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complex analysis - $|f^{(N)}(z)| \leq |z|^{-N}$ in the punctured unit circle, then $0$ is a removable singularity.

Take $f$ analytic in $D_1(0)-\{0\}$ such that: $|f^{(N)}(z)| \leq |z|^{-N}$ then $0$ is a removable singularity.I know that I can write $f(z) = \sum_{n = - \infty}^{\infty}c_n z^n$ Moreover, I can tell that the function $g(z) = z^N f^{(N)}(z)$is such that: $|g(z)| \leq 1$. But how do I use this information to deduce that either all $c_n$ such that $n < 0$ are zero or find that $zf(z) \rightarrow 0$ as $z \rightarrow 0$?Thank you!...Read more

complex analysis - Understanding relation between Laurent Series and Singularities

I am thinking about an example, in order to better understand how Laurent Series help us understand the Poles, Zeros and Essential singularities of a complex function.I am trying to find the singularities of $\frac{1}{sin(z)} - \frac{1}{z}$Individually, 1/z has a pole of order 1 at 0For $\frac{1}{z}$ I am quite confused as to what to do. Given for sin(z), it has zeros at $n \pi$ for all integer $n$, so the inverse has poles of order 1 at those points. But what happens at 0, where $\frac{1}{sin(z)} - \frac{1}{z}$ have poles pushing against each ...Read more

complex analysis - Subsequence of Sequence of Conformal maps

Let $\Omega \subset \mathbb{C}$ and take $\{ f_n \}$ to be a sequence of conformal mappings on $\Omega$. Suppose that there is some $z_0 \in \Omega$ such that $f_n(z_0) \to \omega \in \partial \Omega$. I want to prove that there is a subsequence of $\{ f_n \}$ that converges to $\omega$ in $\mathscr{H}(\Omega)$. I know that if the $f_n$ are conformal then the limit in $\mathscr{H}(\Omega)$ is conformal also, but this problem is really hard. We can perhaps use Arzela-Ascoli, since the holomorphic functions are on a bounded set, therefore bounded...Read more

self learning - Complex Log Function

This problem is giving me quite a bit of grief. Heres my issue I do the first problem step by step and get the correct solution. I then try the second problem step by step and get the wrong answer. Specifically I get them to be equal again. Can someone explain why the process taken in question 1 does not work in 2. Thanks for the help. textbook solution...Read more

self learning - Complex Conjugation Proof

Hello the question is: 1. Prove (z*)^n = (z^n)* where * represents the complex conjugate.This is my proof https://imgur.com/y9HyEgf . I looked online to verify if my proof was right, but all the ones i come across use induction. I'm assuming my proof is wrong i just cant tell where or why its wrong. If someone could let me know why it is that would be great. Thank you!...Read more

complex analysis - How and when to do exercises from books?

Most math books I have seen seem to be structured in the same way; each chapter is a few dozen pages, and after each chapter there are a few dozen exercises. I often find myself not knowing what exercises to do and when to do them. If I have read the first 10 pages, I'm not sure what $x$ is in "the first $x$ exercises should be answerable after reading the first $10$ pages." A prime example is Tristan Needham's "Visual Complex Analysis". It's a very good book; however, the first chapter is about $50$ pages long, after which follows a splatter o...Read more

How to 'analyze' problems in analysis; Computing $\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$

If $a, b \in \mathbb{R}$ with $a > b > 0$, compute this ungodly thing;$$\int_0^{2\pi}\frac{1}{(a+b\cos(\theta))^2}d\theta$$I'm really not a fan of complex analysis... I can't visualize what's going on here. When I look at things from group/number/graph theory I see ideas. When I look at this... all I see a bunch of symbols.Where to begin with this? Maybe someone with analysis background can offer some intuition on 'analyzing' these things to better see whats going on..Because right now, I'm lost looking at this.Thanks guys!...Read more

complex analysis - Using the inversion formula of a Fourier Transform to calculate a limit

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by$$f(x)=\begin{cases}1-|x|/2 \quad \text{if} \ |x|\leq 2, \\ 0 \qquad \quad\quad\ \text{otherwise}.\end{cases}$$Calculate the fourier transform of $f$ and hence, using inversion formula, show that$$\int_{-\infty}^\infty \frac{\sin^2 (t)}{t^2}dt =2\pi.$$I computed the fourier transform to be$$\hat{f}(t) = \frac{1-\cos(2t)}{t^2} = \frac{2\sin^2(t)}{t^2}.$$I'm not sure how to use it now. I thought that the inversion would give us our old function $f$ back... let alone a constant value. The definition of...Read more

fourier analysis - Solving an Infinite Complex Integral with a Singularity and Oscillatory Behavior

I have derived the expression for the motion of a bead in an infinite elastic medium in frequency space, and now I am attempting to take the inverse Fourier Transform of my expression in order to recover the time domain behavior of the bead. Using$$h(t) = \frac{1}{2 \pi}\int_{-\infty}^{\infty}h(\omega)e^{i \omega t}\mathrm{d}\omega$$ as the definition of the inverse fourier transform, I obtain some terms of the form$$\int_{-\infty}^{\infty}\frac{i e^{i \omega t}}{(\omega-\omega_0)g(\omega)}\mathrm{d}\omega$$where $g(\omega)$ is a complex polyno...Read more

complex analysis - Analytical continuation of complete elliptic integral of the first kind

I am dealing with a problem involving the complete elliptical function of the first kind, which is defined as:$K(k)=\int_0^{\pi/2} d\theta \frac{1}{\sqrt{1-k^2\sin^2(\theta)}}=\int_0^1 dt \frac{1}{\sqrt{1-t^2}\sqrt{1-k^2 t^2}} $for $k^2<1$. I am trying to find however how to analytically continue the function for cases where $k^2>1$, and especially when $k^2$ is complex and classify it depending on $\mathrm{Im}(k^2)$ is complex or not.This looks like a very trivial question that must be written somewhere, but i couldn't find it explicitel...Read more

complex analysis - What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I fail to see the intuition behind this. Most of the time, I even think they tend to make calculations harder.Is there a simple interpretation of these operators? What mental picture do you have when you use them?...Read more

complex analysis - Uniform convergence of $\int_{a}^{\infty}{\frac{\sin x}{x^s}}dx$ for $\Re(s)>0$?

For $a>0$, does$$f_b(s)=\int_{a}^{b}{\frac{\sin x}{x^s}}dx$$converge uniformly for all compact subsets of $\{s\in\mathbb C|\Re(s)>0\}$ when $b\to\infty$ ?For $\Re(s)>1$ the integral converges absolutely, and since$g_s(z)=\frac{\sin z}{z^s}$ is holomorphic on $\{x\in\mathbb R|x>0\}$, by applying the Weierstrass theorem (which states that for a family of holomorphic functions converging uniformly, the family of derivative of those functions converges uniformly and equals the derivative of the converging function of the given family) i...Read more