All Questions
110 questions
1
vote
1
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49
views
Is these inequalities a criteria that a real polynomial has only real roots?
Suppose $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in\mathbb R[x]$ is a real polynomial with indeterminates $a_0,\cdots,a_{n-1}\in\mathbb R$.
Let $z_1,\cdots,z_n$ be all complex roots of $f(x)=0$.
Consider ...
- 868
5
votes
1
answer
169
views
Roots of polynomials defined by the Syracuse (Collatz) sequence
I'm a french graduate student, and I stumbled on a problem which seems to surpass my current abilities...
My goal was to study polynomials defined by the Syracuse sequence (or Collatz sequence). By ...
5
votes
0
answers
305
views
How many roots does $\tan(z)-z^n$ have for $n \in \mathbb{N}$, $\frac{-\pi}{2}\le \Re(z)\le \frac{\pi}{2}$?
Now asked on MO here.
I am investigating the number of roots of the equation
$$\tan(z) - z^n = 0$$
within the vertical strip $|\text{Re}(z)| \leq \frac{\pi}{2}$ for positive integers $n$. Numerical ...
- 6,781
1
vote
0
answers
94
views
Hard/Interesting analytical problem to solve: $0 = a_{1} \ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8}$
Is there some trick or approach I'm missing in attempting to solve \ref{1}, or better \ref{2}, analytically.
$$0 = a_{1}\ln(1+a_{2}e^{a_{3}x^{2}+a_{4}x+a_{5}})+a_{6}x^{2}+a_{7}x+a_{8} \tag{1}\label{1}$...
0
votes
0
answers
53
views
Prove that a trigonometric polynomial has 2n roots [duplicate]
$0<a_0<a_1<a_2<...<a_n$
Prove:$$a_0+a_1\cos\theta+...+a_n\cos n\theta $$
has 2n distinct roots in $(0,2\pi)$
This is a question in my textbook,the author leaves a hint.That is we should ...
- 349
0
votes
0
answers
105
views
Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has at least one root in $[0,3)$ - how to show this?
Given $6a+9b+4c\log3=0$, then the equation $2ax^2+3bx+4c=0$ has (A) no root in $[0,3)$
(B) all root in $[0,3)$
(C) exactly one root in $[0,3)$
(D) at least one root in $[0,3)$
In this question I ...
user1309752
3
votes
2
answers
193
views
Tricky Application of Rouche's Theorem
I'm supposed to use Rouche's theorem to solve this problem, but I'm pretty sure it's not possible. Can anyone confirm this? I want to determine how many zeros $e^z-z$ has on $B_1(0)$. The obvious set ...
- 846
4
votes
1
answer
103
views
A function with negative discontinuous derivative and many zeros
I want to construct an example of a function $f:\mathbb{R} \to \mathbb{R}$ with the following properties. By the way, I am not sure if it can exist.
$f(0) = 0$
$f'(0) < 0$
$f$ is continuous at $x=...
- 15.2k
9
votes
1
answer
286
views
When is the square root differentiable?
Let $f$ be a non negative differentiable function (defined on $\mathbb R$) and $g(x)=\sqrt{f(x)}$. Can you characterise the points $x_0$ where $g$ is differentiable at $x_0$?
It is clear that when $f(...
- 91
1
vote
0
answers
237
views
Bisection Method : Estimating Iterations needed to approximate a root within three decimal places
I have been working on a problem involving the bisection method and the estimation of the number of iterations required to approximate the root of a given function to three decimal places. I have made ...
- 11
2
votes
1
answer
86
views
Understanding a proof of the Open Mapping Theorem
In Saff's Complex Analysis book, the following OMT is stated.
If $f$ is non-constant and analytic in a domain $D$, then its range $f(D) := \{ w ~|~ w = f(z)~ \text{for some}~ z \in D \}$ is an open ...
- 385
10
votes
3
answers
623
views
How to prove the equations have only one real solution?
There are $n$ equations.
I need answer for the case $n=3$.
$$
\frac{1}{x_1}(1-x_1)^2+\frac{1}{x_2}(1-x_2)^2+\cdots+\frac{1}{x_n}(1-x_n)^2=0,
$$
and
$$
\frac{1}{x_1}(1-x_1^2)^j+\frac{1}{x_2}(1-x_2^2)^...
