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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Analytic functions can't have more than two periods

Let $f(z)$ be a non-constant analytic function. Show that $f(z)$ can't have more than two periods.
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1answer
679 views

Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.

I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
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1answer
66 views

Continuity of $\text{Im}\sqrt{(x+iy)^3}$

I can use Mathematica to investigate the continuity of the real-valued function $$\text{Im}\sqrt{x+iy}$$ by drawing a density plot or a plot of the surface. Clearly, they are continuous on ...
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1answer
122 views

If $f$ is analytic where $f$ is represented as $f=g.h$ where $g$ is analytic . From here can we conclude that $h$ is analytic?

If $f$ is analytic, where $f$ is represented as $f=g \cdot h,$ where $g$ is analytic. From here can we conclude that $h$ is analytic?
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1answer
135 views

Convergence of $\sum_{n=0}ne^{nz}$

I want to see for what values of z the series $\sum_{n=0}e^{nz}$ and $\sum_{n=0}ne^{nz}$ converges and to find the sum in each case. For the first, it is a geometric series and will converge if ...
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1answer
247 views

Does holomorphic a.e. and continuous imply holomorphic everywhere?

Suppose $D$ is a domain in $\mathbb{C}$, $f:D\rightarrow \mathbb{C}$ is a continuous function. Suppose $f$ is holomorphic outside the zero set $f^{-1}(0)$, and $f^{-1}(0)$ has Lebesgue measure zero. ...
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2answers
360 views

A few Questions about Harmonic Functions

Given a harmonic function $u$, it is the real part of some analytic function $f$ whose imaginary part, $v$, is the harmonic conjugate of $u$. Is this relationship symmetric? That is to say is then ...
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2answers
85 views

calculate for $r>0$ $\frac{1}{2\pi i} \int_{|z|=r}{f(z)g(z)dz}$

Let $f : \mathbb{C}\setminus$ {$0$} $\to \mathbb{C}$ be an analytic function with a simple pole of order $1$ at $0$ with residue $a_1$. Let $g : \mathbb{C} \to \mathbb{C}$ be analytic with $g(0)\neq ...
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1answer
241 views

Complex Contour Integration Contour Deformation

Let C be the unit circle |z| = 1 traversed once counterclockwise and then once clockwise, starting from z = 1. Construct a function z(s, t) which deforms C to the single point z = 1 in any domain D ...
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2answers
70 views

Power series involving complex variable.

Show that $1 + \binom{m+1}{1}z + \binom{m+2}{2}z^2 +...+ \binom{m+n}{n}z^n +... = \frac{1}{(1-z)^{m+1}}$ for non-negative integers $m$ and $|z| < 1$.
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57 views

Complex integral Q

I have this question: Let C be an open (upper) semicircle of radius R with its centre at the origin, and consider $\int_C f(z) \, dz$ where $f(z)=\frac 1{z^2 + a^2}$ for real $a > 0$. Show that ...
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853 views

Radius of convergence and ratio test

My book says that given a power series $\sum_{n = 1}^\infty c_nz^n$ where the $c_n$ are complex the radius of convergence of the series is $\dfrac{1}{L}$ where $L = \lim \sup \sqrt[n]{|c_n|}$. So the ...
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2answers
78 views

complex number power

I have question related to power of i,which is determined by equality $i=\sqrt{-1}$ actually from complex number book I know that $i^2=-1$, as much as i know if we compare physical ...
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1answer
49 views

Complex Tori projection question

My question is based on Miranda's "Algebraic Curves and Riemann Surfaces"; at page 9, it is stated that, for any $z\in\mathbb{C}$, we define the parallelogram $$ P_z=\left\{z+\lambda_1 w_1 + \lambda_2 ...
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1answer
240 views

Using Cauchy's integral formula compute $\int_{\gamma(1;1/2)}\frac{\cos(\pi z)}{z^{2}-3z+2}dz$

How do I compute this? Do I use the factorization of $(z - 2)(z - 1)$ and do two seperate integrals with $n=0$ or do I use the factorization $(z - \frac{3}{2})^{2} - \frac{1}{4}$ and set $n=1$ whilst ...
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0answers
406 views

Finding a radius of convergence

Let $\sum_0^{\infty} a_n z^n$ have radius of convergence $R$ with $0< R< \infty$. Let $\alpha>0$. Find the radius of convergence of $\sum_0^{\infty} |a_n|^{\alpha} z^n$. I tried to start ...
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3answers
153 views

The equation $az+b \bar{z}+c=0$ has exactly one solution if $|a| =|b|$. It is true or false?

