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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102 views

Is a connected Reinhardt Domain which containg $0$ necessarely a polydisc?

I'm studying several complex variables basics. Roughly speaking: call $D\subseteq\Bbb C^n$ the set of points in which a given power series $$ \sum_{\alpha\in\Bbb N^n}a_{\alpha}(z-z_0)^{\alpha} $$ ...
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1answer
164 views

Schwarz Reflection Principle vs. Analytic Continuation

Analytic continuations are unique on simply connected domains: $$F,F':\Omega\to\mathbb{C}:\quad F\restriction=F'\restriction\implies F=F'$$ Schwarz reflection principle offers analytic continuations ...
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2answers
44 views

Analyticity: Uniform Limit

Problem Consider a uniformly bounded sequence over the real line: $$f_n:\mathbb{R}\to\mathbb{C}:\quad|f_n(x)|\leq L$$ Suppose they have analytic continuations with common domain: $$F_n:\Omega\to\...
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2answers
47 views

complex numbers equality question

Let $a$ be given a complex number. Show that $$\left|\frac{z-a}{1-a^*z}\right|=1$$ for $z$ with $|z|=1$ and $a^*z\neq 1$. If $|z|=1$, that means $z$ can be equal to $i$, $-i$, $1$ or $-1$ right? ...
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1answer
83 views

Mean Value Theorem for $f: \mathbb{R} \rightarrow \mathbb{C}$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be a continuous and differentiable function on $[a, b]$. Then does there exists a $c \in (a,b)$ such that $$\frac{|f(b) -f(a)|}{b - a} \leq |f'(c)|?$$
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112 views

Express $\sin(z)$ and $\cos(z)$ in Rectangular Form

"Express $\sin(z)$ and $\cos(z)$ in rectangular form." For $z \in \mathbb{C}$ (complex numbers), we have defined \begin{equation} \sin (z)=\frac{e^{iz}-e^{-iz}}{2i} \end{equation} and \begin{equation}...
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2answers
49 views

Plotting on a complex plane

I'm very confused how you would plot the relationship $|z-4| \leq |z|$. I tried to change it in form which could become $-|z|\leq|z-4|\leq|z|$ and I guess the same can be done for z-4. But I don't ...
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46 views

Computing Principal Logarithm on Different Intervals

Compute the principal logarithm of a complex number $z=\sqrt{3}+i$ using $\mathrm{Arg}(z) \in [0,2\pi)$ and $\mathrm{Arg}(z) \in (-\pi,\pi]$. Wikipedia shows how the answer can be different for the ...
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1answer
86 views

Question about determining accumulation points

So far the way I have determined accumulation points of given sequences or relations has been by drawing them out. However I would like some clarification to see if my thinking is correct or not. a) ...
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70 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
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47 views

Problem with complex derivative

I have to find all points where function $f(z)=\mathbb{Re}z \cdot |z|$ is complex differentiable. CR equations arent satisfied in points $\mathbb{C} \setminus \{0\}$. So Im calculating derivative at ...
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1answer
82 views

Prove $\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$

I have to prove that for $f(x,y)=e^{-xy^2}\sin(x)$ and $\forall b>0$ we have $$\int_0^b \left(\int_{0}^\infty f \,dy\right) dx= \int_0^\infty \left(\int_{0}^b f \,dx\right) dy$$ I've tried to ...
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1answer
60 views

Taylor expansion of $1/(1+z)$

How do I obtain the Taylor expansion of $$\frac{1}{1+z}$$ about $a=i$ please? Do I just expand the series using the binomial expansion?
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2answers
51 views

Trivial complex analysis proof

Question: Prove that if $z, w, v ∈ C$ and $zwv = 0$ then at least one of $z$, $w$ and $v$ must be $0$. My thought was that first, I would assume that $zwv=0$ and that $z,w,v\neq0$ This leads to a ...
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1answer
104 views

Conditions To Make Complex Numbers $z_1, z_2, z_3, z_4$ Vertices of a Square

Let $z_1,z_2,z_3,z_4\in\mathbb C$ be distinct. State conditions in terms of computation of complex numbers, which make $z_1,z_2,z_3,z_4$ vertices of a square (in the counterclockwise direction). ...
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65 views

Limit involving branch point of a complex function

I am having trouble with the following problem : If we restrict ourselves to that branch of $f(z)= \sqrt{z^2+3}$ for which $f(0)=\sqrt 3$ , prove that $$\lim_ {z\to 1}\frac{\sqrt{z^2+3}-2}{z-1} = 1/...
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0answers
182 views

Prove there is no branch of arg $z$ on $0 < z < 1$.

