Intereting Posts

Does $\sin(t)$ have the same frequency as $\sin(\sin(t))$?
Inequalities of expressions completely symmetric in their variables
How many normal subgroups?
Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$
Ring of polynomials as a module over symmetric polynomials
Dimensions analysis in Differential equation
Probability of having $k$ similar elements in two subsets.
How was “Number of ways of arranging n chords on a circle with k simple intersections” solved?
Show that $\mathbb{Q}^+/\mathbb{Z}^+$ cannot be decomposed into the direct sum of cyclic groups.
Bridging any “gaps” between AP Calculus and College/Univ level Calculus II
If a linear map $A$ is injective, then then there exists $c$ such that $|Ax|\geq c|x|\;\;\forall x$
Conjecture $\int_0^1\frac{dx}{\sqrtx\,\sqrt{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt{27})$
differentiablility over open intervals
Number of path components of a function space
Why is a function at sharp point not differentiable?

I’m trying to use Rouché’s theorem to find the number of roots of the function $f(z)=z^5+3z^4+9z^3+10$ on $\vert z\vert<2$. I know that the answer is $3$, but I have been unsuccessful in proving it. I’ve tried nearly every combination of the components of $f$ which also have $3$ roots, applying the theorem up to $3$ times, but to no avail. I’ve even resorted to trying unrelated polynomials, but there’s a lot of those! A hint would surely benefit my sanity.

- Solve $\sin(z) = z$ in complex numbers
- Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
- How to express $z^8 − 1$ as the product of two linear factors and three quadratic factors
- Finding the Laurent series of $f(z)=1/((z-1)(z-2))$
- Is this function holomorphic at 0?
- Conformal map from a disk onto a disk with a slit
- Show that the range of $p(z)e^{q(z)}$ for nonconstant polynomials $p$ and $q$ is all of $\mathbb{C}$.
- What are some general strategies to build measure preserving real-analytic diffeomorphisms?
- $f$ is entire without any zeros then there is an entire function $g$ such that $f=e^g$
- Integral evaluation $\int_{-\infty}^{\infty}\frac{\cos (ax)}{\pi (1+x^2)}dx$

Taking the geometric sequence $1,3,9$ in the leading coefficients into account, multiply with $(z-3)$ to get

$$

g(z)=(z-3)f(z)=z^6-27z^3+10z-30

$$

which has the same number of roots inside the disk $|z|<2$. Now

$$

g(2w)=64w^6-216w^3+20w-30

$$

on $|w|=1$ has clearly the third order term as the dominating one.

- Constructing the reciprocal of a segment
- Finding Eigenvectors with repeated Eigenvalues
- Difference between root, zero and solution.
- Prove that the Gaussian Integer's ring is a Euclidean domain
- What is the Euler Totient of Zero?
- Non-zero prime ideals in the ring of all algebraic integers
- Integral equation
- What is a necessary conditions for Urysohn Metrization theorem?
- Is a CW complex, homeomorphic to a regular CW complex?
- Rain droplets falling on a table
- Existence of a sequence of continuous functions convergent pointwise to the indicator function of irrational numbers
- Probability of two people meeting during a certain time.
- Integral involving Gauss Hypergeometric function, power, exponential and Bessel Function
- How to find the smallest $n$ such that $n^a\equiv 1 \pmod p$
- Find the Vector in the New Position Obtained by Rotation