The equation $az+b \bar{z}+c=0$ has exactly one solution if $|a| =|b|$. It is true or false?
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1answer
158 views

If$f(z)$ is analytic , then what about $f'(z)?? $

If$f(z)$ is analytic , then what about $f'(z)$? can we conclude that $f^{(k)}(z)$ is analytic for any k$\in $$ \mathbb{N} $
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1answer
292 views

Convergence radius of power series

I am trying to solve an exercise, but i am not sure that the result i get at the end is correct...May i kindly ask you for a little help or a remark? Find the radius of convergence of the following ...
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3answers
457 views

Proving that a complex number $z$ is real.

A problem I have in my book is to prove that $z$ is real if and only if $\bar{z} = z$. So far I have got that for $z = x + iy$, if $z$ is real, $y = 0$ and thus $z = x = \bar{z}$ as $\bar{z} = x - ...
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2answers
158 views

$p(z) = 1 + 2z + 3z^2 + \dots + (n+1)z^n$ has no zeroes in a disk for sufficiently large $n$.

Let $0 < r < 1$. I need to prove that $p(z) = 1 + 2z + 3z^2 + \dots +(n+1)z^n$ has no zeroes in the disk $|z| < r$ if $n$ is sufficiently large. I'm thinking Rouche's theorem might be ...
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1answer
100 views

How to apply the cauchy theorem to evaluate an integral?

can someone please explain/show me how to apply the cauchy interal formula? Here is a question: $$\int_{\gamma=(a,a)} \frac{z}{z^4-1} dz$$ where $a\in\mathbb{R}$ and $a>0$ and $a\not= 1/2$. ...
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1answer
78 views

Show the mapping $z^2$ is a homeomorphism?

If $f: \mathbb{C} \rightarrow \mathbb{C}$, $f(z) = z^2$, and $\mathbb{C}$ has the standard Euclidean metric, is $f$ a homeomorphism?
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607 views

Finding the Taylor series expansion of $f(z)=\frac{e^{z}-1}{z}$ around $0$

Find the Taylor series expansion of $f(z)=\displaystyle\frac{e^{z}-1}{z}$ around $0$. I have no idea where to start.
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1answer
101 views

branch point(s) of $\ln (\tan z)$

I was ridiculed for thinking that $\ln (\tan z)$ has infinitely-many branch points on the real axis. My reasoning is that if you expand $\ln (\tan z)$ in a series about $\frac{n\pi}{2}$ (where $n$ is ...
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2answers
2k views

Evaluate improper integral $(\cos(2x)-1)/x^2$

Consider the following improper integral: \begin{equation} \int_0^\infty \frac{\cos{2x}-1}{x^2}\;dx \end{equation} I would like to evaluate it via contour integration (the path is a semicircle in ...
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1answer
278 views

Complement is connected iff Connected components are Simply Connected

Let $G$ be an open subset of $\mathbb{C}$. Prove that $(\mathbb{C}\cup \{ \infty\})-G$ is connected if and only if every connected component of $G$ is simply connected.
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1answer
478 views

How to solve using Cauchy Integral formula?

Let $C$ be the positively oriented boundary of the square whose sides lie along the lines $x=+/-2$ and $y=+/-2$. I am supposed to use the Cauchy Integral formula to evaluate $$\int_C ...
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1answer
2k views

3D Fourier Transform

I'm trying to calculate the inverse of the following 3D Fourier transform. $$ \widetilde{f}= \frac{1}{(k^6-\alpha*k^2-\alpha*k_3^2)} $$ where $k = (k_1^2+k_2^2+k_3^2)^{1/2}$ the fourier transform is ...
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2answers
243 views

Find a branch for $(4+z^2)^{1/2}$ such that it is analytic in the complex plane slit along the imaginary axis from $-2i$ to $2i$

Find a branch for the multiple-valued function $(4+z^2)^{1/2}$ such that it is analytic in the complex plane slit along the imaginary axis from $-2i$ to $2i$ Also, isn't this function already ...
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2answers
150 views

branch cuts of the logarithm

This question is related to a question I asked a few days ago that wasn't really answered. Let $f(z) = \ln(-z)$ If we choose the branch where $0 \le \arg(-z) < 2 \pi $, the cut is on ...
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1answer
213 views

Extended Liouville's Theorem with non-integer exponents

In Bak-Newman's "Complex Analysis", there are two versions of Liouville's Theorem given: 1) A basic version: An entire function bounded by a constant $M$ is constant. and 2) An extended version: An ...
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2answers
76 views

Finding a sequence of complex numbers ${z_n}$ such that $\sin z_n$ is real for all $n$ and tends to $\infty$ as $n→\infty$

