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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3
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20 views

Entire function f(z) bounded above by 1/|Im(z)|

How do I show that an entire function $f(z)$ that satisfies $$ |f(z)|\leq\frac{1}{|Im(z)|} \quad \forall z\in\mathbb C $$ is a constant function?
2
votes
0answers
37 views

Roots of $z^6+6z+10=0$

Find the number of roots of $z^6+6z+10=0$ in each quadrant. I want to use the argument principle. In the first quadrant, $f(z)=z^6+6z+10$ can be written as $f(R e^{i \theta})= R^6 e^{i6\theta}\{ ...
0
votes
2answers
27 views

Why is $t=\frac{1}{2}$ a root for $\tan 4\theta= \frac{4t-4t^3}{1-6t^2+t^4}=\frac{-24}{7}$, where $t=\tan \theta$

Show that $(2+i)^4=-7+24i$ $$\cos 4\theta = \cos^4 \theta - 6\cos^2 \theta \sin^2 \theta + \sin^4 \theta$$ $$\sin 4\theta= 4\sin \theta \cos^3 \theta- 4 \sin^3 \theta \cos \theta$$ ...
3
votes
1answer
16 views

Estimate the Cauchy integral for matrix-valued functions

Recently, I became familiar with the concept of the "matrix function via Cauchy integral", i.e., $$f(A):=\frac{1}{2\pi i}\int_\varGamma f(z)(zI-A)^{-1} \mathrm{d}z$$ Furthermore, it can be shown that ...
0
votes
1answer
8 views

finding equation of circle in complex plane

So i was asked to find the equation of the circle going through 1, i, and 0 Now i know that the equation of circle in complex form is: | z - (Zo) | = r where r is radius. So based on these values, ...
1
vote
1answer
18 views

Meromorphic function with simple pole at prescribed points

The problem is : Construct a meromorphic function on $\mathbb C$ with simple poles at $log(n)$ ; $n \geq 1$ & the principal part being $\frac{1}{z-log(n)}$. What I am thinking is: to use the ...
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votes
1answer
18 views

Can the derivative of a conformal homeomorphism onto a bounded domain be unbounded?

Consider $f$ being a holomorphic homeomorphism of the closed unit disk into the complex plane (ie. $f$ must be a homeomorphism - which also imply bijectivity to its image - of the closed unit disk, it ...
2
votes
1answer
46 views

How to find the roots of $(\frac{z-1}{z})^5=1$

Write down the fifth roots of unity in the form $\cos \theta + i \sin \theta$ where $ 0 \leq \theta \leq 2\pi$ Hence, or otherwise, find the fifth roots of i in a similar form By writing ...
3
votes
1answer
14 views

Equal integrals, circles, opposite directions

I've found this equality in my complex analysis book, but I don't see why it is true. Could you help me understand it? $$\int _{\partial D(1,1)} \frac{dz}{(z-1)(z+1)} = \int _{- \partial D(-1,1)} ...
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votes
0answers
14 views

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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votes
0answers
11 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 ...
0
votes
0answers
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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votes
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 ...
3
votes
1answer
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 ...
0
votes
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 ...
0
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0answers
33 views

Inequality about complex plane.

$z_1$ and $z_2$ are two point in the complex plane and $|z_i|<1(i=1,2)$. how to prove in a more elegant way: ...
0
votes
0answers
21 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. ...
2
votes
2answers
26 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 ...
2
votes
3answers
78 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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0answers
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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vote
2answers
37 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 ...
7
votes
1answer
45 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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votes
0answers
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 ...
0
votes
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 ...
0
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0answers
35 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 ...
0
votes
1answer
24 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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vote
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 ...
1
vote
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, ...
1
vote
1answer
38 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 ...
0
votes
0answers
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 ...
1
vote
1answer
57 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 ...
0
votes
2answers
32 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} = ...
3
votes
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 ...
0
votes
0answers
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 􏰀 ...
0
votes
0answers
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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vote
0answers
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 ...
2
votes
0answers
12 views

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 ...
1
vote
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 ...
1
vote
1answer
26 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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vote
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 $ $\| f'(z)\| > 0$ ?
1
vote
1answer
44 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 ...
0
votes
0answers
13 views

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 ...
1
vote
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 ...
-1
votes
1answer
27 views

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

What is the winding number of $e^{ix}$ where $x\in \mathbb{R}$?
5
votes
2answers
103 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 ...
4
votes
3answers
47 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 Theorem 2.15 (Hurwitz Theorem) Let $\{f_j\}$ be a ...
1
vote
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}} = ...
1
vote
1answer
28 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 ...
0
votes
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 ...
1
vote
1answer
37 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} = ...