- 133
0
votes
1
answer
137
views
solving $\frac{d}{dx}(2^x+3^x\cos(\pi x))=0$ [closed]
is there any "elementary" method to solve this equation
\begin{align}
\frac{d}{dx}\big(2^x+3^x\cos(\pi x)\big)=0
\end{align}
I tried to use wolfram alpha for solutions but it couldn't do it.
- 69
2
votes
1
answer
227
views
Complex zeros of modified Bessel functions of first kind with order zero
I want to know about zeros of $$I_0(z)=\frac{1}{\pi}\int_0^{\pi}e^{z\cos(t)}dt=\sum_{m=0}^{\infty}\frac{(z^2/4)^{m}}{(m!)^2}.$$
From the formula $I_0(z)=J_0(iz)$ and the fact that $J_0$ has infinitely ...
- 1,175
-2
votes
1
answer
71
views
Explaining the zeroes of $x+\exp x$ [closed]
I want to know how can I explain why the function $x+ \exp x$ has a zero in $\mathbb{R}$? Where the equation $x+e^x=0$ can also be rewritten as $x=-e^x$
If anyone knows anything on how to begin such ...
user1128264
0
votes
1
answer
116
views
Are the real root of a polynomial continuous to its real coefficients?
I have the following question.
Given a polynomial with real coefficients, are the real root of the polynomial continuous to the real coefficients of the polynomial when the number of real roots does ...
- 11
0
votes
1
answer
63
views
Estimate the number of real roots
Given the function $f(x)=a-bx+ce^{\gamma x}+dxe^{\gamma x}$ with $x \in [0,\infty)$,
where $a,b,c,d$ are real parameters,$\gamma$ is strictly positive and that $f(0)=-1$,
Is it possible to know the ...
- 75
-5
votes
1
answer
326
views
Prove that a strictly decreasing function has only one root in a given interval
Let $G(s)$ be a function who is strictly monotonic decreasing in $s\in[0,1)$ with $G(0)>0$ and $G(1)=0$. How to prove that $s=1$ is the only root of the function in $s\in[0,1]$?
My problem in ...
- 119
1
vote
0
answers
120
views
Smallest and largest magnitude root of a polynomial
Suppose that we have a polynomial with real coefficients like
$P(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$.
Is there a way to write another polynomial whose largest magnitude root is equal to the ...
- 189
0
votes
0
answers
50
views
Prove that an equation has only one complex solution
I need to prove that the equation $z - \varepsilon \sin z = a$ has only one solution for small $\varepsilon$. The problem doesn't state anything specifically but I assume that $a$ is an arbitrary ...
- 599
2
votes
2
answers
138
views
If equation $f(x)=0$, and it has roots $α$ of degree m, then prove equation $h(x)=f(x+f(x))-f(x)=0$ has roots of degree $2m-1$.
If equation $f(x)=0$, and it has roots $\alpha$ of degree $m$, then prove equation $h(x)=f(x+f(x))-f(x)=0$ has roots of degree $2m-1$.
When $\alpha$ is root of degree $m$, that means
$$
f(\alpha)=f'(...
- 23
0
votes
0
answers
67
views
sum over $\frac{x^{k^2}}{k!}$ from $k=0$ to $\infty$
One question that I recently am the following:
suppose I got the series
$
F(x)=\sum_{k=0}^{\infty}\frac{x^{ak^2}}{k!}
$
where $a \in \mathbb{C}.$ I used Mathematica and check that this $F$ can not be ...
- 11
2
votes
1
answer
56
views
$\forall x\in \mathbb{R}, |f'(x)| \leqslant C |f(x)|$ implies $f=0$
Question
Let $f\in \mathcal{C}^1$ a function such that $f(0)=0$ and $\forall x\in \mathbb{R}, |f'(x)| \leqslant C |f(x)|$.
Prove that $f=0$
My attempt
If there exists $a$ such that $f(a)\neq 0$, ...
- 2,341
2
votes
1
answer
81
views
How do I determine further solutions of the equation using Rolle's theorem?
I gave this equation $2^x=1+x^2$ with the $1$st zero is $x_1=0$ and the $2$nd zero is $x_2=1$. (easy reading)
Now I want to calculate more zeros using Rolle's theorem, and I've rearranged the function ...