Find a sequence of complex numbers ${z_n}$ such that $\sin z_n$ is real for all $n$ and tends to $\infty$ as $n→\infty$ ? I get an example as $\log 2in$ . I want to verify that am I right or wrong. ...
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1answer
231 views

Maximize absolute value of complex logarithm

I'm trying to solve exercise 9 in chapter 14 of Real & Complex Analysis of Walter Rudin: Suppose $g \in H(U), |\Re(g)|<1$ in $U$, and $g(0)=0$. Prove that ...
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1answer
55 views

Intuition for generalized complex exponentials (like $i^i$)

I understand complex exponential function $e^z$ and its geometric meaning, but when we expand complex exponentiation to $z^w$ for arbitrary complex z and w, $z \neq 0$, I have no intuition what that ...
2
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1answer
224 views

bonus on stereographic projections

This is one of the bonus question that we are given. Its alright if you guys don't get it because I certainly don't. It will be nice if some one could tell me whats going on in this question. Or a ...
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1answer
257 views

finding an image of a linear transformation

I am so confused in how its asking of finding an image of infinity. I am in my complex class and we have a test and this was part of the past midterm, I feel like if I do one all of the other ones ...
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1answer
169 views

Inversion in complex variables.

Hey guys I am given this question on my past midterm, and I cant come about the solution, i know its a mapping of all complex numbers minus the 0 and it maps to itself. So i tried to graph the points ...
4
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1answer
162 views

Is there any specific formula for $\log{f(z)}$?

Let $f(z)$ be a nonvanishing analytic function on a simply connected region $\Omega$. Then there is an analytic function $g(z)$ such that $e^{g(z)}=f(z)$. Is there any specific formula for $g(z)$? ...
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1answer
143 views

Can I use Schwartz's Lemma to prove that $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ implies $f(z)=0$ for all $z\in\mathbb{C}$?

Problem. Suppose that $f(x)$ is an entire function satisfying $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ as $|z|\rightarrow \infty$. Show that $f(z)=0$ for all $z\in \mathbb{C}$. The ...
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1answer
72 views

Open map in complex numbers

Is a function $f: \mathbb{C} \rightarrow \mathbb{C}$ where $f(z) = z^2$ an open map if $\mathbb{C}$ has the metric topology $d(z,a) = |z-a|$ ? I can think of several reasons why $f$ should map open ...
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0answers
155 views

Give an example function that is solution for Dirichlet - problem in unitdisk D.

Give an example function that is solution for Dirichlet - problem in unitdisk D. I have tried border function $f:\partial D \rightarrow \mathbb{R}$, such that $f(z)=Re(e^{i\theta})$, but $\theta \in ...
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1answer
120 views

Why isn't this a counter example to "Linear Fractional Transformations are Automorphisms of $\mathbb C \cup \infty$

$\frac{Z-1}{2Z-4}$ is a linear fractional transformation, but it cannot take on the value $\frac{1}{2}$ -- so how can it be an Automorphisms of the Extended complex plane?
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2answers
447 views

The boundary of a region for complex-valued functions

From Chapter $3$ of Stein and Shakarchi's complex analysis book, we have the following problem ($15$): Show that if $f$ is holomorphic in the unit disc (open), bounded, and converges uniformly to ...
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1answer
211 views

Existence of an entire function with algebraically independent derivatives

Let $\mathbb{A}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. A collection of functions $F=\lbrace f_i:X \rightarrow\mathbb{C}\rbrace$ is said to be algebraically independent over ...
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0answers
42 views

For compact $K \subset \mathbb C$, show that $u(z) = -\log(\mathrm{dist}(z,K))$ is subharmonic

Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic? Subharmonic, here, is ...
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2answers
706 views

Order of the pole

What is the order of the pole at $z=0$: $$f(z)=\frac{1}{(2\cos(z)-2+z^2)^2}$$ and find and classify the isolated singularities of: $$\frac{1}{e^z-1}$$ My attempt: If I let ...
0
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1answer
1k views

Using Cauchy's integral formula to evaluate a function

This problem is from Brown/Churchill Complex Variables and Applications, 8th edition 2009. Section 52, exercise 2, subsection (a) How do I show that the integral of the function $g(z) = ...
2
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3answers
157 views

why can infinite product only be zero if one of the factors is zero?

I was reading about the Riemann zeta function in the region Re(Z) > 1, where it can be represented by the Euler product formula. And the book mentioned that there can be no zeros in this region, since ...
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1answer
168 views

Magnetic fields and the complex plane

The electrostatic potential $\varphi$ must satisfy Laplace's equation in regions without charge: $$\nabla^2 \varphi = 0.$$ If there is no $z$ dependence in the problem we are solving, we can choose ...