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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32 views

Understanding a calculation deduced for the function $\pi^{-s/2}\Gamma(s/2)\zeta(s)$

With my current knowledges I don't know if this is a bad question, but since I am interesting in this kind of calculations I want to ask you, if I was wrong or if if my statement is obvious. From ...
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1answer
25 views

Maximal value of real part of holomorphic function

Let $f:U \rightarrow C$ be a non-constant holomorphic function. $U$ is open, connected and $D(0,1+\epsilon) \subset U$. I'd like to show that there exists $z_0 \in \partial D(0,1)$ such that $Re(f(z))...
2
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1answer
38 views

All power series has a point that is not regular.

Definition: Let $f = \sum_{n \geq 0} a_n z^n $ a power series and $0<R< \infty$ its convergence ratio. We say that $z_0 \in \mathbb C, |z_0| = R$ is a regular point if $\exists r > 0$ such ...
3
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1answer
59 views

Why Differential Forms on Riemann surfaces?

I am working with Rick Miranda's "Algebraic Curves and Riemann Surfaces". Right now I am in chapter four "Integration on Riemann Surfaces" and struggle with it a lot!:( It starts with the definition ...
2
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0answers
37 views

How to apply the Identity Theorem to this function?

Given the function $f(z)=\exp\left(z^2-\cos\left(iz\right)-4\right)$ with the domain $|z|<10$, if we try to apply the Cauchy integral formula, we'll see that f(2) "will be" $$\frac{1}{2\pi i}\int_\...
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1answer
54 views

How does one compute this heavy integral?

The integral is $$\frac{1}{2\pi i}\int_\Gamma\frac{\exp(z^2-\cos(iz)-4)}{z-2}dz$$ where $\Gamma$ is the unit circle. Here's how I tried to parametrize it: $z=e^{i\theta}$ on $\theta\in [0, 2\pi]$, ...
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25 views

Existence of analytic function from A to B . [duplicate]

Does there exists a non constant analytic map $f:A\to B$ . Where $A=\{z\in \mathbb C~:~ |z|\neq 0\}$ and $B=\{z \in \mathbb C ~:~ |z|>1\}$. I am unable to construct one
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1answer
28 views

Minimum modulus principle - looks like a counterexample?

The minimum modulus principle states that if $f$ is holomorphic within a bounded domain D, continuous up to the boundary of D, and non-zero at all points, then $|f (z)|$ takes its minimum value on the ...
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27 views

Minimum norm of analytic function may not be achieved on the boundary of its domain

I need to show that the minimum modulus of an analytic function may not be achieved on the boundary of its domain. I'm stuck with this question, would appreciate if someone could help me with it. I ...
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17 views

Construction of continuous/analytic maps

Let $A=\{z\in C ~:~ |z|>1\}$ , $B=\{z\in C ~:~z\neq 0\}$. Which of the following is true? 1.There exists a continuous onto map $ f:A\to B. $ 2.There exists a continuous one to one map $ f:B\to ...
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22 views

$|f(x)-f(y)|\leq c|x-y|^{\alpha}$ uniformly continuous, while c>0 and $\alpha\in \mathbb{Q}\cap (0,1]$

Let $f:\mathbb{C}\supset X\rightarrow\mathbb{C}$ be a function with the property that c>0 and $\alpha\in\mathbb{Q}\cap (0,1]$ exist such that for all $x,y\in X$ following holds: $|f(x)-f(y)|\leq c|x-y|...
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1answer
55 views

Find the regular points.

If $f(z)$ is a power series, i.e., $f(z) = \sum_{n \geq 0} a_nz^n$, and this function is define in $B(0,R)$, where $ 0 < R < \infty <$ and $R$ is the convergence radius. We say $z_0 \in \...
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1answer
20 views

existence of an automorphism F(A)=A

Let G be a bounded domain in $\mathbb{C}$ and let $A\subset G$ be finite.I got to show that for every $f\in Aut(G\backslash A)$ there is a unique $F \in Aut (G)$ with $F\vert_{G\backslash A}=F$ and ...
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0answers
21 views

convergence to a non-injective function

Let G be a simply connected domain, $G \not \neq \mathbb{C}$ and $z_0 \in G$, I got to show that for every $n \in \mathbb{N}$ there is a holomorphic and injective mapping: $f_n:G \rightarrow D_1(0)$, ...
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1answer
27 views

holomorphic injective function

Prove: Let $z_o \in G\;$. There exists a holomoprhic injective function for every $n \in \mathbb N$ $\;f_n:G\rightarrow D_1(0)$ such that $f_n(z_0)=1-\frac{1}{n}$ I don't know how to find such a ...
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3answers
70 views

$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$?