If $G$ is an open connected subset of $\mathbb{C}$ that does not contain the origin, we call a continuous function $\alpha$ satisfying $\alpha(z) = \text{arg} z$ for all $z \in G$ a branch of arg $z$. ...
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1answer
45 views

What does “$C^{\infty}$” convergence mean?

I'm studying first notions about several complex variables. As a consequence of the (generalized form) of the Cauchy esteem for holomorphic functions, the book says that in the space $\mathcal H(\...
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1answer
258 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
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90 views

Isolated singularity is removable iff $\lim\limits_{z\to z_0} (z-z_0)f(z)=0$

Could someone explain a step in the following proof? Theorem An isolated singularity $z_0$ of $f$ is removable if and only if $\lim\limits_{z\to z_0} (z-z_0)f(z)=0$. Proof ($\Leftarrow$) ...
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2answers
92 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
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1answer
42 views

Holomorphic function is zero on an analytic set then $df=0$.

Assume we have an homomorphic function $f:U\rightarrow \mathbb{C} $ which is holomorphic on the open set $U$ of $\mathbb{C}^n$. Assume there is $V\subset U$ analytic and that $f$ restricted to $V$ ...
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1answer
194 views

Green's function for Dirichlet problem on a half disk

Let $D=\{z=(x,y):x^2+y^2<R^2, y>0\}$ be the half disk with radius R. Then if we consider the Dirichlet problem on this domain, i.e., we want to find $$ \Delta u=0, ~~z\in D,\\ u=f,~~z\in\...
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51 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is $\...
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1answer
27 views

Deriving definition of the complex logarithm

Given that: $$z = Re^{i\theta} = R(a + bi) = R\left( \cos(\theta) + i\sin(\theta) \right)$$ In its polar form. $$\log(z) = \log(R) + i\theta$$ $$|z| = \sqrt{(Ra)^2 + (Rb)^2} = R\sqrt{a^2 + b^2} ...
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0answers
32 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
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2answers
284 views

Exercise: Evaluating integration $\int_{|z|=r} \frac{1}{(z-a)(z-b)}dz$, $|a|<r<|b|$

This is an exercise from Stein-Shakarchi's Complex Analysis: evaluate integration $$\int_{|z|=r} \frac{1}{(z-a)(z-b)}dz, \,\,\,\, |a|<r<|b|. $$ The problem I am facing is the following. It is ...
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0answers
44 views

Compute the complex integration [duplicate]

Let, $f(z)$ be an analytic function. Then the value of $$\int_{0}^{2\pi}f\bigl(e^{it}\bigr)\cos t dt= ?$$ (a) 0 (b) $2\pi f(0)$ (c) $2\pi f'(0)$ (d) $\pi f(0)$. $\mathcal{My}{Attemt}:$ ...
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1answer
43 views

Complex analysis basics

If I z = x + yi and w = f(z), describe the image R of D in the w-plane when $$0<x<\pi/2, 0<y<\infty;w = e^{iz}$$ Wouldn't this mean that in the w-plane the argument arg(w) = $\infty$ ...
3
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1answer
76 views

using Taylor's Theorem to find region of convergence of series

!(http://imgur.com/0fDL4KZ) I am a third year Electrical engineering student, and I was going through one of the example from my math module lecture notes but couldn't understand the solution printed ...
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65 views

Complex integration misconception?

Playing around with the complex integretion I encountered the following: Consider a holomorphic function $f(z)$ on $\Omega$. Let's say this holomorphic function has a primitve $F(z)$ such that $F'(z) ...
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3answers
117 views

Lagrange inversion theorem application

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show ...
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1answer
77 views

Prove that indicator function of integer numbers is positive semidefinite

How to prove that the function $\mathbb{1}_{\mathbb{Z}}(x)$ is positive semidefinite? I.e. to show that for any $n = 2, 3, ...$ and $x_1, ..., x_n \in \mathbb{R}$, $z_1, ..., z_n \in \mathbb{C}$ $$...
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0answers
80 views

A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
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3answers
83 views

Find the real and imaginary part of the following

I'm having trouble finding the real and imaginary part of $z/(z+1)$ given that z=x+iy. I tried substituting that in but its seems to get really complicated and I'm not so sure how to reduce it down. ...
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1answer
53 views

