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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Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
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1answer
25 views

Complex Integration and Uniform continuity

Let $R=[a,b]\times [c,d]$ be a closed rectangle and for $\epsilon>0$, let $R_{\epsilon}$ be the rectangle $[a+\epsilon, b-\epsilon]\times [c+\epsilon, d-\epsilon]$. Consider this situation in ...
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1answer
27 views

Describe geometrically what the function $f(z)=z^k$ for $k$ some positive integer does to the points on a line through the origin

Describe geometrically what the function $f(z)=z^k$ for $k$ some positive integer does to the points on a line through the origin or to the points on a circle with center at the origin. We are able ...
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31 views

All rotations about a point in the complex plane also Möbius maps

Is my claim true? If so, is my "proof" correct? Claim: All rotations about a point in the complex plane also Möbius maps. Proof: We have shown that the set of Möbius maps is a group. So we can do the ...
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25 views

What is $\sup_{z\in \mathbb{D}} |\frac{1}{1-\bar{\lambda}z}|$ for $\lambda \in \mathbb{D}$?

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. I tried computing $\inf_{z\in \mathbb{D}}|1-\bar{\lambda}z|$ by noting that $|1-\bar{\lambda}z|=|\lambda|\cdot|z-\lambda|\lambda|^{-2}|$, and ...
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61 views

Sketching an inequality in the complex plane

so I have to sketch this inequality on the complex plane, $$\frac {|z-a|} {|1- \bar az|}<1$$ where $|a| < 1$ is a complex number. I know typically when there is just $z$'s and $i$'s you ...
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58 views

How to solve this real-analysis problem?

Since $f(x)$ is bounded we have: $f(x) \le M$ for some $M \in \mathbb{R}$ Also, we have to prove: $$\exists N\in \mathbb{N}\;\;\;\forall n>N\;\;\;:\;\;\; \left| \int_{0}^{1} f(x)\cdot x^n ...
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40 views

Differentiablility of f(z) using Cauchy-Riemman and First Principles

If $f(z)=z|z-1|^2$ where $z=x+iy$ i need to show where it is differentiable, and then from first principles find its derivative at each point. I have started by saying $f=(x+iy)((x-1)^2+y^2)$ After ...
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1answer
37 views

Why is this function holomorphic?

consider a locally integrable function $f:(0,\infty)\rightarrow \mathbb{R}$ and then the function $\phi(z):= \int\limits_{1/n}^n t^{z-1}\cdot f(t) dt$ for some fixed $n\in\mathbb{N}$. I'm wondering ...
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91 views

$f(z)=\bar{z}$ has no primitive

As a consequence of Goursat's Theorem, we can prove that every holomorphic function on an open disk has primitive. Question: Is it true that every continuous function $f\colon D\rightarrow ...
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1answer
20 views

Trying to prove the sequence is cauchy

If $\Omega_1 \supset \Omega_2 \supset ... \supset \Omega_n \supset ... $ is a sequence of non-empty compact sets in $\mathbb{C}$ with that property that $$ \lim_{n \to \infty} diam( \Omega_n ) = 0 ...
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1answer
85 views

Does such an analytic function exist?

Consider the following region in the complex plane $$ R=\{re^{i\theta}:r>0, 0<\theta<\gamma<\frac{\pi}{2}\} $$ Then does there exist a function $f(z)$, which is analytic in $R$ and ...
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1answer
54 views

Evaluate $(5-i)^4 (1+i)$ then prove that $ \pi/4=4\arctan(1/5)-\arctan(1/239)$

Evaluate $(5-i)^4 (1+i)$ then prove that $ \pi/4=4\arctan(1/5)-\arctan(1/239)$ I did the first part of evaluating it manually and I got 956-4i. However I am having trouble seeing how I can use ...
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21 views

Question about complex polynomials

Let $P(z,\overline{z}) = \sum a_{lm} z^l \overline{z}^m $ be a polynomial such that $$ \frac{ \partial P}{\partial \overline{z} } = 0$$ Qs: Show that $P$ contains no term with $m > 0 $ I dont ...
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1answer
54 views

Quasiconformal Mappings in Fluid Dynamics

I know that conformal mappings can be used to study 2 dimensional fluid flows. But I was wondering how quasiconformal mapping have been applied in this respect?
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57 views

Singularities of $ {1}/{\cos(\frac{1}{z})}$

I would like to determine the singularites of $f$, given by $$f(z) = \frac{1}{\cos(\frac{1}{z})}.$$ It is clear to me that $z = 0$ and $z = \frac{2}{(1+2k)\pi}$ for $k\in\mathbb Z$ are singularities. ...
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105 views

