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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Double periodic entire function

Suppose f is entire and $f(z)=f(z+1)=f(z+\pi)$. Does this imply $f$ is constant? I want to prove that it is constant.I see that it is enough to consider the value of $f(z)$ in between the lines $z=1$ ...
0
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1answer
20 views

Difference between branches $[-\pi, \pi)$ and $[\pi, 3\pi)$ of the complex logarithm

I think that both branches just exclude the negative part of the real line. So what's the difference between them then?
5
votes
3answers
1k views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| \...
0
votes
2answers
540 views

How can I show that circles in the complex plane correspond to circles on the Riemann sphere? How about lines?

Suppose $ T \subset \mathbb{C} $. Show that the corresponding set $ S \subset \Sigma $ is a. a circle if $ T $ is a circle. b. a circle minus (0, 0, 1) if $ T $ is a line. Here we are defining $ \...
4
votes
1answer
57 views

Classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$

I have to classify entire functions satisfying $|f(z)|\leq (1+|z|)^2$ for all $z\in \mathbb{C}$. Using Cauchy integral's formula, I've shown that $f^{(3)}=0$. Thus $f(z)=a+bz+cz^2$ for some $a,b,c \...
17
votes
3answers
469 views

Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$

Hi I am trying to prove the relation $$ I:=\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}. $$ I tried expanding the log argument by using $\sin x/ \cos x=\tan x,$ and than used $\log(...
5
votes
1answer
248 views

Show that $\int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2$

Hi I am trying to prove this $$ \int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2,\qquad a>0. $$ What a ...
8
votes
2answers
252 views

Show $\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\left( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\...
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0answers
23 views

A problem about the property of limit of holomorphic function

Suppose $G\subset\mathbb{C}$ is open and connected,let $\left\{ f_{n}:n=1,2\ldots \right\}$ be a uniformly bounded sequence of holomorphic functions on $G$ that convergences uniformly on compact ...
16
votes
1answer
234 views

Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
0
votes
1answer
31 views

Rewriting a complex matrix as a real matrix and changing basis to $(z,\bar z)$

If we have a complex $n\times n$ matrix $A$ and we rewrite it as a real $2n\times 2n$ matrix $B$ by using the identity $z=x+iy$. I don't get how if we change basis from $(x,y)$ to $(z,\bar z)$ that is ...
0
votes
1answer
20 views

Proper maps and their codomains

A continuous map $f:X\to Y$ is called proper map if for every compact $K\subset\subset Y$ the set $f^{-1}(K)$ is compact. Now, if $\mathbb D=\{z\in \mathbb C;|z|<1\}$. Why the map $f:\mathbb D\to ...
0
votes
1answer
14 views

find a holomorphic function satisfying specific equality

Let $h$ be holomorphic function on a simply connected domain $\Omega$ with no zero in $\Omega$.Show in detail that there exists a holomorphic function $g$ on $\Omega$ where $h\left(z\right)=e^{g\left(...
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vote
1answer
14 views

Calculating the convergence radius of a Taylor expansion

Find the radius of convergence for the Taylor series $$\left(\cot\dfrac{\pi}{100}z\right)=\sum^{\infty}_{n=0}a_{n}\left(z-20\pi\right)^{n}$$ The singularities of this function are the $100n$ where $n$...
4
votes
1answer
50 views

For $f$ analytic on $|z|<1$, $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ zeros of $f$ show that $|f(0)|\le M \prod |a_j|$

For $f$ analytic in unit disk $\Bbb{D}$ where $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ such that $f(a_1)=\cdots=f(a_n)=0$ show that $|f(0)|\le M \prod |a_j|$. I have tried many approaches ...
3
votes
2answers
93 views

Find the value of $\sum_{r=0}^{\left\lfloor\frac{n-1}3\right\rfloor}\binom{n}{3r+1}$

Show that $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\ldots=\dfrac{1}{3}\left[ 2^{n-2} + 2\cos{\dfrac{(n-2)\pi}{3}}\right]$$ My solution:- $$(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\...
0
votes
0answers
14 views

Domain of convergence is complete Reinhardt

Let $W\subset \mathbb C^n$ be a domain such that $W$ is the domain of convergence for a certain power series at the origin. How to show that the interior of $W$ is a complete Reinhardt domain?
0
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1answer
33 views

Is $\int_{\gamma} \sec ^2z \ \mathrm{d}z=0$?

