5
votes
3answers
117 views

How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$?

If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4 = 0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + ...
3
votes
1answer
53 views

If a function is smooth is 1 over the function also smooth

If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
1
vote
3answers
102 views

Finite number of points inside a disk

Let $n\ge 2$ and suppose that $z_1, z_2, \ldots, z_n$ are distinct points in the interior of some disk $D$ in the plane. Why is it true that there exists a smaller disk $D'\subseteq D$ such ...
7
votes
2answers
2k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
-1
votes
1answer
164 views

Buddhabrot Sewing machine [closed]

The Buddhabrot fractal traces the orbits of the points outside the Mandelbrot set. What design considerations need to be taken into account to create a computerised sewing machine that traces out ...
232
votes
6answers
6k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...