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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77 views

Calculate: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$

I would like to calculate this integral: $F(x)=\int_{0}^{+\infty}\frac{e^{i xt}}{t^{\alpha}}dt\quad \text{avec}~x\in \mathbb{R}~\text{ et }~0<\alpha<1$ I calculated : $\displaystyle ...
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0answers
21 views

Answered: Converting between Schwarz-Christoffel formulas for disk and half-plane [duplicate]

Is there a way to conveniently use change of variables (e.g. with a Möbius transformation) in order to convert from a Schwarz-Christoffel integral of the form $$ C_1 + C_2 \int _0^w \prod _{k=1}^n ...
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0answers
21 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
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0answers
22 views

Curves composition with holomorphic function

Statement $(i)$ Let $\gamma:\mathbb R \to \mathbb C$ a $C^1$ curve. Let $v={\gamma}'(t_0)$ the complex number that one obtains from translating to the origin the tangent vector to $\gamma$ at ...
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1answer
32 views

$Res(f,z_0)=0\implies f $ is analytic at $z _0 $?, $z _0 $ is a simple pole.

Can you explain (or refer to results from which this follows ) that if $\operatorname{Res}(f,z_0)=0$ then $ f $ is analytic at $z _0 $? Edit: Also $f $ has at most simple poles as singularities. ...
6
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1answer
144 views

Looking for different proofs of “Discrete Liouville's Theorem”.

Good day. There is a question I have already encountered twice, in very different contexts, that is relatively simple looking, but both solutions I know involve some pretty advanced theorems from the ...
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1answer
35 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
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2answers
148 views

Integration of exponential and square root function

I need to solve this $$\int_{-\pi}^{\pi} \frac{e^{ixn}}{\sqrt{x^2+a^2}}\,dx,$$ where $i^2=-1$ and $a$ is a constant.
2
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3answers
49 views

Harmonic Maximum modulus

So, i am starting to solve some exercises of complex analysis, and i am a little rusty, so if anyone could help me with this exercise. I think that if i just can prove the mean value theorem for ...
0
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1answer
24 views

Proving a region is pathwise connected

I am having problems trying to prove the following statement: Let $\Omega \subset \mathbb C$ be a region (i.e., an open, nonempty, connected subset of $\mathbb C$). Prove that for all $z_0,z_1 \in ...
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0answers
18 views

Proof Strategy for Proving an Inequality Involving Products

I'm working on a proof in my complex analysis course that involves showing that $$ A \cdot B \leq C \cdot D $$ ($A$, $B$, $C$, and $D$ are expressions involving the moduli of complex numbers). My ...
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0answers
15 views

Function $u(x,y) \in C^2$ that admits harmonic conjugate

Problem 1) Prove that if the real and imaginary part of a holomorphic function are of class $C^2$, then they are harmonic. 2)Deduce from 1) that if $u(x,y) \in C^2$ is a function that admits a ...
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4answers
82 views

Sums of solutions to $z^n-1 = 0$ that equal 0

Consider the solutions of the equation $z^n - 1 = 0$, where $z$ is a complex number: ${z_1,z_2...z_n}$. What are ALL the possible sums $\sum_{i=1}^n a_iz_i$ over these n solutions, where $a_i$ are ...
3
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1answer
49 views

Finding Möbius transformations that satistfy certain conditions

Problem Find Möbius transformations that send $(i)$ the circle $|z|=2$ to $|z+1|=1$, and $-2$ to $0$, $0$ to $i$. $(ii)$ the upper half-plane $Im(z)>0$ to $|z|<1$ and $\lambda$ to $0$ (where ...
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1answer
33 views

Complex conjugate root theorem question

From the Complex conjugate root theorem we get that if a polynomial in one varaible with real coefficients has as solution $a + bi$ , than $a-bi$ must also be a solution...however, what happens if ...
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1answer
50 views

What is $\frac{\partial \overline{z}}{\partial z}$?

