The bookUnit 2 · Zeros and Mappings36 / 83

Argument principle, Rouché's theorem, open mapping

Why this is asked: Rouché is a counting tool: split the polynomial into a dominant term and the rest, verify the strict inequality on the circle, and read off the zero count. Watch for the extra factor when the integrand is (zf)′/(zf).

Argument principle, Rouché and zero counting

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See it move

Rouché in action: counting the zeros of z⁵ + 3z + 1interactivefree

Step 1 / 5The question

How many zeros does $p (z) = z^{5} + 3 z + 1$ have inside $∣ z ∣ < 1$ ? And inside $∣ z ∣ < 2$ ?

No formula finds these roots — but Rouché's theorem counts them without finding them.

The trap here

“f′(z) ≠ 0 everywhere implies f is injective” — false

f(z) = eᶻ on $\mathbb{C}$

f′ = eᶻ never vanishes, yet $f(z) = f(z + 2\pi{}i)$ . Non-vanishing derivative gives only local injectivity.