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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Complex function, analyticity

Consider a function from $\mathbf{C}^2$ to $\mathbf{C}$ is defined as $$ f(z,w)=\frac{\alpha}{z} + \frac{\beta}{w} $$ where the parameters $\alpha, \beta$ are complex numbers. Is it analytic in ...
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10 views

Derivation of Poisson - Jensen formula

While reading the book of Conway, I found this version of Poisson- Jensen's formula which states: Let $f$ be analytic in a region $\overline B(0,R)$ . Let $a_{1} , a_{2} , ... , a_{n}$ be the zeros ...
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13 views

mobius transformation with 2 points

How find mobius transformation w, if known that w(-1)=-2 and w(i-2)=1+3i (z: Im z > 0 and w: Im w > 0) I know how do it for 3 known points, but have no idea for 2. Thanks.
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1answer
9 views

Deriving an expression for an n-th composition of some Mobius transformation

Let $T(z)=\dfrac{1-3z}{z-3}, T^2(z)=T(T(z)),..., T^{n+1}(z)=T(T^{n}(z)),n=1,2....$ Could anyone advise me on how to find an expression for $T^n(z) \ ?$ I'm trying to make use of the fact that there ...
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51 views

Is $\left\{ e^{ \frac{2\pi i }{n}}: n\in \mathbb{N}\right\}$ compact in complex plane?

Is $\{ e^{ \frac{2\pi i }{n}}: n\in \mathbb{N}\}$ compact in complex plane? My answer is yes. It is bounded as $$ \left|e^{\frac{2\pi i}{n}}\right|=1$$ and the set is closed because it contains its ...
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1answer
22 views

Proving the reverse triangle inequality of the complex numbers

I'm having trouble understanding this proof for the reverse triangle inequality of the complex numbers. Suppose for any $z, w \in \mathbb{C}$, we have $|z + w| \leq |z| + |w|$ (the triangle ...
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27 views

Inequality about complex plane.

$z_1$ and $z_2$ are two point in the complex plane. how to prove in a more elegant way: ...
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20 views

problem on sum and difference of entire function

Let $\mathbf {f:C \rightarrow C} $ be an entire function and let $\mathbf{ g:C \rightarrow C }$ be defined by $g(z)=f(z)-f(z+1)$ for $z \in C$ Which of the following statements are true? justify. ...
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24 views

maximum, complex quadratic function, Is my solutions correct?

I'm trying to compute $\max_{|z| \le 1} |(z+2)(z-1)|$. Here's how I do it: $\{z \in \mathbb{C} \ | \ |z| \le 1 \}$ is compact and $f(z) = (z+2)(z-1)$ is continuous, so it suffices to look for ...
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67 views

How to find the roots of $(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$

Write down, in any form, all the roots of the equation $z^5 − 1 = 0$ Hence find all the roots of the equation $$(w−1)^4 +(w−1)^3 +(w−1)^2 +w=0$$ and deduce that none of them is real ...
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13 views

how to find the branch points and cut

for $\sqrt{z^2+1}$, how can I find the branch points and cuts? I let $z=re^{i\theta+2n\pi}$ and substitute into $$\sqrt{r^2 e^{i(2\theta +4n\pi)}+e^{2k\pi}}=$$ then, I don't know how to deal with ...
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2answers
36 views

Property of a given entire function

Let $f(z)$ be an entire function and let $\lvert f(z)\rvert \le \lvert z\rvert$, for all complex z. Show that then $f(z) = \alpha z$ for some constant $\alpha$. I feel like I need to use the maximum ...
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1answer
44 views

How do I find a constant for a polynomial so its roots are reflective around a linear function?

How can I find all complex numbers $w$ so that the roots of the following polynomial are reflected around a linear function $f(x)$ $$p(q) = q^2-4q+w = 0$$ If I want to find all the complex numbers ...
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15 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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1answer
13 views

Finding hermitian conjugate and inverse of a complex matrix

I have the following matrix: $$ F = [e^{i\frac{2\pi kl}{n}}]^{n-1}_{k,l=0} \in \mathbb{C}^{n,n} $$ for $n = 1,2,3,...,i$ I need to find $F^HF$ and $F^{-1}$ where $F^H$ is a hermitian conjugate ...
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34 views

Generalized circles on $\mathbb{C}\cup\{ \infty\}$

A generalized circles on $\mathbb{C}\cup\{ \infty\}$ are circles or line. If $C_1$, $C_2$ and $C_3$ are circles (only circles) such that they intersect tangentially out two to two, then show that ...
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1answer
23 views

Variant Rouche's theorem

Set $\mathbb{D}=D_1(0)$. Let $f$ & $g$ holomorphic functions in a neighborhood of the disc $\mathbb{D}$ such that $f(z)\not=0$ and $\frac{g(z)}{f(z)}\notin(-\infty,0]$ for all ...
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1answer
19 views

Third degree polynomial with unknown coefficients $q^3-3aq^2+b^2q+c = 0$

For an equation $q^3-3aq^2+b^2q+c = 0$ we know the roots $c, (a+b), (a-b)$. What is a good place to start with such equations? I've tried setting up a system of equations, but this is supposed to be ...
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1answer
34 views

Flaw in this proof that the union of two open sets is open?