Series does not converge [closed]

How would I go about showing that the series$$\sum_{n + m\tau \in \Lambda} {1\over{{|n + m\tau|}^2}}$$does not converge, where $\tau \in \mathbb{H}$?
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2answers
681 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
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1answer
78 views

Describe the set of all $z$ such that $Im(z+5)=0$

Describe the set of all $z$ such that $Im(z+5)=0$. So here is what I have so far. $$ z = a+bi $$ $$ Im(z+5) = Im(a+bi+5)= b$$ Now does this imply that $b=0$ because if we have $$Im(z+5)=b$$ and $$Im(...
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1answer
66 views

False proof of 0=1 using Laurent series

I found the following proof that 0 = 1: \begin{align*} \sum_{n=-\infty}^{\infty} 0\cdot z^n = 0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}} + \frac{1}{1-z} \\ = \frac{1}{z} \...
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2answers
78 views

Understanding complex functions in w - and z - plane

I have a difficulty understanding the basics of complex functions. My exercise looks like this: "The $z$-plane region $D$ consists of the complex numbers $z = x + yi$ that satisfy the given ...
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2answers
50 views

Confusion with Complex Equations

I have the equation $z^6+8=0$. So what I did was I turned it into $z=\sqrt2(\cos(\frac{\pi}{6})+i\sin(\frac{\pi}{6}))$ Now here is where I get confused. Do I simply input this into De Moivre's formula ...
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1answer
134 views

Using contour integrals to evaluate $\int_{-\infty}^{\infty} \frac{e^{ax}}{\cosh x} dx$

Problem: Suppose that $a \in \mathbb{C}$ and that $ Re(a) \in (-1,1)$. Evaluate $\int_{-\infty}^{\infty} \frac{e^{ax}}{\cosh x} dx$ by considering the rectangular contour with vertices $\pm R$, $\pm ...
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1answer
82 views

Locally Bounded Sequence of Functions

Let $f: U\to U$ be an analytic function with $0$ in $U$, where $U$ is an open connected subset of the complex plane. Let $f(0)=0$ and $|\frac{d}{dz}\,f(z)|=|f'(z)|<1$ for all $z$ in $U$. Define ...
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1answer
57 views

How many analytic functions are there on a given set

Consider the set $S=\{0\} \cup \bigl\{\frac{1}{4n+7}:n=1,2,...\bigr\}.$ Then the number of analytic functions which vanishes only on $S$ is (A) Infinite (B) $0$ (C) $1$ (D) $2$ I think, the ...
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1answer
96 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
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2answers
97 views

Laurent series expansion, can one simplify this?

I have to expand $f(z)=\frac{z-1}{(z^2+1)z}$ in an annulus $R(i,1,2)$. $$f(z)=\frac{1}{z-i}\frac{1}{z+i}-\frac{1}{z-i}\Big(\frac{i}{z+i}-\frac{i}{z}\Big)$$ $$\frac{1}{z-i}\frac{1}{z+i}=\frac{1}{...
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1answer
465 views

Find annulus of convergence of Laurent series

Find annulus of convergence of Laurent series $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^3}$ My answer: $0<|z-i|<\infty$ $\sum_{-\infty}^{\infty}2^{-n^2}(z-i)^{n^2}$ My answer: $|z-i|&...
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2answers
145 views

how to write eqn of line in complex form

Write the given equation of a straight line in complex notation: Straight line through 1 and (-1 - i) Attempt: So i treated this initially just like a set of coordinates in the set of R thus (1,0) ...
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1answer
36 views

Prove the inequality $| \frac{z}{|z|}-1| \leq |arg(z)|$

Prove the inequality $| \frac{z}{|z|}-1| \leq |arg(z)|$ Here is what I got $z=r(cos \theta +i sin \theta)$. So $LHS= ((\cos \theta -1)^2 +sin^2 \theta)^{1/2}=(2-2\cos \theta)^{1/2}$ Note that $-1 \...
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1answer
44 views

Difficult Limit involving digamma function

Evaluate: $$\lim_{z \to 0} \psi(-z)\cdot \bigg ( 1 - 2z(z+1) \bigg) - z\cdot\psi'(-z) $$ If we simply substitute in $0$ that gets us infinity, and problems. The answer is $-2 - \gamma$ How do we ...