Articles of complex analysis

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 […]

Region of convergence of $\int_0^{\infty} x^s e^{-\frac{|\log(x)|}{2}}dx$ where $s \in \mathbb{C}$

I am interested in finding the region of convergence of the integral \begin{align} \int_0^{\infty} x^s e^{-\frac{|\log(x)|}{2}}dx \end{align} where $s \in \mathbb{C}.$ How do we approach this type of proablem?

Harmonics functions take minimum value on the boundary

Let $u$ be harmonic on the bounded region $A$ and continuous on the closure of $A$. Then $u$ takes its minimum only on $\partial A$ unless $u$ is a constant. How can we prove this result? Do we have to use the Maximum modulus principles? in my notes, the proof only says that considering $v […]

Find all harmonic functions $f:\mathbb{C}\backslash \{0\}\to \mathbb{R}$ that are constant on every circle centered at 0.

“Find all harmonic functions $f:\mathbb{C}\backslash\{0\} \to \mathbb{R}$ that are constant on every circle centered at 0.” This is one of the past qualifying exam problems that I was working on. I was thinking to deal with $\frac{1}{f}$ so that $\frac{1}{f}$ is defined at 0 and use Schwarz lemma or something like that. Any help or […]