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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Meromorphic function with pole of order 1

Let f(z) be a meromorphic function having pole of order 1 , does for every $\tau \in \mathbb{C} $ there exist a $z_o$ such that $f(z_o)= \tau$ ? If not in $\mathbb{C}$ then does it hold on a riemann ...
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17 views

Mittag-Leffler Proof. Rudin notation question and some basic real analysis/topology

Proof is below What is meant by $\sum_{\alpha \in A_n}$? I thought $P_\alpha$ is a sum already. What is this open set he is talking about? Because $A_n$ isn't open. Oh he means ...
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1answer
6 views

No Generalization of Mean Value Property for harmonic functions?

The Mean Value Property for harmonic functions tells us that the value of a harmonic function evaluated at the center of $D(P,r)$ equals its weighted integral over $\partial D(P,r)$. I am wondering if ...
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3answers
36 views

Finding the sum of the trigonometric serie:

There are two series: $$1) 1+\dfrac{\cos{x}}{p}+\dfrac{\cos{2x}}{p^2}+...+\dfrac{\cos{nx}}{p^n}=\sum_{k=0}^{n}{\dfrac{\cos{kx}}{p^k}}$$ $$2) ...
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21 views

Laurent's expansion of a given complex function

A function $f(z)=\frac{\sin{z}}{(z-\frac{\pi}{4})^3}$. Find the laurent's expansion of this function. The annulus is given $0\lt |z-\frac{\pi}{4}|\lt 1$.
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17 views

What is $P$ and $X$ is supposed to be in this analysis question?

Source page 626. Can someone explain what is $\| P\| $ mean? Is that partition or what? Also why bother with $\epsilon/2$ if the giant expression in the middle proves the lemma. Finally, ...
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31 views

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?, being $A,B \in \mathbb C$ How can it be described?
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2answers
39 views

Question related to Liouville's Theorem

Prove or disprove: Let $D$ be an unbounded domain in $\mathbb{C}$. If $f$ is a bounded, analytic function on $D$, then must $f$ be constant on $D$? This is clearly related to Liouville's Theorem, yet ...
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13 views

A curious map on the complex plane

Given a fixed $z\in\mathbb C$, I am considering the map $$ g_z:w\in\mathbb C\mapsto \frac{\bar w-z}{w-z}. $$ Does anybody have seen such a function somewhere ? Any interesting properties ? I feel ...
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19 views

Existence of nice exhaustion - Rudin.

This is taken from Rudin's Complex Analysis/Real Analysis Can someone tell me why $K_n \subset \Omega$? I agree it is compact, but why does it follow that it is a subset of $\Omega$? WLOG, I ...
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1answer
29 views

Calculating the residue of a function

Let $f(z) = \frac{1+z}{1-\cos(z)}$ I wish to calculate the residue of $f$ at $0$, $2\pi$ and $-2\pi$. I believe this can be done by the following since $f$ has simple poles at these points $Res(f, ...
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1answer
24 views

Why Riemann Mapping Theorem is not valid for $U=\mathbb{C}$

If we take simple connected domain $U=\mathbb{C}$, in the statement of Riemann mapping theorem, then why is it not valid. What is the proper justification?
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26 views

Growth rate of an infinite product

I have an (infinite) Blaschke product in the upper half plane, $$ f(z) = \prod_k \frac{z-z_k}{z-z_k^*}. $$ The zeros $z_k$ of the function are complex numbers in the upper half plane. Suppose the ...
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1answer
29 views

Evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle.

I need to evaluate $\int_\gamma z^ne^{1/z}dz$, where $\gamma$ is the unit circle traveled in the counterclockwise direction. I'm thinking about writing the function as a Laurent series and then ...
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43 views

Line integrals of $\frac{1}{z^2}$ and $\frac{e^z}{z}$ without calculations.

I have learned that : 1) if $f$, a holomorphic function on $U\subset \mathbb{C}$, open and simply connected, then $f$ has a holomorphic antiderivative on $U$ 2) if $f$, a holomorphic function on ...
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1answer
15 views

sub mean value property of plurisubharmonic function

It is well known that a plurisubharmonic function $\varphi$ defined in a domain $\Omega\subset \mathbb C^n$ satisfies the sub mean value property. Now if $\varphi$ is defined on a complex manifold ...
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1answer
22 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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23 views

continuity of the complex square root function

I want to show that there is no continuous square root function in the complex plane, i.e. a function $f:\mathbb{C}\rightarrow\mathbb{C}$ with $f(w)^2=w$ for all $w \in \mathbb{C}$. I already ...
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1answer
32 views

A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
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3answers
52 views

How to show there exists $E$ such that $E \cap K_n$ is dense for every $n$?

