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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4
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1answer
169 views

Roots of some modified Bernoulli polynomials

Update The polynomials are generated as follows: Where $B_n(x) = \sum_{k=0}^n {n \choose k} b_{n-k} x^k$ is used to generate standard Bernoulli polynomials, top plot is generated as follows: ...
0
votes
5answers
337 views

Why doesn't $1/x=0$ have any solution?

Just out for curiosity ! Why $1/x=0$ doesn't have any solution? Or is it that the solution takes you to $1=0$ situation which would nullify mathematical principle that we stood for years Educate ...
3
votes
1answer
139 views

$n \approxeq k + 2^{2^k}(k+1)$. How can one get the value of $k(n)$ from this equation?

I am trying to find approximation for this sum. Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$ Doing following way. Let $a_k(n) = \frac{n\choose k}{2^{2^k}}$. I tried to ...
2
votes
0answers
72 views

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?

For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...
1
vote
4answers
348 views

Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots.

Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots. Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.
1
vote
1answer
40 views

Is it possible to solve this nonlinear equation analytically?

Is it possible to solve the following equation analytically? $B_1\exp(\beta_1 x) + B_2\exp(\beta_2 x) = C_1\exp(\alpha_1 x) + C_2\exp(\alpha_2 x)$ where, $B_1$, $B_2$, $C_1$, $C_2$, $\beta_1$, ...
1
vote
2answers
80 views

Let $a,b,c \in \mathbb{R^+}$, does this inequality holds $\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$?

Does the following statement/inequality holds for $a,b,c \in \mathbb{R^+}$? $$\frac{a}{na + kb} + \frac{b}{nb+kc} + \frac{c}{nc + ka} \ge \frac{3}{k+n}$$ I've been thinking for hours and I ...
0
votes
2answers
66 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
1answer
45 views

At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
0
votes
2answers
63 views

roots of polynomial equation

How to find the roots of $x^5-2^5$ by hand. I see that we get a root of $x=2$ and 4 complex roots (should come in pairs). Not sure how to work out the complex roots. Do we need to convert to polar? ...
1
vote
3answers
61 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 ...
0
votes
0answers
59 views

Finding Roots of 2 Variable Inequalities

If I happen to have a two variable inequality such as $x+y<xy$ what is the most efficient way of finding out the critical points/roots since I cannot plot 3d functions in my head. For example, in ...
0
votes
3answers
90 views

What are the methods to find approximatly the 5th roots of an equations

By which method, can I find the nearest root of : $x^5−2x+1.1=0$ ? Thank you.
1
vote
3answers
250 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 ...
5
votes
2answers
179 views

Find asymptotics of $x(n)$, if $n = x^{x!}$

Find the asymptotic for $x(n)$, if $n = x^{x!}$. I've tried 1) to take a logarithm: $x! \log{x} = \log{n}$. 2) to find $n'(x)$, using gamma-function for factorial $\Gamma(z) = \int_0^\infty ...
2
votes
2answers
48 views

Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+…+x^{2008}\right)=2010x^{2009}$

Solve for $x$ $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$ solution should be by hand
6
votes
3answers
222 views

Solve $\lfloor{x}\rfloor$+$\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$

Solve for $x$ $$\lfloor{x}\rfloor+\lfloor2x\rfloor+\lfloor4x\rfloor+\lfloor16x\rfloor+\lfloor32x\rfloor=12345$$ I tried to put $x$=$I$+$f$ where $I$ is integer part and $f$ is fractional part ...
1
vote
3answers
143 views

How to solve $e^{ax}+e^{bx}+e^{cx}+d=0$

How to solve an equation like $e^{ax}+e^{bx}+e^{cx}+d=0$ (i.e. to write $x=...$) where $a,b,c,d$ are fixed non-zero real numbers. I have tried assuming that $x=ln(y)$ for $y>0$ but it goes ...
4
votes
1answer
103 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
146 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. ...
1
vote
2answers
50 views

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ …

Problem : If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ then the value of $(1-\alpha_1)(1-\alpha_2)(1-\alpha_3)(1-\alpha_4)$ is (a) 2$\sqrt{3}$ ...
2
votes
1answer
104 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} + ...
5
votes
1answer
2k views

How to get the roots of a quartic function when given a quadratic factor

We have the function $$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$ $x^2 -x - 6$ is a factor of this function. Find all the roots of the polynomial. So we have $(x-3)(x+2)$, and since it is a quartic we ...
2
votes
4answers
150 views

Find number of solutions of $2^x$+$3^x$+$4^x$=$5^x$

Find number of solutions of $$2^x+3^x+4^x=5^x$$ I tried using graphs but don't know how to draw graph of L.H.S.
0
votes
1answer
857 views

Calculation of Chebyshev coefficients

The Chebyshev polynomials can be defined recursively as: $T_0(x)=1$; $T_1(x)=x$; $T_{n+1}(x)=2xT_n(x) + T_{n-1}(x)$ The coefficients of these polynomails for a function, $\space f(x)$, under ...
4
votes
1answer
79 views

What's wrong with my conjecture?

I was doing math homework, and I formulated the following conjecture from one of the questions: If $f(x)$, $g(x)$ and $h(x)$ are continuous functions and the equations $f(x) = h(x)$ and $g(x) = h(x)$ ...
1
vote
0answers
91 views

Analytical solution(root) for a tenth order polynomial?

is it possible to develop an analytical solution (root) for such a polynomial: $f(x)=\left(x^{10}-c_1^2\right)*\left(c_2-x\right)^2-0.2*\left(x^2-1\right)*c_1^2$ with $c_1$ and $c_2 >0$. Numerical ...
4
votes
1answer
98 views

Bernoulli polynomial root symmetry

New @ Antonio Vargas - Many thanks - feeling a little foolish! Old Can anyone point me in the direction of anything that might explain the sudden change in near-symmetrical complex roots of ...
4
votes
4answers
123 views

Comment upon nature of the roots

How many roots are there of the following polynomial? How many are real, and how many are complex? ...
1
vote
0answers
43 views

what are the possible solutions to this equation?