Problem on Integration: $\Bbb R-\Bbb C$ split and pull back of forms

This post is not short. However I'm sure that a guy who good handle these concepts, could read and answer in five minutes. I only want to write my attempt, in order to understand where I'm wrong. Let ...
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1answer
17 views

There is a 1-1 correspondence between circles on the riemann sphere and lines or cirles on the plane

There is a 1-1 correspondence between circles on the riemann sphere and lines or cirles on the plane attempt : Let $P \subset \mathbb{R}^3 $ be the plane $a x_1 + b x_2 + c x_3 = d $. We denote the ...
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30 views

Proving an Inequality Involving the Modulus of the Difference of Moduli

Prove the following inequality and give necessary and sufficient conditions for equality. $\left| |z|-|w| \right| \leq |z-w|$ for complex numbers $z$ and $w$. I have the following: By definition ...
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3answers
48 views

Partial fractions of $1/(z^2+2)$

How does one split $1/(z^2+2)$ into partial fractions? Normally I would factorise, but I cannot spot the solutions of $z^2+2=0$.
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1answer
82 views

Cauchy riemann eq for anti-holomorphic function.

I face a problem asking for CR equation for anti-holomorphic function. They ask for three forms: in rectangle coordinate, polar coordinate and complex coordinate. My approach is that : Let $f = u + ...
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What is the description of the range of the complex function $f(z)=2z-1$ for $\operatorname{Im}(z)<0$?

As the description says, I need to find the range. I got the range to simply $\operatorname{Im}(z)<0$, but I really am not too sure. Thanks.
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32 views

ML inequaility problem, finding M

I have attempted the following problem, is my method and logic correct? I am looking for a upper bound of; $f(z)=z^2$. My line is from $1-2i$ to $1+2i$. Since we have no change in $x$, length is ...
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77 views

Simplify $\frac{\Gamma(n)}{\Gamma(n+a)}$ with $a\in\mathbb C$.

How can simplify the following expression? $$\frac{\Gamma(n)}{\Gamma(n+a)}\sim \cdots\text{ ?}$$ Where $a\in\mathbb C$, $n\in \mathbb N$. Any suggestions please? I propose the following. We have ...
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18 views

Extending Newman's proof of PNT to bound the error term

Recently I've been getting into Newman's proof of PNT (explained nicely in this paper). Later I have been trying to find similar results which prove an error term in PNT, but all of them seem to ...
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1answer
97 views

Prove that $f(z)=z^2$ is continuous.

Prove that $f(z)=z^2$ is continuous for all complex and real values of $z$. What I've got so far is: Given $ \epsilon >0$ and $|z-z_0|<\delta$ after some calculations (which I've checked with ...
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1answer
41 views

Complex Analysis Branch cuts

Take the branch of $\log(z)$ to lie in $(-\pi, \pi]$. With complex numbers when does $\sqrt{z^2} = z$ hold and when doesn't it? If we take $z=-1$, this equality holds but then $1=-1$. I have a feeling ...
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1answer
24 views

How can I prove that $\max_{\partial K}\varphi$ does exist?

Let $\varphi:\Omega\to[-\infty,+\infty[$, where $\Omega\in\Bbb C$ is a domain and $\varphi$ is upper semicontinous, i.e. $\varphi(z_0)\ge\limsup_{z\to z_0}\varphi(z)\;\;\;\forall z_0\in\Omega$. How ...
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Coupling real functions

I ended up with the following two real functions, that are actually the cosine and sine Fourier transform of other more complicated functions: ...
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26 views

Roots of of the derivative of a polynomial are in the convex hull of the polynomial

$(def)$: The $\mathbf{Convex}$ $\mathbf{Hull}$ of a set $\{ z_1,\ldots,z_k \} \subset \mathbb{C}$ is the set $$ CH[z_1,\ldots,z_k] = \left\{ z \in \mathbb{C} : z = \sum_{j=1}^k \lambda_j z_j, \; \; 0 ...
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1answer
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Let $B = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| \leq |z_2| \}$. Show that B is balanced, but that its interior is not.

I have the following definitions. The interior $E^o$ of $E$ is the union of all open sets that are subsets of $E$. A set $B \subset X$ is said to be balanced if $\alpha B \subset B$ for every ...
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1answer
74 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
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1answer
18 views

What is the range of this complex function: $f(z) = 2x^2+(1-x^2)(1+i)$ defined on $|z|\leq1$?