Let $\gamma = \gamma(0;2)$. Is $$\int_{\gamma} \sec ^2z \ \mathrm{d}z$$ equal to $0$? I'm trying to answer this question using only tools like Cauchy Theorem or the Deformation Theorem since ...
0
votes
1answer
29 views

A property about the automorphisms of $\mathbb{D}$

I want to prove the next proposition: if $T$ is a Möbius transformation from $\mathbb{D}$ to $\mathbb{D}$, then \begin{equation} \left|\frac{T(z_1) - T(z_2)}{1 - \overline{T(z_1)} T(z_2)}\right| = \...
4
votes
6answers
3k views

Applications of Complex Numbers

For my Complex Analysis course, we are to look up applications of Complex Numbers in the real world. The semester has just started and I am still new to the complex field. I want to get a head start ...
1
vote
1answer
45 views

self-homeomorphism of the circle

$|z|=1$ is the unit circle in the complex plane. Suppose $g$ is a self-homeomorphism of this circle of order $n$, $n \in \mathbb{N}$, and $g$ acts freely. Is that true that $g$ must be defined by $z ...
0
votes
0answers
39 views

theorem 2 of perfect powers with all equal digits but one

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets $a$ and $c$ not equal to ...
1
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1answer
25 views

A related Mean Value theorem result for complex functions

Let $f:\mathbb{R}\rightarrow\mathbb{C}$ a differentiable function and I wonder if I can affirm that $$\forall a,b\in \mathbb{R}\,\,\text{we have}\,\,\left|\frac{f(b)-f(a)}{b-a}\right|\leq |f'(c)|\,\,\...
0
votes
0answers
14 views

Minimum vertex cover exact algorithm analysis

An exact algorithm to find a minimum vertex cover in a simple, undirected graph would be based on the following recursive idea: "either a vertex v is in the minimum cover, or all of its neighbors are"....
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votes
1answer
26 views

Laurent expansions of $\frac{1}{z-1}$

I want to calculate Laurent expansion of $\frac{1}{z-1}$ thtat are valid in the annuli $\begin{align} (a) & \;\;1<|z|<3\\ (b) & \;\;0<|z-3|<2 \end{align}$ For part $(a)$ since $|...
0
votes
0answers
24 views

Products of Laurent Series

I'm trying to find the Laurent expansion for $$\frac{e^{1/z^2}}{z - 1}$$ about $z_0 = 0$. Writing the series for $e^{1/z^2}$ and $1/(z-1)$ individually gives $$\frac{e^{1/z^2}}{z - 1} = -\left(\...
1
vote
1answer
40 views

Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
1
vote
1answer
28 views

Nevanlinna–Pick interpolation of two sets of points on the unit disk

Given that $z_1, z_2, w_1, w_2 \in D = \{z: |z| < 1\}$ and $\left| \frac{w_1 - w_2}{1 - w_1 \overline{w_2}} \right| \leq \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| $, how do you ...
0
votes
0answers
22 views

Removable singularity of derivatives

If a function f has a removable singularity at $z_0$, is $z_0$ always a removable singularity of the derivative of f?
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0answers
36 views

Fourier transform of $1/z$

What is Fourier transform of the complex function $f: \mathbb C \to \mathbb C$ defined by $f(z)=1/z$? I want to know how to interpret the integral. Is it equal to $\hat{f} (\xi)= \int_\mathbb C \frac {...
0
votes
0answers
31 views

Alternative derivation of Euler's product formula for sine

Euler's product formula states that: $$\sin(x)=x\prod_{n=1}^{\infty}\left[1-\frac{x^2}{\pi^2n^2} \right].$$ There is also a very simple formula for another product representation for the sine ...
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votes
0answers
25 views

Fourier transform in complex analysis

What does convolution means in complex analysis? In particular, I want to calculate $\varphi \ast 1/z$, where $\varphi$ is the characteristic function of unit ball in $\mathbb{C}$, i.e. $\varphi= \...
0
votes
1answer
12 views

Linear functions vs Linearithmic functions complexity

Can we say Linear functions complexity is lower than Linearithmic functions? ie: ...
-4
votes
1answer
43 views

Power series(e^) in complex analysis [on hold]

How to prove the sum $\sum_{n=1}^{\infty}\frac{e^{2\pi inx}}{n}$ converges for any x $\notin$ $\mathbb{Z}$ ? Thanks.
1
vote
2answers
40 views