I'm trying to find some angles for my characteristic equation , I need to know the roots or possible answers to cosine equation $$1-\cos(u)\cdot\cosh(w)=0,$$ and $$u=\sqrt{\lambda_1}\cdot L.$$ ...
1
vote
1answer
40 views

Plot implicit equation

I'm working with a frequency-response curve of a nonlinear oscillator and came across the following equation (Kovacic & Brennan 2011, p. 179): $$ A^2 = \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - ...
3
votes
0answers
64 views

Roots of the derivative as symmetric (?) functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...
4
votes
2answers
915 views

Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros

I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
0
votes
1answer
93 views

Finding root between two points

The function $f:[0,1]\to \mathbb{R}$ is continuous, $f(0)<0$, $f(1)>0$ and there is one root in between. Using $f(0)$ and $f(1)$, the expression $\frac{1\cdot f(0)-0\cdot f(1)}{f(0)-f(1)}$ would ...
4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
1
vote
1answer
60 views

Problem in harmonic series.

If the sum of roots of the quadratic equation $ax^2 +bx+c=0$ { $(a,b,c)$ not equal to zero} is equal to sum of squares of their reciprocals , then $a/c$ , $b/a$, $c/b$ are in? Actually the ...
0
votes
3answers
248 views

$\log x =Cx^4$ has only one root. Find C

$\log x =Cx^4$ has only one root. Find C. I don't know how to solve this problem. Do you take derivative on both sides? I am thinking C equals 0. Am I correct on that?
3
votes
3answers
85 views

Number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$?

The number of irrational roots of the equation $(x-1)(x-2)(3x-2)(3x+1)=21$ is (A)0 (B)2 (C)3 (d)4 Actually im a 10 class student i don't know any of it,but my elder brother(IIT Coaching) cannot ...
2
votes
3answers
67 views

Roots of the equation?

If p,q.r are real numbers satisfying the condition $p + q + r =0$, then the roots of the quadratic equation $3px^2 +5qx +7r=0$ are (A)Positive (B)Negative (C)Real and distinct (d)Imaginary ...
6
votes
2answers
253 views

Can we prove that all zeros of entire function cos(x) are real from the Taylor series expansion of cos(x)?

Q1: Can we prove that all zeros of cos(x) are real from the following Taylor series expansion of cos(x)? $$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^k}{(2k)!}x^{2k} $$ The Riemann $\xi(z)$ function is ...
0
votes
3answers
65 views

Find the roots of 2 equations

Show that the equation $e^{-x} = x^2$ has a root between $x=0.70$ and $x=0.71$. I think you have to use natural logs to get rid of the $e$ however after that, i'm not sure how to solve for $x$
1
vote
0answers
101 views

Broyden's Method Failing to Converge

I have a matrix equation $$ \textrm{det}(\mathbb{M}) - \textrm{tr}(\mathbb{M}) + 1 = 0 $$ where $\mathbb{M}(z)$ is a matrix function of a complex number $z$ that I want to solve for. Because I have ...
0
votes
0answers
59 views

Finding the roots of this function

I have the following special function. $$f(x) = \sum _{i=1}^n \left\{\frac{(x - z_i)_+^2}{1+ 2*z_i+(x - z_i)_+^2}\right\} - \left\{(\frac{x^3}{3} - \frac{(x - z_i)_+^3}{3})\right\} $$ which ...
0
votes
2answers
116 views

Simplifying equation into Newton Raphson form

Given the equation $\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$: a. Simplify the integral using Gauss method with 3 points. b. Solve given equation by Newton Raphson iterative ...
0
votes
1answer
93 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 ...
1
vote
3answers
3k views

cubic equations which have exactly one real root

Question is to check : For any real number $c$, the polynomial $x^3+x+c$ has exactly one real root . the way in which i have proceeded is : let $a$ be one real root for $x^3+x+c$ i.e., we have ...
11
votes
1answer
210 views

Solving $x^2+bx^{1+\varepsilon}+c =0$

Let $x \in \mathbb{R}$. Is it possible to find the roots of $x^2+bx^{1+\varepsilon}+c =0$ where $b,c \in \mathbb{R}$ and $\varepsilon$ is small. I am guessing that an explicit expression might not be ...
3
votes
2answers
129 views

Root of the function $f(x)=xe^x-R$

How can we find the root of the function $f(x)=xe^x - R$ for a general R where $R>=-1/e.$ I don't have any idea as to how to even approach this. Came across this problem during my self-study in ...
3
votes
2answers
87 views

Number of real roots of the equation $2^x = 1+x^2$

Find the number of real roots of the equation $2^x = 1+x^2$ My try: Let we take $f(x) = 2^x-1-x^2$. Now for Drawing Graph of given function, we use Derivative Test. $f'(x) = 2^x \cdot \ln ...
0
votes
2answers
52 views

Number of real roots of $2^x = 1-x^2$ for $x\in (0,1)$

How can I found no. of real roots of $2^x = 1-x^2$ in $x\in (0,1)$ I did not found a method by which i can draw graph of two curve in the interval $x\in (0,1)$ please help me , Thanks Sorry ...