Let $\Omega$ be a region (nonempty connected open subset of the complex plane). Let $K_n$ be a sequence of compact sets whose union is $\Omega$, such that $K_n \subset \mathring{K_{n+1}}$ (the ...
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2answers
35 views

Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$

Actually I need only the $res(f;0)$ where $f = e^{e^{\frac{1}{z}}}$ I thought of finding the Laurent series of $e^{e^{\frac{1}{z}}}$ around $z=0$ Any other Ideas if you have ?
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28 views

Suppose $a \in \mathbb{C}$, $|a| < 1$, and $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. How to prove dependence of $|f(z)|$ on $|z|$? [duplicate]

Let $a \in \mathbb{C}$, $|a| < 1$. Also let $f(z) = \dfrac{z - a}{1 - \overline{a}z}$. I am asked to prove that $|f(z)| < 1$ if $|z| < 1$ and that $|f(z)| = 1$ if $|z| = 1$. What is a good ...
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43 views

Complex/Real Analysis mysterious quantity.

The following is a lemma to prove the Runge (only excerpt) page 629 in the link Can someone explain how they got the $$\frac{b_1 - b}{(z-b)}.$$ I believe he took $$\frac{1}{z - b}$$ and the other ...
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1answer
11 views

Heine borel theorem on the complex plane

I'm trying to understand this proof of the Heine-Borel theorem on the complex plane. I'm reading Lang's Complex Analysis (page 22): I didn't understand the converse. Why there is a convergent ...
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1answer
20 views

Please could someone check my results for principal values of the complex logarithm?

I solved an exercise in my book and would greatly appreicate it if someone would check my result and tell me if it is correct:. The exercise: Find the principal values of the logarithm for the ...
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1answer
24 views

Showing this is analytic and finding its derivative $f(z)= \frac{4z+1}{z^3 - z}$

How to show the following is analytic and find it's derivative? $$f(z)= \frac{4z+1}{z^3 - z}$$ I am having trouble solving the above, since I am not sure how to break this into terms of $u,v$ for my ...
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2answers
24 views

Determining whether $f(z)=\ln r + i\theta$ (with domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$) is analytic [duplicate]

Define $$f(z)=\ln r + i\theta$$ on the domain $\{z:r\gt , 0\lt \theta \lt 2\pi\}$. This domain is just a punctured disk of radius $\ln r$, correct? How does one determine whether this is ...
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Proving $\prod_k \sin \pi k / n = n / 2^{n-1}$

I am stuck trying to prove $$\prod_{k=1}^{n-1} \sin {\pi k \over n} = {n \over 2^{n-1}}$$ and I'd appreciate help. What I have done so far: $z^n - 1 = \prod_{k=1}^n (z - \xi^k)$ where $\xi = ...
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1answer
18 views

IS $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable?

Where is $f(z) = x^3 + i(1-y)^3$ analytic and where is it differentiable? I have taken Cauchy-Riemann equations as follows: $$u(x,y) = x^3$$ $$v(x,y) =(1-y)^3$$ $$\frac{\partial u}{\partial ...
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1answer
25 views

Using de l'Hopital for complex functions?

I was wondering if de l'Hopital's rule also applies to complex functions. Some background information: This question came up as I was trying to calculate $\displaystyle \lim_{z \to 1} {z^n - 1 \over ...
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1answer
13 views

Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$

Assume that a sequence $(z_n)$ of complex numbers converges to a nonzero limit. Then Prove that there is a postive integer $n_0$ such that all the $z_n$ are nonzero for $n \le n_0$ I know I should ...
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1answer
29 views

Prove that $|\int_C f(z)dz| \le M |z_2 - z_1|$ where $M \gt 0$ such that $|f(z)|\le M; \ \forall \ z \in \Omega$

Let $z_1$ and $z_2$ be any two points in $\Omega$ and let $C$ be any oriented contour in $\Omega$ from $z_1$ to $z_2$. Also, assume that $f:\Omega \to \Bbb{C}$ is analytic on an open convex set ...
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16 views

Use of Cauchy's intergral theorem (and consequences).

