# Tagged Questions

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### 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 ...
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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}] ... 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} $$Is it safe to say that n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^* 1answer 21 views ### Problem in showing that contours \gamma_2 is equivalent to  \gamma  Let \gamma_2(t)= e^{-it^2}, t\in[0,\sqrt{2\pi}] and \gamma(t)=e^{2\pi it}, t\in[0,1] Show that \gamma_2 \sim \gamma . I think that for the latter to be true \gamma_2 should be ... 1answer 38 views ### Does \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] represent the unit circle?$$ \begin{align} \gamma(t)=e^{it^2} , t \in [0,\sqrt{2\pi}] \end{align} $$Is my thinking correct that \gamma represents the unit circle correct? 1answer 39 views ### Image of a parametrised curve and its geometrical meaning \gamma_1(t)= t^2 + i\, t^4 , t\in [ -1, 1] which is the Image of this curve and / or what is the geometrical description of the set of points  z : z=\gamma_1(t) I am helping a friend study ... 2answers 61 views ### Finding singularities of a projective curve For w \in \mathbb{C} we define the projective curve$$p(x,y,z):= x^3+y^3+z^3+wxyz.$$Now I have to find all w \in \mathbb{C} for which the projective curve p(x,y,z) is singular and show that ... 0answers 36 views ### Singular point of a plane curve: the geometrical meaning Consider a plane curve C\subset\mathbb C^2 where$$C=\{(z,w)\in\mathbb C^2\,:\, P(z,w)=0\}$$A singular point of C is a point (z_0,w_0) such that \frac{\partial P}{\partial ... 2answers 57 views ### Rotations around the origin The problem is to find the number of rotations around the origin for the function$$f(z)=z^{2013}+2z+1 $$when z moves through \left\{|z|=1\right\}. I tried to solve it with the help of argument ... 0answers 52 views ### Evaluate \|f\| on complex domain Given a function  f\colon \mathbb{C} \rightarrow \mathbb{C}, how do I find ||f||_{\operatorname{Trace}(\gamma)}, where \gamma is a curve in the complex plane? In particular, I need to do this ... 2answers 313 views ### General way to find out whether a curve is positively oriented I have a general question, that deals with the question how I am able to find out whether a particular curve(in \mathbb{C}) is positively oriented? Take e.g.  y(t)=a+re^{it}. Obviously this one ... 1answer 54 views ### Finding a curve with a condition on winding numbers I want to find a continuous and closed curve \gamma so that the map \nu_{\gamma}:\mathbb{C}\Im(\gamma)\to \mathbb{Z} takes infintely many values. Here \nu_{\gamma}(a) is the winding number ... 1answer 269 views ### Problem with calculating a winding number I have a problem with calculating the winding number n\left ( \gamma ,\frac{1}{3} \right ) of the curve \gamma :\left [ 0,2\pi \right ]\rightarrow \mathbb{C}, t \mapsto \sin(2t)+i\sin(3t). ... 2answers 166 views ### A problem on Residue Theorem Today I had a problem in my test which said Calculate \int_C \dfrac{z}{z^2 + 1} where C is circle |z+\dfrac{1}{z}|= 2. Now, clearly this was a misprint since C is not a circle. I tried to find ... 1answer 930 views ### Fractal behavior along the boundary of convergence? The complex power series$$\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, ...
It asks me to graph the region in the complex plane. $Re(z+iz) \le 1$