x^2-(r+3)x+(5r-3)=0$ is twice the other root. Find the two possible values of r. [closed]

One root of the equation $x^2-(r+3)x+(5r-3)=0$ is twice the other root. Find the two possible values of $r$. I need help with this question, thank you.
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votes
2answers
62 views

Form a quadratic equation whose roots are $\sqrt{3}+2$ and $\sqrt{2}+3$ [closed]

Form a quadratic equation whose roots are $\sqrt{3}+2$ and $\sqrt{2}+3$ I need help with this question, thank you
7
votes
3answers
182 views

Upper and lower bounds for the smallest zero of a function

The function $G_m(x)$ is what I encountered during my search for approximates of Riemann $\zeta$ function: $$f_n(x)=n^2 x\left(2\pi n^2 x-3 \right)\exp\left(-\pi n^2 x\right)\text{, ...
4
votes
2answers
127 views

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$?

How to solve $\displaystyle x=\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}$ for $x$? I tried this way: Let $$f(x)=\sqrt{4+\sqrt{4-x}}$$ So, $x=f^2(x)=f^{2n}(x)$ where $n\in\mathbb{N}$. Then, I tried to ...
0
votes
1answer
41 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
8
votes
1answer
149 views

Do perfect polynomials of degree $4$ exist?

I asked this question already, but I cannot find it anymore. If it is a duplicate, I will delete it. Is there a polynomial $$p(x)=x^4+ax^3+bx^2+cx+d$$ such that p and all the derivates upto the ...
3
votes
1answer
45 views

Find zero of sum of 4 modified Bessel functions

I am trying to find the (positive) root of the function $f(x) = I_{-3/4}(x) + I_{3/4}(x) - I_{-1/4}(x) - I_{1/4}(x)$ where $I_\alpha(x)$ denotes the modified Bessel function of the first kind. ...
1
vote
0answers
35 views

Counting Zeros of complex functions in the upper half plane

I have a question about counting zeros. Here it goes Given $f(x)= i z^5+z-2010$. Find the number of zeros of $f$ in the upper half plane $\operatorname{Im}(z)>0$. I have tried to use the Argument ...