We can write $$x=\frac{z+\overline{z}}{2}$$ where $z$ is a complex number. Here $z=x+iy$. What is $\frac{\partial x}{\partial z}$? My book says that it is $\frac{1}{2}$. But is $\frac{\partial ...
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1answer
36 views

A question about an assertion in Ahlfors.

Let $ f(z)=u+iv $. Ahlors (pg. 25) says that if $f$ is analytic, then so are $u$ and $v$. Using the fact that $u_x=v_y$ and $u_y=-v_x$, we get $$\Delta u:=\frac{\partial^2 u}{\partial ...
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1answer
36 views

Can I differentiate $Log(|f(z) |)+ i Arg(f(z))$

If we define the complex logarithm as $log(f(z))=Log(|f(z)) |)+ i Arg(f(z))$, where $Log(f(z)) $ is the common logarithm defined for real numbers, how can I differentiate it? If I assume f(z) is ...
2
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3answers
56 views

Recommendations for books on complex analysis and on measure theory?

I'm looking for a book on complex analysis that has a similar writing style to either Terry Tao's Analysis II or Nathan Jacobson's Basic Algebra series. I have found both of these extremely easy to ...
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1answer
36 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
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1answer
42 views

Analytic continuation of given power series

Studying for Analysis qualifier, specifically complex analysis. Problem: Find an analytic continuation for the function $$f(z) = \sum_{k=0}^\infty \frac{z^{k+1}}{3^k}$$ at the point $a = 3-4i$. My ...
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3answers
163 views

Integrate $1/(x^5+1) $from $0$ to $\infty$?

How can I calculate the integral $\displaystyle{\int_{0}^{\infty} \frac{1}{x^5+1} dx}$?
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0answers
37 views

Is residue may be equal to infinity?

Is residue may be equal to infinity? Is it possible?
3
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0answers
67 views

Why do fields seem to be a prerequisite for calculus?

I was in my Complex Analysis class, and the professor said that we should look for a field, rather than a group, to do calculus over. Why is this the case? I understand that we gain another operation ...
3
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1answer
73 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
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0answers
20 views

curvilinear integral: $\oint_{C^{+}}F dx $ and $F(x_1,x_2,x_3) = [\frac{-x_2}{x_1^2+x_2^2}, \frac{x_1}{x_1^2+x_2^2}, 0]$

$ C=(\cos t \cos\sin(nt), \sin t \cos \sin(nt), \sin \sin(nt)): t \in [0,2\pi]$ a) Find $$\oint_{C^{+}}F dx $$ and $F(x_1,x_2,x_3) = \left[\dfrac{-x_2}{x_1^2+x_2^2}, \dfrac{x_1}{x_1^2+x_2^2}, ...
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1answer
16 views

Symmetric points in $\overline{\mathbb C}$ problem

Statement Let $z_1,z_2,z_ 3$ be three distinct points in $\overline{\mathbb C}$, show that there is a unique line or circle $C$ such that $z_1 \in C$, and $z_2$ and $z_3$ are symmetric with respect ...
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1answer
36 views

A problem on complex polynomials

Suppose p(z) is a polynomial of degree $n$ having no zeros in $|z|<1$ and $q(z)=z^n \overline{p(1/\overline{z})}$ then, is $|p(z)|<|q(z)|$ in |z|<1 true? May I know why?
2
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2answers
51 views

Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^{\infty}\frac{e^{itx}}{\pi(1+x^2)}dx$ using residue theorem. Is there any other way to calculate this integral (for someone who ...
0
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1answer
29 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
1
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1answer
15 views

Find a conformal mapping from lens to first quadradrant

Consider the disks of radius 1 centered at 0 and 1 in the complex plane. Their intersection forms a lens shape. I want a complex function which is a conformal map from this lens to the first quadrant. ...
6
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1answer
164 views