Can we compute the integral ? $$\int_{|z|=1} \frac{1}{\sqrt{z}} dz$$ Actual problem asks to compute: $$\int_{|z|=2} \frac{z^n}{\sqrt{z^2+1}} dz$$ To compute this I need to solve the integral: $\...
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0answers
48 views

Xi Function on Critical Strip - Mellin Transform

Story I'm trying to prove the following identity $$\int_0^\infty \frac{\Xi(t)}{t^2 + \frac{1}{4}} \cos(xt) dt = \frac{1}{2} \pi (e^{\frac{1}{2}x} - 2e^{-\frac{1}{2}x} \psi(e^{-2x}))$$ where $$\psi(...
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0answers
22 views

Complex conics as a Riemann surface

Consider the complex curve defined by $\{(x,y) \in \hat{\mathbb{C}}^2 | ax^2 + bxy + cy^2 + dx + ey + f = 0 \}$ for some complex numbers $a,b,c,d,e,f$ (here $\hat{\mathbb{C}}$ is the Riemann sphere). ...
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2answers
52 views

Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right) $$ where I taken ...
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1answer
21 views

Confused with the reexpression of a Hamiltonian in eigenbasis

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring of spin chain. To compute the complexity of ...
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2answers
31 views

show that Biholomorphism operates transitively

I got to show two statements for the following domains: $G:=\mathbb{C}, \mathbb{C\backslash\{0\}}$ and $D_1(0)$ (the circle around zero with radius 1): (i) the group of all biholomorphic function,$...
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1answer
26 views

complex integration cauchy theorem

I need to find the integral of the following assuming a simple closed path. $f(z) = e^z - \frac{1}{z^2}$ along the lower half of the unit circle with center at the origin traversed in the clockwise ...
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1answer
80 views

Is it true that $ \sqrt{-z} = i \sqrt z $?

Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\...
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1answer
29 views

Complex bilinear transformation.

Let $H=\{z=z+iy\in\mathbb{C}:y>0\}$ be the upper half plane and $D=\{z\in\mathbb{C}:|z|<1\}$ be the open unit disc. Suppose that $f$ is a Mobius transformation, which maps $H$ conformally onto $...
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0answers
58 views

Double integral in complex variables form. [closed]

Rewrite $\displaystyle \iint f(x,y) dx \, dy$ in complex variable form of $\displaystyle \iint g(z, \bar{z}) dz \, d\bar{z}$? Where $z=x+iy$, $\bar{z}=x-iy$ and $x$ changes from $0$ to $a$ and $y$ ...
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1answer
42 views

Complex Analysis - Integration [closed]

I am trying to evaluate the integral for $f(z) =\frac{z} {(z^2-1)}$ , along the path $|z-\pi|=1.$ It is a simple closed path and positively oriented. Any help would be greatly appreciated.
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27 views

Plotting $\sum_{n\geq 1}\frac{1}{n}z^{n}$ for $|z|<1$ (the natural boundary): Coding

As part of my tutorial, I would like to show the plots of a)$-\sum_{n\geq 1}\frac{1}{n}z^{n}$ and b)$Log(1-z)=log(|z|)+iArg(z)$, to drive home the point of analytic continuation. Plotting the Log is ...
2
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2answers
102 views

Find limit of $\lim_{z\rightarrow 0}\left(\frac{\sin z}{z}\right)^{1/z^{2}}$ [closed]

where $z$ is a complex number. Please help me to solve it. I have no idea how to solve this but I have little bit knowledge of limits. my textbook's answer is $e^{-1/6}$. I am very confused at this. ...
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0answers
39 views

Exciting applications of the Riemann-Roch-theorem for Riemann-surfaces

This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch-theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course ...
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1answer
18 views

Solve the Cauchy-Riemann Equations for $u_x$ and $u_y$

I know the Cauchy Riemann Equations in Polar Coordinates are as follows: $$u_r= u_xcos\theta + u_ysin\theta$$ $$ u_\theta= -u_xrsin\theta + u_yrcos\theta$$ I need to solve the following equations ...
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1answer
36 views

Calculate the residues of this complex function

Calculate the residues of this complex function $$\frac{1}{z^2\sin(z)}$$ I can notice that we have singularities at $z=n\pi$, where $n=0,1,2,3,\dots$ But, how to find the residues?
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2answers
66 views

Solve this complex integral [closed]

Solve this complex integral $$\lim_{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi i}\frac{e^{-i\omega x}}{\omega + i\varepsilon}$$ Where $\varepsilon > 0$ and $x$ is real....
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0answers
28 views

Determine the nature and situation of the singularities of this function [closed]

Determine the nature and situation of the singularities of this function $f(z) = \frac{1}{z(e^z -1)}$ and show their residues.
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2answers
50 views

