Articles of complex analysis

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 […]

Coefficients of Quotient of two power series

So I have two functions $f$ and $g$ which are holomorphic on some disk $D(a,r)$ and such that $g$ is never zero on that disk. Where we can represent the two function as power series given by $f(z)=\sum\limits_{n=0}^\infty a_n(z-a)^n$ and $g(z)=\sum\limits_{m=0}^\infty b_m(z-a)^m$ Let $f_N=\sum\limits_{n=0}^N a_n(z-a)^n$ and $g_N=\sum\limits_{m=0}^N b_m(z-a)^m$ then by carrying out the long division […]

Automorphisms in unit disk

Let $\mathbb{D}=\{|z|<1,\ z\in\mathbb{C}\}$. Are there any other automorphisms in $\mathbb{D}$ except the Blaschke factor $\displaystyle B_{a}(z)=\frac{z-a}{1-\overline{a}z},\ a\in\mathbb{D}$? I denote with $\overline{a}$ the complex conjugate of $a$. Thank you for your time, Chris

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t – 1)}}{s(s^2 + 3s + 2)}ds\\ &= \lim_{s\to 0}\frac{2e^{s(t – 1)}}{s^2 + 3s + 2} + \lim_{s\to – 1}\frac{2e^{s(t – 1)}}{s(s + 2)} + \lim_{s\to -2}\frac{2e^{s(t – 1)}}{s(s + 1)}\\ &= 1 – 2e^{-(t-1)} + e^{-2(t […]

Showing that a function is not meromorphic on $\mathbb{C}$

Please, I need some advices to solve this exercise: Let $f$ be meromorphic on $\mathbb{C}$ but no entire. Let $g(z)=e^{f(z)}$. Show that $g$ is not meromorphic on $\mathbb{C}$. I appreciate your help. Thanks.

How to find limit of imaginary recursive sequence?

Let $z_0 = x_0 + i y_0$ be a given complex number. I previously calculated the limit of $$ z_{n+1} = {1\over 2}( z_n + z_n^{-1})$$ for $x_0 <0$ and $x_0>0$ respectively. Now I am trying to do the case $x_0 = 0$ but the previous method does not work in this case and I […]

How to establish $\lim\limits_{r\rightarrow 0}\int_C f(z)dz=i\lambda(\theta_2-\theta_1)$?

I got stuck in it. Please help me. Let $\lim\limits_{z\rightarrow a}(z-a)f(z)=\lambda$ and let $C$ be the arc $\theta_1\leq \theta\leq \theta_2$ of the circle $|z-a|=r$. Prove that $\lim\limits_{r\rightarrow 0}\int_C f(z)dz=i\lambda(\theta_2-\theta_1)$. Can you please tell me how to proceed in this one?

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge’s theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic on an open set $U$ containing $K$. Let $A_f$ denote the set of poles of $f$ in $K$. Then there is a sequence […]

Ring of analytic functions

We know that the ring of analytic function on a connected open set is a Bezout domain. Do we know what happens if we remove the hypothesis of connectedness? It is no longer a domain, but does the ring share the same nice properties like: every finitely generated ideal is principal?

Show that a Lissajous curve has incommesurate frequencies iff it isdense in a rectangle

If we have a Lissajous curve given by the parametric equations $$x(t)=A\sin(\omega_x t)$$ $$y(t)=B\sin(\omega_y t + \delta),$$ how can we show that the curve is dense in the rectangle of sides $A,B$ if and only if $\omega_x$ and $\omega_y$ are incommensurate (i.e. their ratio is irrational)? What I tried: Suppose it is not dense, so […]