Let $f$ be an analytical function, with $|f(z)|\leq\displaystyle\frac{1}{1-|z|}$ for $|z|<1$. I have to prove that : ...
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Find the set of points on which the maps of $e^z$ and $\log(z-1)$ are expanding and contracting.

I understand that $e^z$ is has a domain $\Omega$ such that $\Omega = \Bbb {C}$ and is analytic on the whole complex plane, but I have never been tasked with understanding the map of a function that is ...
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1answer
34 views

$f$ holomorphic, calculate $f(1+i)$ with two informations about $f$

Let $f(x+iy)=u(x,y)+iv(x,y)$ be a holomorphic function, knowing that : 1) $Im(f'(x+iy))=6x(2y-1)$ 2) $f(0)=3-2i$ Find $f(1+i)$. There is nothing in my notes, but I have read online that ...
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A metric such that every Blaschke factor is an isometry

Let $\mathbb{D} := \{z \in \mathbb{C} : |z| < 1\}$. Define $d : \mathbb{D}\times \mathbb{D} \rightarrow \mathbb{R}$ to be $$ d(z,w) := \left| \frac{z-w}{1-\overline{w}z} \right| $$ I am supposed to ...
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2answers
33 views

Find image of complex set:

Find image of set: $$ \{ z \in C : 0 \le Im (z), 0 \le Re(z) \}$$ and $$f(z)=\frac{i-z}{i+z}$$ I caclulate $ w=\frac{i-z}{i+z} $ and then $z=\frac{i(1-w)}{w+1}$ and don't know what to do next... I ...
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Analytic nowhere - C-R equations satisfied on line $(0,y)$

Analytic nowhere - C-R equations satisfied on line $(0,y)$ I have the C-R equations satisfied on the above line, but I imagine it is still analytic nowhere, since there is no open neighborhood where ...
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18 views

Fourier Series for a conformal map on unit disk

Given that a conformal map on the disk $\mathbb{D}$ will always have the form $f(z)=\lambda \displaystyle\frac{z-w}{1-\overline{w}z}$ for some $\lambda\in \partial \mathbb{D}$ and some $w\in ...
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21 views

Solving an equation with complex numbers

I want to use complex numbers to solve the following problem: $x^2 = 95 - 168i$. I am sure there are a few ways of doing this but the way I want to do it is to let $x = a + bi$ and then solve for $a$ ...
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1answer
29 views

Image of Möbius transformation

What's the image of the first quadrant $Rez\ge0$ and $Imz\ge0$ under transformation $f(z)=(i-z)/(i+z)$? I know that real axis is mapped to the unit circle, $f(0+i*0)=1$ and $f(\infty)=-1 $.
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Showing $f(z) = e^ye^{ix}$ is defined on all $\Bbb C$

Now I asked how to determine if a function was defined on all of $\Bbb C$ in my previous question, but since that function was a polynomial, the answer was: Yes because its a polynomial, what if it ...
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22 views

Geometric solution for $\frac{1-R}{| 1 - R e^{j\theta} |} = k$

Given: $\frac{1-R}{| 1 - R e^{j\theta} |} = k$ How to solve for R? (Suppose R is the only unknown quantity -- the task is to rearrange with R as subject). I encountered this problem in an academic ...
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27 views

Showing $f(z)=2xy+i(x^2+y^2)$ is defined on all of $\Bbb C$

What does it mean to be defined on all of $\Bbb C$? That is has no points at infinity? How do I show the below is defined on all of $\Bbb C$? $$f(z)=2xy+i(x^2+y^2)$$ Is it something to do with ...
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1answer
55 views

$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$

How can I make this rigorous? $$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$$ Proof: $$\lim_{z\to0}\frac{x+iy}{x-iy} \text{ taking } y\ne 0, x\to 0 \implies ...
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25 views

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $. This problem is in pg 137 in Classical Topics in Complex function ...
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18 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
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2answers
32 views

How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$

Let $f : C \mapsto R $, $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ?
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23 views

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
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3answers
69 views

How to finish proof of $ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I'm trying to prove the identity $$ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I've done so far: From geometric series ...