The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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Do these intervals have the same image under the $e^z$ transformation?

Is the image of $D_1=\{z \in C: 0 \lt Re(z) \lt \infty, 0 \lt Im(z) \lt \pi\}$ the same as the image of $D_2=\{z \in C: 0 \lt Re(z) \lt \infty, \alpha \lt Im(z) \lt \beta \}$ as $0 \lt \alpha \lt ...
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1answer
15 views

Determine the order of entire function $G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$.

Let $$G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$$ for $\alpha>0$. Prove that it's an entire function and determine its order. Any suggestions please?
2
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1answer
17 views

Find all holomorphic functions with the following property

Let $D$ be the unit disc. Find all holomorphic functions $f:D\to D$ such that $f(\frac14)=\frac14$, and $f'(\frac14)=\frac7{15}$. I guess that we should use Schwarz lemma. And I guess that the only ...
2
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1answer
20 views

Complex differential operators

Consider the differential operators $\dfrac{\partial}{\partial z}$ and $\dfrac{\partial}{\partial \bar{z} }$ defined by $\frac{\partial}{\partial z} = \frac {1}{2} (\frac{\partial}{\partial x} - ...
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2answers
19 views

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$ then $f(z)=ce^{\alpha z}$

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$, $\alpha$ being a constant, for every $z \neq 0$ then $f(z)=ce^{\alpha z}$, $c \in \mathbb C$. So what I tried doing is defining ...
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0answers
20 views

Problem with a proof of Mittag-Leffler theorem

I've been going through Rudin's Real and Complex Analysis (3rd edition) but I got somehow stuck at the proof of Mittag-Lefler theorem (Theorem 13.10, page 273). The problem is I can't see why Theorem ...
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0answers
26 views

at least one coefficient of the power series is zero [on hold]

Is the following statement is true or false. If $f$ is an analytic function on the disc $\{z\in \mathbb C :|z-1|<1\}$ and $f(z)=f(z^2)$ then $f$ will be constant function. If it is true then why ...
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2answers
19 views

Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions?

Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions? I tried writing $z$ as $e^{i \phi}$, but I didn't know how to continue from ...
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2answers
15 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
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1answer
15 views

The decomposition of a meromorphic function

I want to solve the following problem, taken from the book of T.W Gamelin, Complex Analysis, so it goes like this: Suppose $f(z)$ is a meromorphic on the disk $\{|z|<s \}$ with only a finite ...
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2answers
40 views

Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.

Let , $f$ be entire function such that $|f\left(\frac{1}{n}\right)|\le \frac{1}{n^{3/2}}$ for all $n\in \mathbb N$. Then prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded. From the ...
2
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0answers
30 views

Textbook +reference book in complex analysis

Which book can be used as an introductory textbook in complex analysis? I have some choices (more suggestions are welcomed) Marsden & Hoffman J.B. Conway Ahlfors Palka Lang Stein & ...
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1answer
16 views

Fourier Series of Eisenstein series [on hold]

$$G_{2k}(\tau)= 2\zeta(2k)+2\frac{(2\pi i)^{2k}}{(2k-1)!}\sum_{n\geq 1}\frac{n^{2k-1}q^n}{1-q^n}$$ where $q =e^{2\pi i \tau}$ and $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $L(\tau ...
2
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1answer
29 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
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2answers
21 views

Holomorphic curve with unit norm

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \parallel \gamma(z) \parallel =1$. if the answer is yes, can the ...
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1answer
16 views

Radii problem in a power series

I was studying some basic matters of several complex variables (here $\Omega\subseteq\Bbb C^n$, open): After this, before the proof, the author pointed what follows: So I'm going to tell you ...
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0answers
27 views

‘Integral’ of a Weierstrass $ \wp $-function.

