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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162 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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3answers
1k views

Find the principal argument of a complex number

I have a text book question to find the principal argument of $$ z = {i \over -2-2i}. $$ I know formulas where we find using $$ \tan^{-1} {y \over x}$$ but I am kinda stuck here can somebody please ...
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1answer
101 views

Complex analysis proof with maximum

$f(z)$ is analytic in {$z:|z|>1/2$}. $|f(z)|\le1$ for every $|z|=1$ and $\lim_{z\to\infty}\frac{f(z)}{z^3}$ it final. Prove that $|f(z)|\le|z|^3$ for every $|z|\ge1$. After that, if in addion ...
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votes
3answers
486 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
97 views

Complex numbers

I am a newbie to complex numbers so please bear with me if i ask some very naive question., So i was trying to solve my class tutorials and the very first question is, Show that ...
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1answer
67 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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2answers
567 views

$f(z)$ is analytic in the unit circle - can you prove it's a constant function?

$f(z)$ is analytic in the open unit disk and continuous on its edge. Can you prove that if $f(z)=1$ on the upper half of the unit circle (for $z=e^{i\theta}, 0\le\theta\le\pi$) then $f(z)$ in ...
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2answers
162 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
106 views

An aproximation of the lambertw function for a complex number

Here is my problem, I used the fact that $W(x)=\ln(x)-\ln(W(x))$, replacing $W(x)$ by $\ln(x)-\ln(... $ a lot amount of times and it seems to works for simple $x$ but when I try with, for example, ...
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1answer
101 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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2answers
262 views

Cauchy Integral formula question

$$\int_{\gamma=(i,1)} \frac{z^3}{(z-i)^n} dz$$ for any $n\in\mathbb{N}$. Can someone please help me answer this question as I cannot seem to get the right answer! Please note that the Cauchy ...
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vote
1answer
79 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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3answers
619 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
103 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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1answer
207 views

Using the Cauchy integral formula to evaluate $\int_{\gamma=(a,a)} \frac{z}{z^4-1} dz$.

I'm trying to understand how to use the Cauchy integral formula, but a bit confused as to how to use it as I cant seem to get the right answer! $$\int_{\gamma=(a,a)} \frac{z}{z^4-1} dz$$ where ...
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1answer
294 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
512 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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votes
2answers
158 views

Branch cut for $\log (-z)$

I'm trying to understand the location of the branch cut for 2 particular branches of $\log (-z)$. Supposedly if we restrict $\arg (-z)$ to $0 \le \arg(-z) < 2 \pi $, we need to omit the half-line ...
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1answer
236 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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4answers
228 views

Find all entire functions $f$ such that $f^{(n)}(z) = z$ for all $z$, $n$ being a given positive integer

Find all entire functions $f$ such that $f^{(n)}(z) = z$ for all $z$, $n$ being a given positive integer. I can not think such a function exist or not.can somebody help me please
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2answers
77 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
56 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 ...
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1answer
228 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
262 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 ...
2
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1answer
176 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 ...
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2answers
252 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 ...
2
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1answer
146 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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2answers
557 views

All the zeroes of $p(z)$ lie inside the unit disk

Let $p(z) = c_0 + c_1z + c_2z^2 + \dots + c_nz^n$ where $0 \le c_0 \le c_1 \le \dots \le c_n$. I would like to show that all zeroes of this polynomial lie inside the unit disk by applying Rouche's ...
2
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1answer
714 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 ...
2
votes
1answer
74 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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1answer
48 views

Continuous complex funtion

I have this function $$F(z)=\frac{1}{\alpha-i\sqrt{z}}$$ with $\alpha>0$ and the determination of the square root with $\Im z>0$. I have to study its continuity in the set $$A=\lbrace z|a\leq\Re ...
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0answers
157 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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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 ...
2
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2answers
746 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) = ...
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votes
3answers
161 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
173 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 ...
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1answer
150 views

Exponential sum identity

How do I show that $$\sum_{|j| \leq J} (J-|j|) e(j \alpha) = \left| \sum_{j=1}^J e(j \alpha) \right|^2,$$ where $e(n)=e^{2 \pi i n}$ and $\alpha \not \in \mathbb{Z}$? Thank you very much in advance! ...
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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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1answer
145 views

change of variables in contour integration problem

On this answer: http://math.stackexchange.com/a/282675/65097, we see that $$\int_{-\infty}^{\infty} \: \frac{t^2}{t^4+1} dt = \int_0^{\infty} \frac{\sqrt{x}}{x^2+1} dx$$ from the change of ...
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2answers
85 views

How to show that $f-g$ is imaginary constant in $\mathbb{D}$?

How to show that $f-g$ is imaginary constant in $\mathbb{D}$? Let $f$ and $g$ be continuous functions in $\bar{\mathbb{D}}$ and analytic in $\mathbb{D}$. Show that if $\mathfrak{R}f=\mathfrak{R}g$ at ...
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1answer
123 views

Is $z^2-i(x^2-y^2)$ analytic in $\mathbb{D}$?

Is $f(z)=z^2-i(x^2-y^2)$ analytic in $\mathbb{D}$? I think it is because it is complex differentiable at $\mathbb{D}$. In other words it has unique complex derivative at all points of $\mathbb{D}$. ...
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1answer
103 views

Question on harmonic conjugates and Liouville's theorem

Suppose that $f(z)=u+iv$ is entire, and the harmonic function $u(x,y)$ has an upper bound. Then how to show that $u(x,y)$ must be constant throughout the plane?
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187 views

Question on application of Liouville's theorem

Let $f$ be an entire function such that $\mid f(z) \mid \leq A \mid z \mid$ for all $z$, where $A$ is a fixed number. Show that $f(z)=a_1z$, where $a_1$ is a complex constant.
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1answer
53 views

Is this estimation correct?

I have to estimate the following quantity $$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$ in $\mathbb{R}^3$ ($\lambda>0$) where ...
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2answers
172 views

A problem on Residue Theorem

Today I had a problem in my test which said Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$. Now, clearly this was a misprint since C is not a circle. I tried to find ...
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2answers
483 views

functions with multiple branch points

EDIT: My original question was poorly worded and thus confusing. So I'm going to edit it and then give a brief answer. $ $ Let $f(z) = \displaystyle \sqrt{1-z^{2}}= \sqrt{(1+z)(1-z)} = ...
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1answer
49 views

determining complex function

Problem Let $f$ be holomorphic in $D=\{|x+iy|< 1\}$, with $|f|\le |y|^{-1/2}$ $\lim_{r\rightarrow 1} f(re^{i\theta}) = 0$, for any $\theta\in[0,2\pi]$ Prove $f = 0$. It is an old qualify ...
2
votes
3answers
126 views

Cauchy Integral theorem

Let $f(z)=\sum^{\infty}_{k=0}\frac{k^3}{3^k}z^k$, compute $\int_{|z|=1}\frac{f(z)}{z^4}dz$ and $\int_{|z|=1}\frac{f(z)sinz}{z^2}dz$. I do not know how to do these problems. I know it is a ...
2
votes
2answers
138 views

Cauchy's Integral Formula

Please can someone help me understand how to use the cauchy's integral formula? I have put a picture of a question which i am struggling to get the correct answer for! I have the formula but i am a ...