Using normal families to bound a complex integral

I am trying to prove that $$\int_{\partial T(Q)} |F'(z)| \,ds(z) \lesssim \int\int_{T(Q)} |F'(z)| |\varphi'(z)|^2 \log \frac{1}{|z|} \,dx\, dy$$ This is an estimate on page $6$ of this paper by ...
0
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1answer
20 views

Laplace's equation periodic in one dimension, from boundary values

I'm trying to solve Laplace's equation in a domain that is semin infinite in one ordinate and periodic in the other. That is, we consider a pair of functions $x(\xi,\nu),y(\xi,\nu)$ such that we ...
24
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8answers
2k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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2answers
24 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
0
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1answer
54 views

How to show that if möbius transformation has an inverse, then it is injective?

Let $f(z)$ be möbius transformation. How to show that if möbius transformation has an inverse, then it is injective? I mean why don't you use this definition to show injectivity of möbius ...
1
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2answers
21 views

Find the coefficients $a^{−1}, a_0, a_1$ in the Laurent expansion $\frac{1}{e^z − 1} = ···+a_{−1}z^{−1} +a_0 +a_1z+…$

Find the coefficients $a^{−1}, a_0, a_1$ in the Laurent expansion $\frac{1}{e^z − 1} = ···+a_{−1}z^{−1} +a_0 +a_1z+...$ on $2π < |z| < 4π.$ I know this should be a very easy problem, but not ...
0
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1answer
51 views

How to solve these two differential equation?

I try to solve these two difference equation ; $$ \frac{dq}{dz} = -j\left(b_1q - kp\right),\\ \frac{dp}{dz} = -j\left(b_2p - kq\right) $$ where $j$ stands for $\sqrt{-1}$, and $b_1$ ,$b_2$ and k are ...
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0answers
78 views

Question on inequality

Is it true that $$\left|np(z)+(\alpha.K-z)p'(z)\right|\geq \left|np(z)+(\frac{\alpha}{K}-z)p'(z)\right|$$ on $|z|=1,$ where $p(z)$ is a polynomial of degree $n$, $p'(z)$ is the derivative of $p(z)$ ...
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2answers
52 views

using geometric series rules when solving Laurent Series

when looking at Laurent Series expansions, I sometimes see something like: $$ \frac{1}{\sin z}= ...
0
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2answers
99 views

Argument principle and Abel-Plana formula

I find proofs of Abel-Plana formula $\sum_{n=0}^{\infty} f(n)-\int_{0}^{\infty} f(x)\text{d}x=\frac{1}{2}f(0)+\text{i}\int_{0}^{\infty}\frac{f(\text{i}t)-f(-\text{i}t)}{e^{2\pi t}-1}$ where $f$ is a ...
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0answers
28 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
0
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1answer
18 views

Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
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2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
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2answers
47 views

Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
2
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1answer
27 views

Liouville's theorem and holomorphic function

I'm working on some practice exams and in one I am looking at the following question: Let $f$ be a function holomorphic on $\mathbb{C}$. Suppose that there exist [real] constants $A$ and $B$ ...
0
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3answers
33 views

convergence of an infinite series of complex number

there is the series $\sum\limits_{k=1}^{\infty}\frac{(k^2+i)}{(k+i)^4}$.I wonder how it can be proved of convergence with ratio test
0
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1answer
23 views

Conformal maps on the boundary

Let $\Omega$ be a domain bounded by a closed smooth curve,(i.e $\Gamma=\partial\Omega$), $\mathbb{D}_+=\{z: |z|>1\}$ and $\Omega_+=\mathbb{C}\setminus{\Omega}$. Suppose ...
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0answers
58 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$
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1answer
43 views

Why doesn't the functional equation imply that $\zeta(s)=0$ for positive even integers?

The Riemann Zeta Function satisfies the functional equation $\zeta(s)=2^s\pi^{s-1}\sin\left(\dfrac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$. But when $s$ is a positive even number, $\sin\left(\dfrac{\pi ...