I'm revising for my finals and I've seen a question which asks: Is there a meromorphic function $f: \mathbb{C}/\Lambda \to \mathbb{P}^1$ such that $f' = \wp$? There is a hint which says consider the ...
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3answers
57 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
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0answers
12 views

Classify singularities (poles)

Consider the following function $\frac{1}{\sqrt{h(p-\log(\frac{h}{1-h}))}}$ on where $p$ is real. Are the singularities at $0$ and $\frac{e^{p}}{e^{p}+1}$ removable? essential? Thanks.
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3answers
27 views

Bolzano-Weierstrass theorem (complex case)

I'm trying to prove Bolzano-Weierstrass Theorem to the complex case, i.e., if $(z_n)$ a complex sequence is bounded, then there is a subsequence of $z_n$ which converges. I'm trying to use the real ...
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1answer
22 views

an exercise from “Representation Theorems in Hardy Spaces”, J. Mashreghi [on hold]

$\textbf{Exercise 7.3.2 (page: 165):} \ \text{Let} \ z_0 \in \mathbb{D}, \ \text{and let} \ 0 \leq r \leq 1. \ \text{Show that}$ $$\max_{|z|=r} \bigg| \frac{z_0-z}{1-\overline{z_0}z} \bigg| \leq ...
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2answers
46 views

Riemann-esque sums (complex analysis)

I have been struggling to prove the following statement: "Let $\gamma:[t_0,t_1] \rightarrow \mathbb{C}$ be a $C^1$ curve. For any $N \in \mathbb{N}$ and $k \in (0,N]$ define $t_N^k := \bigg(1 - ...
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0answers
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Does a relationship exist between the anharmonic absolute elliptical invaliant and the Lattes map?

In a paper in " The Beauty of Fractals" page 153 ,Benoit B. Mandelbrot said his first view of the Mandelbrot set like complex dynamics was in the Samuel Lattes Map which he gives as: ...
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0answers
14 views

What does it mean to be a “family of parametrizations?”

I am reading this book and there is a definition of a curve as follows: So, according to the authors what is a curve? (I am getting confused by "family of all parametrizations" part of the ...
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2answers
24 views

Understanding Maximum Principle

Understanding Maximum Principle One of the point of that theorem is: If $f$ is analytic on the open connected set $\Omega$ and $|f|$ assumes a local maximum at some point in $\Omega$, then $f$ is ...
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2answers
32 views

How to calculate $\int_{C(0,1)}\frac{\sin z}{z^4}dz$

$\displaystyle\int_{C(0,1)}\frac{\sin z}{z^4}\:\mathrm{d}z$, where $C(0,1)$ is the circle around $0$ with radius $1$ $\displaystyle\int_{C(0,1)}\frac{\sin ...
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0answers
25 views

Contour Integral Example

In the linked lecture notes below https://math.nyu.edu/faculty/childres/lec12.pdf I don't understand the part where the professor writes $$ \lim_{R\to\infty}\lim_{\epsilon \to ...
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0answers
78 views

A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...
0
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1answer
24 views

non constant holomorphic functiom

If $f$ is a non-constant holomorphic function in the unit disc $|z|<1$ that satisfies $f(0)=1$ , then prove that there are infinitely many points $z$ lying inside the disc such that $|f(z)|=1$. How ...
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0answers
10 views

Bounds of Zeros of a Higher Degree Polynomial Equation

The Equation is follows: $$\lambda^{k+1}+\frac{A}{A+B} \lambda^{k-l}+\frac{B}{A+B}=0$$ Where $l$ and $k$ are positive integers such that $l<k$. Also A and B are complex numbers. I would like to ...
3
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3answers
42 views

$n$ such that $|\sin in | > 10 000$

An exercise in my book asks me to find $n\in \mathbb N$ such that $|\sin in |> 10 000$. Could someone please check my solution? I wrote $$ |\sin in |^2 = \cosh^2 n + \sinh^2 n = 1 + 2 ...
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0answers
20 views

Proving a property of a holomorphic function $ f $ on the unit disk that satisfies $ f(0) = 1 $. [on hold]

If $ f $ is a non-constant holomorphic function in the unit disc $ |z| < 1 $ that satisfies $ f(0) = 1 $, then prove that there are only finitely many points $ z $ lying inside the disc such that $ ...
2
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1answer
26 views

Computing Residues Confusion

For $C := \left \{ |z| = 2\right \}$, $\int_{C}\frac{e^{\pi z}}{4z^2 + 1}dz$ the isolated singularities are $\pm \frac{1}{2}i$. By Cauchy Residue Theorem,$\newcommand{\Res}{\operatorname{Res}}$ ...
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0answers
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Analytic continuation at polygon vertices

