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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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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70 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
27 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
20 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,\\ ...
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42 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
19 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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28 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
49 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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37 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
35 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) = $\infty$ ...
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1answer
15 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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43 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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1answer
31 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
35 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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25 views

complex taylor series

I have a series $ f(z)= 1 - z + z^2 - z^3 $ and i want to substitute $ z=b + (z-b) $ into the equation, (where $b=1/2+i/2$) and find the first two coefficients. Wont they just be the same as before ...
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37 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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19 views

complex function and change of a rectangular region to a circle region with mapping

I was wondering if anyone knew a mapping that would take the region inside a rectangle of length "a" and width "b" and map it into the region inside a circle with radius R.
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3answers
22 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
35 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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55 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
28 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 ...
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1answer
38 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
33 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
46 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
24 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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33 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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33 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) infinity (b) 0 (c) 1 (d) 2 I think, the answer is ...
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1answer
48 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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1answer
66 views

Laurent series expansion, can one simplify this?

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

Is $x = 2014$ is the solution of inequality $|z_3-z_1|^x$+$|z_3-z_2|^x \le |z_1-z_2|^x$?

Let $z_1,z_2,z_3$ be distinct complex numbers such that $Re \frac {z_1-z_2}{z_2-z_3}=0$. Is $x = 2014$ is the solution of inequality $|z_3-z_1|^x$+$|z_3-z_2|^x \le |z_1-z_2|^x$?
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1answer
20 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: ...
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2answers
21 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
21 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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0answers
30 views

Explicit analytic continuation of the zeta function and shift operators

Is there a way to compute the radius of convergence of the expansion of the zeta function, e.g. around $a=2+2i$? We have $\zeta(a)=\sum_{n=1}^\infty n^{-a}\approx 0.867.. - i\,0.275.. $, but I ...
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1answer
18 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 ...
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0answers
16 views

How to determine the order of the poles of this particular function?

Consider the function $$f(z) := \frac{1}{z^{2n}+1} \quad .$$ The denominator is equal to zero when $z^{2n} = -1$, so its zeros $z_{k}$ are located at $z_{k} = e^{\frac{\pi i}{2n}+\frac{k \pi i}{n}} $. ...
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2answers
28 views

Find all functions $f\in O$ ( in some neighbourhood of $0$) such that $|f(z)|^2=1+|z|^2$

Find all functions $f\in O$ ( in some neighbourhood of $0$) such that $|f(z)|^2=1+|z|^2$. I was thinking about maximum modulus principle, but $|f|$ does attain minimum in $0$, not maximum. Any ...
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1answer
22 views

Residue of $g(z)$ at z=0 simple pole

Find the residue of: $$g(z) = \frac{\psi(-z)}{z(z+1)^2} \space \text{at} \space z = 0$$ My Attempt: Because $z=0$ is a simple pole, I thought of using the definition. $$\mathrm{Res} \space _{z=0} ...
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1answer
40 views

Let P be a polynomial in R^2 and $\oint$ P(x,y)(dx + idy) = 0 for every circle $\gamma$ in $\mathbb{C}$. Prove that P is $\mathbb{C}$-differentiable.

I approached this question by first proving the Cauchy integral formula and then using that to get the taylor series expansion. But i am facing difficulty in getting the taylor series expansion.
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1answer
18 views

Finding the limit of complex functions

I'm having trouble with finding the limit of the following complex function (and complex functions in general): $$\lim_{z\to0}\frac{ze^{i/z}}{(z^2+a^2)^2}$$ What I've tried is writing $z$ as ...
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1answer
16 views

Computation - dominated convergence theorem

Consider the following equality: $\sum_{m\in\mathbb{Z}}\sum_{n\in\mathbb{Z}} \int_{n}^{n+1}f(u)e^{-2\pi im u} du =\sum_{m\in\mathbb{Z}}\int_{-\infty}^{\infty} f(u)e^{-2\pi im u}$ From my notes I ...
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20 views

Let $f \in O(D)$ for some domain $D$. Prove that if $|f(z)|$ is a constant, then $f(z) =$ const on $D$. [duplicate]

Let $f \in O(D)$ for some domain $D$. Prove that if $|f(z)|$ is a constant, then $f(z) =$ const on $D$. It seems to me it is the direct application of the following version of maximum modulus ...
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0answers
18 views

Residue of a rational function

In this answer by Jack D'Aurizio, which is fantastic, I do understand that: If $f(z) = (\psi(-z) + \gamma)^2$ where $\psi(-z)$ is digamma, and $H_n$ (following) is harmonic number: $$\mathrm{Res} ...
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1answer
22 views

Mappings from the z plane into the w plane

I have solved the following problem , but I am not sure if my solution is correct : A Square $S$ has vertices $(0,0), (1,0), (1,1), (0,1)$. Part 1: Determine the region in the $W$ plane which $S$ is ...
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0answers
21 views

How to visualize bilinear transform of the form $S(z) = \frac {T}{2} \frac {z+1}{z-1}$

Note that this is the bilinear transform from a z-domain as appears in Z-transform to s-domain in Laplace transform Recall that bilinear transform has form $M(z) = \frac{az+b}{cz+d}$ with and has to ...
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4answers
92 views

Considering $ (1+i)^n - (1 - i)^n $, Complex Analysis

I have been working on problems from Complex Analysis by Ahlfors, and I got stuck in the following problem: Evaluate: $$ (1 + i)^n - (1-i)^n $$ I have just "reduced" to: $$ (1 + i)^n - (1-i)^n = ...
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1answer
19 views

Could someone point me in the right direction for this complex analysis equation?

I'm supposed to show that the maximum value of $|z^2+1|$ on the unit disk $|z|\leq1$ is 2. My teacher's hint was "triangle inequality". I've been racking my brain how to tie the triangle inequality ...
1
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1answer
23 views

How to factor polynomials with complex coefficients

Factor completely the polynomials $$p(x)=5ix^4-(9+2i)x^3+7x+6-i$$ and $$q(x)=9x^5-x^3+7x+6$$ First, I tried to use the Fundamental Theorem of Algebra but it did not work out. ThenI tried plugging in ...
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2answers
44 views

How would one prove this flavour of the triangle inequality?

I have to prove $|z_1 - z_2| \leq |z_1|+|z_2|$ where $z_1,z_2$ are in $\mathbb{C}$. What I wrote down is: $$|z_1| = |z_1+z_2-z_2| \geq |z_1-z_2|-|z_2|\implies |z_1|+|z_2|\geq |z_1-z_2|,$$as desired. ...
0
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1answer
35 views

Determine Radius of Convergence

I am trying to determine the radius of convergence for the series $$\sum_{k=0}^\infty\frac{z^{k^m}}{k!}\qquad(m\in \mathbb{N})$$ with no success. I tried to used the root test to determine it but ...