These range problems, I just don't get it. I tried to get this into a form where I could use the fact that $0\leq\theta\leq2\pi$, but I'm just not sure how to get it to that point. Any ${hints}$ ...
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Does $\frac{1}{z^n}$ have a primitive on $\mathbb{D} - 0$ for $n < 1$?

I'm studying complex analysis from Stein and Shakarachi. There is a question that asks you to evaluate: $\int_{\gamma} z^{n}\ \mathrm{d}z$ for all integer n, where $\gamma$ is any circle centered ...
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31 views

evaluating a sequence of complex numbers

Let $\{z_n\}$ be sequence of complex numbers such that $$ |z_n - z_m| < \frac{1}{1+ |n-m|} $$ for all $n,m$ Given this information, can we compute $\lim_{n \to \infty} z_n $? Attempt: For sure ...
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1answer
29 views

Sketch $ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $

Sketch the following $$ z\in \mathbb{C}:0 < arg(z-(1+i)) < \frac\pi3 $$ I have considered this geometrically and ended up thinking that the complex numbers $z$ must satisfy $$0 < ...
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3answers
133 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
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74 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
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Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number.

Determine the set of points that satisfy the condition $Im \frac{z-z_1}{z-z_2 }=0$ where $z_1$ and $z_2$ are fixed complex number. Since $Im\frac{z-z_1}{z-z_2 }=0$, then there is no imagining part in ...
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Find taylor of $\psi (z)$ where $(e^z-1)^2=z^2 \psi (z)$ - first 3 terms

I was asked to find the first three terms in the taylor series of $\psi (z)$ around $z=0$ where $(e^z-1)^2=z^2 \psi(z)$ and I'm having a few difficulties. My original idea was to say $\psi ...
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1answer
24 views

Show that a complex map is onto

I consider $\mathbb{C}$ as a real vector space. For $(a,b) \in \mathbb{C}^{2}$, consider the map : $F_{a,b} \, ; \, \mathbb{C} \, \rightarrow \, \mathbb{C}^{\ast}$ such that : $$ \forall z \in ...
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Continuous complex function from Rudin's Real and complex analysis

Lemma 10.29 from Rudin's Real and Complex Analysis, p. 314 of the third edition states that "if $f \in H(\Omega)$, then $g:\Omega \times \Omega \to \mathbb{C}$ defined by \begin{equation} g(z, w) = ...
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45 views

Understanding Green's Theorem

When looking at Goursat's theorem in complex analysis, I came across the Wiki proof which involves beautiful application of Green's theorem. I saw Greens theorem simply as "connection between line ...
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1answer
51 views

How to indentify the power series I need to use?

Let $$ f(z) = \frac{1}{(z - 4)(z + 8i)} $$ a) Find the domains where f(z) is valid b) Find its power series at such domains Considering three singularities, I believe the domains are: $$ D_{1} = ...
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1answer
83 views

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$

Find all holomorphic functions $f:\mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$ such that $$\limsup_{z\rightarrow 0}\left|\frac{f(z)}{\sin z}\right|<\infty$$ and $$\limsup_{z\rightarrow ...
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votes
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246 views

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$

Determine the set of points $z$ that satisfy the condition $|2z|>|1+z^2|$ I tried to redo this problem and got to this point $|2z|>|1+z^2|$ $\Rightarrow$ $2|z|>1+|z^2|$ $\Rightarrow$ ...
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45 views

Show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$

Let $a\in\mathbb{C}, |a|>e, n\in\mathbb{N_1}$ I have to show that $e^z-az^n=0$ has exactly $n$ zeros in $B(0,1)$ First, $f(z)=e^z$ and $g(z)=-az^n$ are entire. On $\partial B(0,1)$ we have ...
2
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1answer
27 views

Evaluation of an integral using nonrigorous methods

I was trying to solve the following integral $$ G(\alpha,m,n)=\int_0^{\infty}\cos(2nx)e^{-\alpha x}x^{m-1}dx;n\in N,\alpha>0,m\ge1. $$ By doing a change of variable I brought it to the integral $$ ...
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2answers
40 views

Complex Analysis holomorphic function question

I have a Complex Analysis assessment question about holomorphic functions: Let f be a function on a plane and satisfies $f'(z) = f(z)$ and $f(0) = 1$ i) Give an example of a function with this ...
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18 views

Coupling complex functions

After several calculations I end up with two complex functions: $$g(z)=zA(z)+\overline{z}A(\overline{z})+z^{-1}B(z)+\overline{z^{-1}}B(\bar{z})$$ and ...