Automorphism of unit disk without zero

Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. $g(z)= e^{i{\...
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1answer
24 views

Radius of convergence of power series of $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ at $0.$

The function $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ has a power series expansion in a neighborhood of the origin. What is its radius of convergence. I believe I have to use the ratio test and show that ...
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0answers
31 views

$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ with a pole $a\in\mathbb{R}$

Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)...
0
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1answer
53 views

Hamiltonian differential equation involving complex logarithm

Consider the differential equation $$\begin{pmatrix}\dot p \\ \dot q \end{pmatrix} = \frac{1}{p^2+q^2}\begin{pmatrix} p \\ q \end{pmatrix}$$ where $(p,q)^T\in \mathbb R ^2 - \{0\}$. I want to show ...
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2answers
69 views

Integrate logarithmic derivative of a periodic function

Given $f$ a $p$-periodic function over $\mathbb{C}$, how to show that : $$\frac{1}{\mathrm{i}p}\int_a^{a+p}\frac{f'(t)}{f(t)}dt \in \mathbb{Z}$$ Is there any elegant method ?
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3answers
39 views

Express $y^2=4ax$ using complex numbers $z$ and $\bar{z}$ [closed]

Express $y^2=4ax$ using complex numbers $z=x+iy$ and $\bar{z}$ (Hint: Use appropriate substitution for x and y)
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1answer
38 views

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\setminus \{0\}$ [closed]

Prove that $u(x,y) = \ln(x^2 + y^2)^{\frac{1}{2}}$ is harmonic on $\mathbb{C}\ {0}$, then find a harmonic conjugate $v(x,y)$ of $u(x,y)$ so that $f(z) = u(x,y) + iv(x,y)$ is analytic on $\mathbb{C}\...
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2answers
39 views

An example of a power series that has a radius of convergence of 3

The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if ...
2
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1answer
30 views

Formula for the graph of 3 hyperbolas

I recently was doing some complex number work and found this guy: Is there a formula for a graph like this in two - dimensions? I know that the values are the same on every curve, but they are ...
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0answers
20 views

How to define a variable which is an integral involving cauchy principal value inside?

How to define a variable which is an integral involving cauchy principal value inside in any computer programming language? I want to know how to break down the procedure step by step from a ...
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1answer
67 views

What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $ e^{i\theta} $ versus $ e^{a+bi} $? Am I correct in saying that these give the same information but offer convenience in different situations? ...
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0answers
7 views

Complex Valued Autocorrelation in an Autoregressive Process.

Suppose we have an autoregressive process $$y_{n+1} = \alpha y_n + \epsilon_n$$ Here $y_n,\alpha, \epsilon \in \mathbb{C}$. Suppose that $\alpha = \exp(\lambda)$ for complex $\lambda$. If $\Re(\...
1
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1answer
61 views

Zeros of $f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+…+\frac{1}{n!z^n}$ are in $B_{\varepsilon}(0)$

I want to prove that for every $\varepsilon >0$ there is a $N\in\mathbb{N}$ so that for every $n\ge N$ all zeros of $$f_n(z)=1+\frac{1}{z}+\frac{1}{2!z^2}+...+\frac{1}{n!z^n}$$ are in $B_{\...
0
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1answer
24 views

$\log \inf_{x\in \partial U} f_x(z)=\inf_{x\in \partial U} \log f_x(z)$?

Let $U$ be an open set of $\mathbb{C}$. Fix $x\in \partial U$. Define $f_x(z)=|z-x|$. Is the following true? $\log \inf_{x\in \partial U} f_x(z)=\inf_{x\in \partial U} \log f_x(z)$. My try: $\...
3
votes
3answers
59 views

If $x= m-m^2-2$ then find $x^4+3x^3+2x^2-11x+6$ where m is a cube root of unity

If $$x= m-m^2-2$$ then find $$x^4+3x^3+2x^2-11x+6$$ where $m$ is a cube root of unity. My try: Since $ m+ m^2+1=0$ the value of $x$ is $-1$. Let $f(x)=x^4+3x^3+2x^2-11x+6$ then $ f(-1)=5$ So the ...
1
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0answers
40 views

Is there only one complex structure on complex plane $\mathbb{C}$? [duplicate]

There is a trivial complex structure on $\mathbb{C}$. Do we have other complex structures on complex plane $\mathbb{C}$? If not, how to prove it?
3
votes
2answers
71 views

Solution of $e^{-z}+z=\lambda$

Let $\lambda\in\mathbb{C}$ with $\Re(\lambda)>1$ be given. I want to show, that $e^{-z}+z=\lambda$ has exactly one solution in $U=\{z\in\mathbb{C}:\Re(z)>0\}$. I think that exercise can be ...