Notes

Complex Equation Solutions

Count of Solutions

The equation $z^n=w$ for $z,w\in \mathbb{C}$ always has a solution > algebraic closure of $\mathbb{C}$.

• For $w=0$, it has exactly 1 unique solution, with multiplicity $n$
• For $n=0$ it gives $1=w$ so only 1 distinct as well

For the rest, the n-gon around the circle describes all $n$ solutions (n-th root), around a circle of radius $|w{|}^{1/n}$.

Limit Proofs

The Limit $l$ trick

First state the existance of the limit.
We need to clearly state that the function is continuous, which is why we can pull it in and then still compare the two limits.

Delta-Epsilon

Uniform Continuity

The delta has to depend on the epsilon.
Simply use $\delta :=\frac{ϵ}{L}$ when working with Lipschitz continuity!