I'm trying to show that if U, V are open sets of $\mathbb{C}$, then U $\cup$ V is an open set of $\mathbb{C}$. My attempt at proving this is as follows: If $U$ is open, $\forall x$ $\in$ U, ...
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31 views

What's the inverse of the Weierstrass-Mittag-Leffler-Transform $\exp\left[\int_\mathbb C f(y)\ln(z-y)\,dy\right]$?

As mentioned in another post, as a consequence of Mittag-Leffler's theorem combined with the Weierstrass factorization theorem, after reducing to the common denominator, any meromorphic function can ...
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20 views

Classification singularity

I have to classify the singularity of the complex function $$f(z) = z \sin(1/z).$$ I already saw that zero is a essential singularity of $f$. But I can't determine the $$f(\{z \in \mathbb{C} : 0 ...
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40 views

Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.

This is a problem from my complex analysis textbook. The hint is to consider $g(z)=(z-iR)^2(z+iR)^2 f(z)$ and to show that $|g(z)| \leq 8R^2$. This is what i have tried: Consider $Re z \geq 0$, then ...
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2answers
30 views

Real part of a complex number divided: $\Re\frac{z+1}{z-1}=0$

$\Re\frac{z+1}{z-1}=0$ I've tried so many methods, they all end up with two variables $a, b$. I tried setting $z= a+ib$. This give me the equality $2\cdot\Re\frac{z+1}{z-1} = ...
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1answer
45 views

A suitable integration path for $\cos z/(2 + \cos z)$

I was solving the exercises and got stuck when trying to solve this with tools of residual calculus $$ \int_{0}^{2 \pi} \frac{\cos (z)}{2 + \cos (z)} \, dz = \int_{0}^{2 \pi} f(z) \, dz. $$ Isolated ...
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21 views

Complex number arguments question

Given that u = -3i, how would I go about tackling these questions: (ii) For complex numbers 􏰀 satisfying arg(z􏰀 − u) = 0.25π, find the least possible value of |􏰀z|. (iii) For complex numbers 􏰀 ...
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25 views

Inverse Laplace transform of $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$

I have been desperately trying to find the inverse laplace transform using the complex inversion formula for this question. $\frac{1}{s} \frac{\sqrt{s}-1}{\sqrt{s}+1}$ I have found it extremely ...
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17 views

Fourier transform and conjugate variables

When you make a Fouriertransform of a function of time $f(t)$, it is said that it's Fouriertransform is a function of frequency $\widetilde{f}(\omega)$. The same argument goes for position and ...
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How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
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1answer
50 views

Is there a quicker way to write $\cos (n\theta)$ in terms of $\cos \theta$?

Im writing $\cos 8\theta$ in terms of $\cos \theta$ using De Moivre's Theorem $$\cos 8\theta= \Re {(\cos\theta+ i \sin \theta)^8}$$ Let $s=\sin \theta$ and $c=\cos \theta$ $$=c^8 ...
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1answer
23 views

Determine integral by using the following identity (which is imaginairy)?

I want to determine the following integral: $$\int_{-\infty}^\infty \frac1{x^6+1} dx$$ by using the following identity: $$\frac1{x^6+1} = \Im\left[\frac1{x^3-i}\right]$$ How in the world can I do ...
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2answers
28 views

Is an holomorphic function injective if $\| f'(z)\| > 0 $?

Is there any holomorphic function $f$ which isn't injective, even if $ \forall z $ $\| f'(z)\| > 0$ ?
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33 views

How can I find the complex numbers satisfying this condition?

For a given complex number $a$ with $|a|\ge1,$ I want to find the all complex numbers on the unit circle such that $$\dfrac{z}{(a-z\bar a)^2}\in\mathbb{R}$$ and satisfying the condition ...
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Find the Residue of $\frac{e^z}{sin^2(z)}$ at each finite singularity

The problem states: Find the Residue of $f(z)=\frac{e^z}{sin^2(z)}$ at each finite singularity. The poles are clearly at $z=k\pi (k\in\mathbb{Z})$, and the order are all 2, since: $\lim_{z \to ...
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1answer
25 views

Application of Riemann mapping theorem

Let $\Omega \neq \mathbb{C}, \emptyset$ be a simply connected domain and $a \in \Omega.$ Let $f:\Omega \to \mathbb{D}$ be a conformal map such that $f(a)=0, f'(a)>0.$ Could anyone advise me how to ...
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1answer
25 views

Winding number of $e^{ix}$? [on hold]

What is the winding number of $e^{ix}$ where $x\in \mathbb{R}$?
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2answers
102 views

Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$

Prove using contour integration that $\displaystyle \int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$ I am at a loss at how to start this problem and which contour to pick. I ...
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3answers
37 views

why $f$ is holomorphic if $f(z) = \frac{1}{2\pi i} \int_\gamma \frac{f(\zeta)\, d \zeta}{\zeta - z}$?