Tangent and Circle in Complex Plane

Question:- Three points represented by the complex numbers $a,b$ and $c$ lie on a circle with center $O$ and radius $r$. The tangent at $c$ cuts the chord joining the points $a$ and $b$ at $z$. ...
0
votes
1answer
26 views

Analytic function zero in the given disk

I need to show that f(z)=0 for all z \in D(0,2). From the analyticity of f in D(o,2), I know by Cauchy's theorem it's integral in |z|<2 is zeros. And clearly the integrand has a pole at 1/(n+1) ...
0
votes
0answers
41 views

Find $\int_{-\infty}^{\infty}\frac{\sin x}{x} dx$ by integrating over a semicircle in $\mathbb{C}$.

I'm trying to find $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx$$ by solving $$\oint_{\gamma}\frac{e^{iz}}{z}dz$$ where $\gamma$ is the upper semi-circle, with appropriate choices for the paths. My ...
1
vote
1answer
33 views

Complex Differentiation using definition

The function $f:\mathbb{C}\to \mathbb{C}$, $f(x+iy)=y+ix$, is not $\mathbb{C}$-differentiable in any point, because the Cauchy-Riemann equations do not hold. It's $\frac{\partial u}{\partial y}=1\neq-...
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0answers
32 views

Contour integral of $\sqrt[3]{z^3-1}$ on $|z| = 2$ and branches

This is an exam question that i'm trying to figure out. Apart from the title, there is a note added to the question that says that: The branch of $\sqrt[3]{}$ is the one that has $\sqrt[3]{7} \in \...
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vote
1answer
41 views

Complex convergence of sequence Proof check and help

Prove that, for $z \in C$ , the sequence $(z^{n})$ converges if and only if $|z| < 1$ or $z=1$ Proof. Say $z^n$ converges to some $a$ then there exists $n_o$ such that for all $n\geq n_o$ $|z^n-...
0
votes
1answer
17 views

Differentiability of a complex valued function

Consider : $$f(z)=Arg(z)$$ where $Arg(z)$ is the the principal argument of $z\in \Bbb C$ Show that $f$ is nowhere differentiable in $\Bbb C$ I tried the solve this buy the definition of ...
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vote
2answers
31 views

Existence of such a meromorphic function?

Is there a function $f$ that is holomorphic on $\mathbb{C}-\mathbb{Z} $ and maps into or onto $\mathbb{C}-\mathbb{R}$ ? Into or onto $\mathbb{C}-\mathbb{R}^{+}\cup\{ {0} \}$? All I have been able to ...
2
votes
2answers
70 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
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1answer
10 views

How does the author derive this (Difference of analytic functions evaluated at two points)

The conditions are $f:U\to V$ is holomorphic and injective. I basically have 2 questions: Q1) How did the author get $f(z)-f(z_0)=a(z-z_0)^k+G(z)$? Q2) What does "vanishing to order $k+1$" mean? ...
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votes
1answer
20 views

Broad Question about complex analysis [on hold]

I am currently minoring in math and the classes I decided to take are graph theory, number theory, complex analysis and abstract algebra. I am done already with graph theory and number theory. I took ...
0
votes
1answer
103 views

Question regarding a proof that derivatives of injective holomorphic functions are nonzero.

Proposition: Let $\Omega\subseteq\mathbb{C}$ be an open path connected set and suppose $f:\Omega\to\mathbb{C}$ is holomorphic and injective. Then $f'\left(z_{0}\right)\neq0$ for all $z_{0}\in\...
5
votes
1answer
68 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
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votes
0answers
16 views

find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
3
votes
1answer
80 views

Prove $f$ is identically zero in $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} (z)| < 1,|{\mathop{\rm Im}\nolimits} (z)| < 1\} $

Let $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} z| < 1,|{\mathop{\rm Im}\nolimits} z| < 1\} $ and consider the function $f:\bar\Omega\to\Bbb C$ continuous on $\bar\Omega$, analytic in ...
2
votes
1answer
51 views

Show that for $f$ analytic in $B(0,2)$, $\max_{|z|=1}|\frac{1}{z}-f(z)|\ge 1$?

Let $f:B(0,2)\to \Bbb C$ be an analytic function. Show that $$\max_{|z|=1}\left|\frac{1}{z}-f(z)\right|\ge 1.$$ I tried to write $f(z)$ as power series since it is analytic, it doesn't seem work. I ...