- 23
1
vote
0
answers
227
views
Why does inverse quadratic interpolation converge quicker than inverse linear interpolation
I have used linear inverse interpolation and quadratic inverse interpolation to estimate the root of a function. I found that my linear interpolation procedure required 9 iterations to achieve ...
- 183
2
votes
1
answer
212
views
Does $\exp(-x) \sin(x)$ have infinite oscillations AND an infinite number of real roots?
A Course of Pure Mathematics by G. H. Hardy states that a function oscillates if it does NOT tend to a limit or to $± \infty$. I find this definition difficult to square with $e^{-x}\sin(x)$. It has ...
0
votes
2
answers
153
views
What is the non-trivial solution to $0 = (x-1) \ln x - \ln 2$?
Consider the following logarithmic equation in $x$:
$$
0 = (x-1) \ln x - \ln 2
$$
It is easy to see the trivial solution $x = 2$. There is a non-trivial solution $x \approx 0.346...$. Is it possible ...
- 108
-1
votes
2
answers
72
views
Prove that $x/9 + (\sin x)/8=\cos^2(x/3)+\cos^2x+1$ has a positive solution. [closed]
I'm trying to solve this problem but after several attempts I wasn't able to prove it. I know there is a point where the intermediate value theorem should be used (that's the topic of the unit) but I ...
- 43
3
votes
2
answers
124
views
How to show the following inequality $_2F_1\left(5.5, 1, 5;-x^2\right)>0$?
Consider the function $_2F_1\left(5.5, 1, 5;-|x|^2\right)$ for $x\in \mathbb{R}^n.$ I want to show that this function is positive. I checked that it does not have any roots so can I conclude the ...
- 9,326
0
votes
1
answer
102
views
Classifying the roots of polynomials with integer coefficients
There is no generalized version of the quadratic formula to find the zeroes of polynomials with integer coefficients of degree $n>4$. I am curious about the forms that these zeroes take. More ...
- 1,980
2
votes
1
answer
58
views
Finite intersection of nonempty sets is not empty (Poincaré-Miranda theorem)
I am trying to understand a proof of the Poincaré-Miranda theorem given in this article.
So, the theorem states that if we have a continuous map $f = (f_1, \ldots, f_n) : I^n \rightarrow {\rm I\!R}^n$ ...
- 509
0
votes
0
answers
95
views
Function with given properties
Consider the following function:
$$g(x)=\sin²(πx)+f(x)$$
We have to construct a "non trivial" function $f(x)$ such that $g(x)$ doesn't have any (real or complex) roots in the region $\{x|Re(...
- 916
1
vote
1
answer
43
views
derivative $f(z)=\sqrt [n] {g(z)}$ and $f(z)=\log(g(z))$
How I can compute the derivative at a point of function $f(z)=\sqrt [n] {g(z)}$ and $f(z)=\log(g(z))$ if I know $f(z_0)$ for a fixed $z_0$ and if I have a suitable plane cut?
Is it true that we can ...
user720634
-2
votes
3
answers
46
views
a (bigger than 1) is always bigger than its roots [closed]
how can I formally proof that for a > 1: $$ a> \sqrt a > \sqrt[3]a > \sqrt[4]a ... $$?
Can someone help? ;)
- 393
0
votes
4
answers
320
views
What is the limit of $(3n^2 + n)^{1/n}$ and why? [duplicate]
I am really unsure of how to answer this as I have tried taking out n and using the ratio lemma and am still confused!
user843427
0
votes
0
answers
56
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Root finding with integral constraints - how to classify problem
I have a question about root finding.
The output of the algorithm is a variable $b\in \mathbb{R}$. The equation is of the form $f : \mathbb{R} \to \mathbb{R}$
$$f(b) = c + \int_{g(x)\leq b} h(x) \, dx ...
- 95
0
votes
3
answers
79
views
Proof that there is an $x \in R^+$ such that $\sqrt{x}+\ln(x)=0$
So I was trying to solve this problem and I wasn't able to complete the proof . To translate the problem, let $f(x) = \sqrt{x}+\ln(x)$. Using the Intermediate Value Theorem , if we assume there is a $...
user685013
3
votes
2
answers
773
views
elementary proof that sum of algebraic numbers is algebraic(Real Analysis and Foundations)
How can one prove that the sum of two algebraic numbers is an algebraic number without using theories of algebra? I don't know about (abstract) algebra. Actually, this is an exercise of an analysis ...
