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Complex numbers › Roots of unity and complex roots › Counting complex solutions

Analytic solutions on four quadrants

University · ●●○○○ 20 pts

How many analytic functions f:Ω→Cf:\Omega\to\mathbb{C} satisfy f(z)2+3i f(z)+4=0f(z)^2+3i\,f(z)+4=0 for all z∈Ωz\in\Omega, where Ω\Omega is the complex plane with the two coordinate axes removed? Give the number of such functions.

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quadratic equationscomplex analysis