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 polynomial. Define property $B$ for an entire function $f(z)$ as $1)$ $f(z)=f_1(z)$ with property $A$. $2)$ $f_i(z)= ln(f_{i-1}(z)/z)$ for every positive integer […]

Problem Statement:- If the cubic equation $f(x)=0$ has three real roots $\alpha$, $\beta$ and $\gamma$ such that $\alpha\lt\beta\lt\gamma$, show that the equation $$f(x)+2f'(x)+f”(x)=0$$ has a real root between $\alpha$ and $\gamma$. Attempt at a solution:- If $f(x)$ is a quadratic equation whose roots are $\alpha$, $\beta$ and $\gamma$, then $$f(x)=(x-\alpha)(x-\beta)(x-\gamma)$$ From this we get $f'(x)$ […]

If $x^5+ax^4+ bx^3+cx^2+dx+e$ where $a,b,c,d,e \in {\bf R}$ and $2a^2< 5b$ then the polynomial has at least one non-real root. We have $-a = x_1 + \dots + x_5$ and $b = x_1 x_2 + x_1 x_3 + \dots + x_4 x_5$.

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.

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 $$g(z) = z^3-2z-3+e^{-z^{2}}$$ in the region A. Comments: I think this is a fairly difficult problem. I assume that you have […]

I’m trying to solve a problem, but it involves finding the exact roots of the octic polynomial $$x^8+4x^7-10x^6-54x^5+9x^4+226x^3+125x^2-301x-269$$ How can I find the roots of an octic? Wolfram Alpha just gives me the rounded values. Not the exact ones.

I am wondering whether one could find Riemann zeta zeros iteratively by using relationships such as this one: $$\rho _1=\lim_{s\to 1} \, \frac{\zeta (s) \zeta \left(s \cdot \rho _1\right)}{\zeta ‘(\rho _1)}$$ where $\rho _1 \approx 0.5 + 14.1347i$ is the first zeta zero. Are there better relationships like Newton iteration or something like that? The […]

Let $x^3-(m+n+1)x^2+(m+n-3+mn)x-(m-1)(n-1)=0$, be a cubic polynomial with positive roots, where $m,n \ge2$ are natural nos. For fixed $m+n$, say $15$, it turns out that least root of the polynomial will be smallest in case of $m=2,n=13$ i.e the case in which difference is largest. I checked it for many values of $m+n$ and same thing […]

When solving for roots to a cubic equation, the sign of the $\Delta$ tells us when there will be 3 distinct real roots (as long as the first terms coefficient, $a$, is non-zero.) Namely when $\Delta$ is positive. The equations to find the 3 roots are: $x_1 = -\frac{1}{3a}(b + C + \frac{\Delta_0}{C})$ $x_2 = […]

Consider the recurrence relation: $P_0=1$ $P_1=x$ $P_n(x)=xP_{n-1}-P_{n-2}$; 1). What is closed form of $P_n$? 2). Let $k,j.\in\{1,2,…,n+1\}$ and $Q_k(x)=xP_{2n+1}-P_{k-1}.P_{2n+1-k}$. Then prove that largest root of $Q_k$ is greater than that of $Q_j$ for $k>j$. Another observation is taking $k,j$ to be even and $k>j$, smallest positive root of $Q_k$ is greater than that of $Q_j$. […]

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