I'm reading Gong Sheng's Concise Complex Analysis to get some basic understanding. On $\S 2.4$ page 61 Theorem 2.15 (Hurwitz Theorem) it says For an arbitrary point $z \in U \subseteq \mathbb C$, ...
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1answer
32 views

Complex Integration and deduce that function is constant

Let $f$ be an entire function, $z_{1}$, $z_{2}$ $\in$ $C$, with $z_{1} \neq z_{2}$ and $R>\max{(|z_{1}|,|z_{2}|)}$. Prove that $$2\pi i\dfrac{f(z_{1})-f(z_{2})}{z_{1}-z_{2}} = ...
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1answer
27 views

Find the first three terms of the maclaurin series of $\tanh(z)$ and its radius of convergence

This is my first time dealing with maclaurin series of complex variables. Here is my attempt: Since $\tanh = \frac{\sinh(z)}{\cosh(z)}$, the maclaurin series is valid when ...
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1answer
28 views

Analytic onto maps from D to D

We just characterized using the Schwarz Lemma the conformal self maps of the open unit disk. I am now trying to characterize the holomorphic onto maps from $\mathbb{D}$ onto $\mathbb{D}$. As a ...
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1answer
33 views

Trying to evaluate $\prod_{k=1}^{n-1}(1-e^{2k\pi i/n})$ for my complex analysis homework

For my complex analysis homework, I am trying to show that the integral of the real function $1/(1+x^n)$, for integer $n\ge2$, along the positive real line is $$\int_0^{\infty}\frac{dx}{1+x^n} = ...
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Logarithms, principle logarithms in complex plane

How were these solutions reached? (a) $\log2 = \log|2| + i\mbox{Arg}2 +2\pi n i = \log2 + i2\pi n $ (b) $\log i = \log|i| + i\mbox{Arg}i + 2\pi n i = \frac{i\pi}{2} + i2\pi n $ (c) $\log(1+i) = ...
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2answers
40 views

$\int_{0}^\pi \frac{\sin(nx)}{\sin x} dx$

How do I integrate :$\int_{0}^\pi \frac{\sin(n\theta)}{\sin \theta} d\theta $ I did the following: $\int_{0}^\pi \frac{\sin(n \theta)}{\sin \theta}d\theta = \mbox{Im} \int_{0}^{\pi} \frac{e^{i n ...
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1answer
11 views

Prescribing zeros and poles of a rational function on $\bar {\mathbb C}$ at once

I have to show that for any points $P_1$, $\ldots$, $P_n$ and $Q_1$, $\ldots$, $Q_n$ ($P_i \neq Q_j$ for all $i$, $j$) on $\bar{\mathbb C}$ there exists a rational function $f$ with poles at $P_j$, $j ...
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8 views

Parametrizing shapes, curves, lines in $\mathbb{C}$ plane

I've been struggling with parametrizing things in the complex plane. For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how ...
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8 views

Finding a Mobius transformation

Let $R=\{z\in \mathbb{C}: Re(z)>0, |z-3|>1\}$ and $A=\{z \in \mathbb{C}: 1<|z|<p\}.$ Find a Mobius transformation $f$ and $p$ such that $f$ maps $R$ conformally to $A.$ May I verify if my ...
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0answers
23 views

Equations with modulus of a complex variable

I am struggling a bit to solve equations involving the modulus of complex variables. I am given the equation $|z-z_0|=|1-z_0z^*|$, where $z$ is a complex variable, $z_0$ is a complex number and $z^*$ ...
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1answer
29 views

Question about laurent series

Find laurent series for $f(z)=\dfrac{1}{z^2-1}+\dfrac{1}{z(z-1)};z_0=0$ that converges in $0<|z|<1$. I tried to find the solution for the first fraction like this. \begin{align} ...
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1answer
40 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
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1answer
13 views

Help needed to establish a conformal mapping

Could anyone advise me on how to find a conformal map from $H=\{z \in \mathbb{C}: Re(z)>0\}$ to $A= \{z \in \mathbb{C}:|z|>1, |z-2|<3\} \ ?$ I tried to compose the map in terms of ...