2
votes
0answers
30 views

Tractable indefinite integral of the exponentiation of some function

Consider the function $z(s)\in\mathbb{C}$ defined as $z(s)=\int_0^s \exp\left[i(q u+\lambda(u))\right]du$ for some $q\in\mathbb{Q}-\mathbb{Z}$ and $\lambda(s)$ a $2\pi$-periodic real differentiable ...
0
votes
0answers
18 views

Problem with calculating winding number in sum of curves

Let $$ \begin{align} \gamma= \gamma_1 +\gamma_2+\gamma_3,\\ \gamma_1(t)=e^{it}, t\in[0,2\pi] \\ \gamma_2(t)=-1+2e^{-2it}, t\in [0,2\pi]\\ \gamma_3(t)=1-i+e^{it},t\in [\frac{\pi}{2},\frac{9\pi}{2}] ...
0
votes
1answer
25 views

winding number of $\gamma$ and point exterior to $\gamma$

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$ Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$
1
vote
1answer
21 views

Problem in showing that contours $\gamma_2$ is equivalent to $ \gamma $

Let $\gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}]$ and $\gamma(t)=e^{2\pi it}, t\in[0,1]$ Show that $\gamma_2 \sim \gamma $. I think that for the latter to be true $\gamma_2$ should be ...
0
votes
1answer
38 views

Does $\gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}]$ represent the unit circle?

$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$ Is my thinking correct that $\gamma$ represents the unit circle correct?
0
votes
1answer
39 views

Image of a parametrised curve and its geometrical meaning

$\gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1]$ which is the Image of this curve and / or what is the geometrical description of the set of points $ z : z=\gamma_1(t)$ I am helping a friend study ...
2
votes
2answers
61 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
36 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
57 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
313 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
54 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
269 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
166 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
930 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
552 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 ...