- 731
1
vote
1
answer
93
views
Roots of the derivative of a nice polynomial function (with explicit roots)
Let $n\in\mathbb{N}$ and consider the polynomial function $f:\mathbb{R}\to\mathbb{R}$ given by $$
f(x):=x^2(x^2-a_1^2)^2(x^2-a_2^2)^2...(x^2-a_n^2)^2,
$$
where $a_1,...,a_n\in\mathbb{R}$, all ...
- 691
0
votes
2
answers
27
views
Proving equation has solution for every $c ≥ 0$
Task: Proof that the equation $x^5 − x = c$ has a solution for every $c \ge 0$ in the interval $[0, \infty)$.
No idea where to start, anyone have any suggestions?
Kind regards
Anthony
- 1
1
vote
1
answer
45
views
Finding zeros for $2a + 4bx^2 - qcx^{q-2} $
Does anyone know a free app or site that I can use to solve the following equation:
$ 2a+4bx^2 -q c x^{q-2}=0 $, where, $a, b, c\in \mathbb{R}$, $a, b, c>0$ and $q>2$.
I would appreciate any ...
- 539
0
votes
2
answers
63
views
Show that $x^2 -6 = \csc(x)$ has at least $3$ roots. [closed]
I need to show that $$x^2 -6 =\csc(x)$$ has at least $3$ roots.
How can I show that?
I tried by using mean value theorem...
0
votes
2
answers
213
views
Interval in which a function's roots lie.
I have this problem:
The root of the function $f(x)=\cos(x) -x +2 $, lies in
a. [0,2]
b. [1,2]
c. [-1,1]
3
votes
2
answers
222
views
Please check my work! Question about cubic polynomials
I need some help with this problem. Here is the link. Can you please tell me if there is an easier way to show that cubic polynomials have a real root? The question is in an analysis book from the ...
- 43
0
votes
1
answer
31
views
Growth of the given complex valued function
Question: Let $F(z)=z - \frac{p(z)}{p'(z)}$, where
$$p(z)= (z-\alpha)^k \prod_{j=1}^{s} (z-\alpha_j)~.$$
Show that $|z-\alpha| < \epsilon$ implies
$$|F(z) - \alpha| < |z-\alpha|~.$$
Here $\...
user741114
0
votes
1
answer
80
views
Zeros of a function after a Gaussian convolution
Given an analytic real function $f$ with $n$ zeros, has its Gaussian convolution at most $n$ zeros?
- 3
2
votes
1
answer
2k
views
Roots of $e^{(z^{2})}=1$
I’m looking for complex roots of $e^{(z^{2})}=1$ on the inside of the circle |z|=3.
I have found $0, \sqrt{\pi}\cdot \{1+i,1-i,-1+i,-1-i\}$. Is that it, wolfram alpha doesn’t help:/
Thank you
- 310
0
votes
3
answers
212
views
Behaviour of roots of family of polynomials
Let
$$
P(x)=(1+\frac{w^3}{12})x^4 + w^2 \sqrt{u}x^3+(wu-\frac{2}{u}(1+\frac{w^3}{12}) )x^2-\frac{w^2}{\sqrt{u}}x+\frac{1}{u^2}(1+\frac{w^3}{12})
$$
a polynomial in the variable $x$, where $w,u>0$...
- 311
0
votes
3
answers
106
views
Number of zeros: Do we use the definition of the derivative?
We have that a function $f:\mathbb{R}\to \mathbb{R}$ is twice differentiable in $\mathbb{R}$ and it has infinitely many roots.
To show that the equation $f''(x)=0$ has infinitely many roots, do we ...
- 14.1k
1
vote
1
answer
151
views
Trying to find roots of $\arctan(2(x − 1)) − \ln |x| = 0$ analytically
I have found the roots graphically and also numerically.
Apparently there is also a calculus root to finding them analytically.
I was thinking to use the derivative but it doesn't seem to work.
...
- 21