2
votes
2answers
59 views

Finding singularities of a projective curve

For $w \in \mathbb{C}$ we define the projective curve $$p(x,y,z):= x^3+y^3+z^3+wxyz.$$ Now I have to find all $w \in \mathbb{C}$ for which the projective curve $p(x,y,z)$ is singular and show that ...
0
votes
0answers
35 views

Singular point of a plane curve: the geometrical meaning

Consider a plane curve $C\subset\mathbb C^2$ where $$C=\{(z,w)\in\mathbb C^2\,:\, P(z,w)=0\}$$ A singular point of $C$ is a point $(z_0,w_0)$ such that $\frac{\partial P}{\partial ...
4
votes
2answers
55 views

Rotations around the origin

The problem is to find the number of rotations around the origin for the function $$f(z)=z^{2013}+2z+1 $$ when $z$ moves through $\left\{|z|=1\right\}$. I tried to solve it with the help of argument ...
1
vote
0answers
52 views

Evaluate $\|f\|$ on complex domain

Given a function $ f\colon \mathbb{C} \rightarrow \mathbb{C}$, how do I find $||f||_{\operatorname{Trace}(\gamma)}$, where $\gamma$ is a curve in the complex plane? In particular, I need to do this ...
1
vote
2answers
300 views

General way to find out whether a curve is positively oriented

I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one ...
1
vote
1answer
52 views

Finding a curve with a condition on winding numbers

I want to find a continuous and closed curve $\gamma$ so that the map $\nu_{\gamma}:\mathbb{C}$\Im$(\gamma)\to \mathbb{Z}$ takes infintely many values. Here $\nu_{\gamma}(a)$ is the winding number ...
6
votes
1answer
263 views

Problem with calculating a winding number

I have a problem with calculating the winding number $n\left ( \gamma ,\frac{1}{3} \right )$ of the curve $\gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t)$. ...
3
votes
2answers
164 views

A problem on Residue Theorem

Today I had a problem in my test which said Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$. Now, clearly this was a misprint since C is not a circle. I tried to find ...
42
votes
1answer
916 views

Fractal behavior along the boundary of convergence?

The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, ...
0
votes
1answer
71 views

How can I proceed with this exercise?

It asks me to graph the region in the complex plane. $Re(z+iz) \le 1$
3
votes
1answer
551 views

Polynomials, Rouche's theorem and index of vector fields

In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two ...