Articles of complex analysis

Characterizing complex solutions of $\frac{1}{z_1}+\frac{1}{z_2}=\frac{1}{z_1+z_2}$

What I have done so far isn’t so helpful. If $z_1=x_1+y_1i$ and $z_2=x_2+y_2i$ then these two things must hold: $$x_1^2 -y_1^2 +x_2^2-y_2^2+x_1x_2-y_1y_2=0$$ $$2x_1y_1+2x_2y_2+x_1y_2+x_2y_1=0$$ This also means that $$\Re(z_1z_2)=y_1^2+y_2^2-(x_1^2+x_2^2)$$ $$\Im(z_1z_2)=-2(x_2y_2+x_1y_1)$$ But again, this isn’t very helpful. Are there any simpler solutions that don’t involve this nasty algebra?

Looking for an example of a rationally indifferent cycle.

Let $p \in \mathbb {c} $ be a period point of a rational function $R$ i.e. it is a fixed point for an Iterate $R^n $. If $(R^n)'(p)=e^{2\pi i t} $ for some $t \in \mathbb {Q} $ then p is called rational indifferent. For a minimal $n $, $(p,R(p),R^2 (p),…,R^{n-1}(p))$ is then called rational […]

Does there exist a $C^1$-path path-homotopic to a rectifiable curve?

Related: Let $G$ be an open connected subset of $\mathbb{C}$. Let $\gamma:[0,1]\rightarrow G$ be a rectifiable curve. Then, does there exist a $C^1$-curve $\Gamma:[0,1]\rightarrow G$ such that $\gamma$ and $\Gamma$ are homotopic relative to $\{0,1\}$ in $G$?

Let $f\in Hol(\Bbb{C}\setminus \Bbb{R})\cap C(\Bbb{C})$ then $\int_{\gamma}f(z)dz=0$ for all closed curve $\gamma$

For $f\in Hol(\Bbb{C}\setminus \Bbb{R})\cap C(\Bbb{C})$ show that $\int_{\gamma}f(z)dz=0$ for all closed curve $\gamma$ piecewise-continuous. I am completely clueless but this is primarily because I feel hopeless regarding the winding numbers. Can you give me an advice/guidance?

Evaluating a trigonometric integral using residues

Finding the trigonometric integral using the method for residues: $$\int_0^{2\pi} \frac{d\theta}{ a^2\sin^2 \theta + b^2\cos^2 \theta} = \frac{2\pi}{ab}$$ where $a, b > 0$. I can’t seem to factor this question I got up to $4/i (z) / ((b^2)(z^2 + 1)^2 – a^2(z^2 – 1)^2 $ I think I should be pulling out $a^2$ and $b^2$ […]

Is there an holomorphic function? If that function exists, is it unique?

I am solving a problem that asked me if exist an holomorphic function which satisfies only two condition. both equally: $$ f\left(\frac{1}{\alpha n} \right)=0\ \ \ \ and\ \ \ f\left( \frac{1}{\alpha n+1}\right)=1$$ where $\alpha, n \in \mathbb{N}$ where $\alpha$ is unique I think that I have to use Identity theorem for holomorphic function. Someone […]

separate vs joint real analyticity

Let $$f(x,y) := xy\exp\left(-\frac{1}{x^2+y^2}\right),$$ if $(x,y)\neq (0,0)$ and $f(0,0):=0$. I read the claim that $f$ is (a) separately real analytic on $\mathbb{R}\times\mathbb{R}$ (i.e. for each fixed $y$ the map $x\mapsto f(x,y)$ is real analytic and for each fixed $x$ the map $y\mapsto f(x,y)$ is real analytic), (b) $C^\infty$ on $\mathbb{R}\times\mathbb{R}$, (c) not jointly real analytic […]

Showing that a Möbius transformation exists

I’m trying to show that a specific Möbius transformation exists, where I have some points that map to some other points. (I don’t wanna be too specific here about what goes where, as I don’t wanna run the risk of having the insight in solving this myself spoiled). How does one go about proving the […]

Finding $z\in \mathbb C$ with $\sin z = 2$?

I’m trying to find all $z\in \mathbb C$ with $\sin z = 2$ but I’m stuck. I tried writing $\sin z = {e^{iz} – e^{-iz}\over 2i} = 2$ and then writing $z = x + iy$ but the expression I got was $e^{ix}e^{-y} – e^{-ix}e^{y}=4i$ and I don’t see how it’s possible to solve this […]

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\mathbb{C}:\quad F_n\restriction_\mathbb{R}=f_n$$ Does their uniform limit have an analytic continuation, too? $$F:\Omega\to\mathbb{C}:\quad F\restriction_\mathbb{R}=f\quad(f_n\stackrel{\infty}{\to}f)$$ (By uniform boundedness this seems very likely; but really?) Application An almost modular state is modular: $$A\in\mathcal{A}^\omega:\quad\omega(\sigma^t[A]B)=\omega(B\sigma^{t+i\beta}[A])\quad(B\in\mathcal{A})$$ (Supposed that entire elements are […]