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Questions tagged [rouches-theorem]

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Apply Rouché's theorem to count number of zeros inside disk

Attempt to find the number of zeros of $f(x) = x^5 + 3x - 1$ for $1 < |x| < 2$. I think I need to use Rouché's theorem to solve it. Rouché's theorem states that "suppose $f(z)$ and $g(z)$ are ...
45 views

show $\frac{z}{(1+z)^2}$ is injective in the unit disk

So I need to show that $\frac{z}{(1+z)^2}$ is injective in the unit disk, $\mathbb{D}$. Here is what I have thought so far: If $\frac{z}{(1+z)^2}$ is injective, then $\frac{z}{(1+z)^2} - c$ always ...
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Bound on constant in Polynomial so that zeros are bounded (Rouche)

Statement of the Problem: Let $a\in\mathbb{C}$ be a constant and consider the polynomial $$P(z)=z^{10}+a(z^9+z^8+z^7+z^6+z^5+z^4+z^3+z^2+z+1).$$ Given a radius $\rho>0$, use complex analysis to ...
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