The Riemann Mapping Theorem states that there exists a bijective, biholomorphic mapping from a simply connected set $\Omega \ne \mathbb{C}$ to the unit disk. Schwarz-Christoffel gives a (mostly) ...
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votes
2answers
69 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [on hold]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
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1answer
50 views

Trigonometric integral evaluates to factorial

I would like to prove the integral identity $$\int_{0}^{2\pi} e^{\cos(x)} \cos(nx - \sin(x)) \, dx = \frac{2\pi}{n!}$$ One approach is to interpret this as the real part of a complex exponential ...
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1answer
33 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
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2answers
37 views

Cauchy- riemann equations

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function that is differentiable at the point $z_0 =x_0 + iy_0$. Prove that $f'(z_0)= \frac{\partial u}{\partial x} (x_0,y_0) + i \frac{\partial ...
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0answers
25 views

Meromorphic Function [duplicate]

Let f be a meromorphic function on $\mathbb{C}$ such that $|f(z)| \geq|z|$ at each $z$ where f is holomorphic then f is entire finction such that $f(z)=Az$ for some constant $A \in \mathbb{C}$.
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2answers
44 views

Determining the Laurent Series

I need to determine the Laurent series of this function: $$\frac{1}{(z-1)(z+5)}$$ Inside the annulus: $$\left\{z|1<|z-2|<6\right\}$$ Any help appreciated.
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0answers
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(Churchill Brown) Integration Branch Point Exercise [on hold]

I am having trouble with the following problem in Churchill/Brown text. I am not able to find a concrete answer. The problem is given below. The reference to Fig99 is completely unnecessary. I know ...
0
votes
1answer
37 views

If $f$ has a primitive on $\Omega$, then $f$ is analytic on $\Omega$

If $f$ has a primitive on $\Omega$, then $f$ is analytic on $\Omega$ I don't understand the proof of the corollary $2.2.12$ here. How can one apply corollary $2.2.11$, if it holds only for ...
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2answers
39 views

To find analytic function with given condition

How to find all analytic function on the disc $\{z:|z-1|<1 \}$ with $f(1)=1$ and $f(z)=f(z^2)$ ?.
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0answers
33 views

Expected Value of the absolute value of the sum of random variables

Hi everyone and thanks in advance. Let's say we have a random variable Y which can be expressed as the sum of two other complex random variables X and W, i.e. $ Y = X + W $. $X$ and $W$ are ...
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1answer
42 views

functions that can be written as $g^3$

Let $D$ be a proper sub-domain of $\mathbb{C}$ in which every everywhere nonzero function $f$ can be written as $g^3$ with $g$ being holomorphic, then show that there is a holomorphic embedding of $D$ ...
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1answer
24 views

Removable Singularities

Can we not we define every isolated singularity of a complex analytic function to be a removable singularity? A removable singularity is a point where the holomorphic function is undefined, but it ...
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0answers
30 views

How to find fourier transform of $e^{-x^2}$?

I want to find the fourier transform of $e^{-x^2} = \int_{-\infty}^{\infty}e^{ikx-x^2}\,dx$ using contour integration. I consider the rectangular contour $C$ with verticies $\pm R, \pm R + ik$ Then ...
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0answers
20 views

Applying Cauchy Residue Theorem

For $C := \{z(t) : t(1+i) : t \in [-1,1]\}$, $\int_{C} \frac{dz}{(z-1)}$. The singularities of $\frac{1}{(z-1)}$ is $z_0 = 1$. Note that this singularity (pole?) is contained within the contour. ...
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vote
1answer
41 views

How to find $z$ with $|\sin z | \le 1$?

I am trying to find all $z \in \mathbb C$ such that $|\sin z |\le 1$. What I did so far: Clearly, for all real $z$ this is satisfied. Next I tried to rewrite it like this: $$ |\sin z |^2 = ...
1
vote
2answers
33 views

Simple Question About Residues/Poles/Zeros/Singularities

I'm having a little bit of trouble with residues. If we have the $f(z)=\left(\frac{\cos(z)-1}{z}\right)^2$ at $z_0=0$, we have a zero of order 2 in the numerator and a zero of order 2 in the ...