Complex numbers › Roots of unity and complex roots

Counting complex solutions

Solve the first problem 16 problems

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1. Analytic solutions on four quadrants ●●○○○ 20 ptsHow many analytic functions $f:\Omega\to\mathbb{C}$ satisfy $f(z)^2+3i\,f(z)+4=0$ for all $z\in\Omega$, where $\Omega$ is the complex plane with the two coordinate axes removed? Give the number of such functions.
2. Complex solutions of a rational equation ●●○○○ 20 ptsFind the number of complex solutions to $\frac{z^3 - 1}{z^2 + z - 2} = 0$.
3. Pairs of unit complex numbers ●●○○○ 20 ptsFind the number of pairs $(z_1,z_2)$ of complex numbers such that $z_1 z_2$ is pure imaginary, $\frac{z_1}{z_2}$ is real, and $|z_1| = |z_2| = 1$.
4. Unit complex numbers with a Re-Im condition ●●●○○ 50 ptsHow many complex numbers $z$ satisfy the following two conditions: $|z|=1$ and $\operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right)$?
5. Gaussian integer pairs with product 10 ●●●○○ 50 ptsCompute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.
6. Imaginary part of a complex root pair ●●●○○ 50 ptsThe complex numbers $z$ and $w$ satisfy $z^{13} = w$ and $w^{11} = z$. The imaginary part of $z$ is $\sin\frac{m\pi}{n}$, for relatively prime positive integers $m$ and $n$ with $m<n$. Find $n$.
7. Counting z with integer parts of f(z) ●●●●○ 100 ptsLet $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 1$. How many complex numbers $z$ are there such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$?
8. Largest finite count of small elements ●●●●○ 100 pts$S$ is a set of complex numbers such that if $u, v \in S$, then $uv \in S$ and $u^{2}+v^{2} \in S$. Suppose that the number $N$ of elements of $S$ with absolute value at most $1$ is finite. What is the largest possible value of $N$?
9. Count complex solutions of an exponential equation ●●●●○ 100 ptsHow many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation $e^z = \frac{z - 1}{z + 1}$?

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