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Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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For an analytic function $f$ in a domain $D$, if $\log|f|$ is harmonic in a neighborhood of $\partial D$ then $f\in C(\bar D)$.

I have stuck in one place while reading a paper. If $\phi$ is an analytic function on $D$, where $D$ is bounded multiple connected domains in $\mathbb{C}$. Now given that $\log\lvert \phi\rvert$ is ...
Ravi's user avatar
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1 vote
1 answer
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Understanding definition of degree of a holomorphic function at a point p

I am confused at the following notation: Let $f$ be non constant meromorphic function in a domain $U$ and $D$ a open set whose closure is a compact subset of $U$. Let $q\in \hat{\mathbb{C}}$, we set $$...
Remu X's user avatar
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-4 votes
0 answers
28 views

Need help proving the following relation [closed]

i am tasked with proving the following relation : $\frac{sinx}{x}=\int_{0}^{π/2}J_{0}(xcosθ)\cdot cosθdθ,\forall x\in R^{\ast } $ , but i don't know how to begin ? Can you help me prove it ? Thank ...
savvetos's user avatar
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1 answer
24 views

Freitag-Busam proof of $\sum\limits_{\omega \in L \setminus \{0\}} |\omega|^{-s}$ converges when $s>2$.

One of the main results allowing the possibility of introducing the Weierstrass $\wp$ function is the following: $\sum\limits_{\omega \in L \setminus \{0\}} |\omega|^{-s}$ converges when $s>2$. I'...
TheWanderer's user avatar
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0 votes
1 answer
29 views

Inverse Laplace Transform of $e^{-as}/s^2$ for $a>0$

I am trying to compute the inverse Laplace transform of $$F(s) = \frac{e^{-as}}{s^2}$$ for $a > 0$. I computed it as follows: $$\mathcal{L}^{-1}_{s\to t} \left\{\frac{e^{-as}}{s^2}\right\} = \text{...
idk31909310's user avatar
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0 answers
10 views

Find all elements of Automorphism on a simply connected domain

Find a conformal map from $U = \mathbb{D}_1(0)−[0, 1)$ to the upper half plane, and use it to find all elements of Aut$(U)$ that fix the point $(2\sqrt{2} − 3) \in U$ My approach: First we take a $\...
Remu X's user avatar
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Can somebody please tell me about good resources on Hyperelliptic function and hyperbolic function? I need that for a project. [closed]

If possible please give me a pdf of the book or lecture notes on that topic...
Sayantika Bose's user avatar
1 vote
2 answers
45 views

Scalar complex holomorphic function - same derivative as scalar real function?

My question is about the derivative of holomorphic complex functions. Assume there is a function $f(x) := \mathbb{R} \rightarrow \mathbb{R}$ , and a function $f(z) := \mathbb{C} \rightarrow \mathbb{C}$...
Bastian's user avatar
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1 answer
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How can I show that $f:\mathbb{C} \to \mathbb{C}$ is bounded? [duplicate]

at the moment I am working on the following exercises: Let $v,w \in \mathbb{R}^2 = \mathbb{C}$ be two linear independent $\mathbb{R}$-vector. Now let be $f:\mathbb{C} \to \mathbb{C}$ holomorphic and ...
WomBud's user avatar
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1 vote
1 answer
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Showing the identity log($z^n$) = n log (z) for a particular value of n and z

I was asked to show $\log(z^n) = n \log (z)$ where $z = 1 + i$ and $n = 5$. The worked solutions state that they are not equal for those values but I do not understand why given that we find the ...
Oofy2000's user avatar
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-3 votes
1 answer
31 views

Is $|1-e^w| \leq c|w|$ true? [closed]

Let $w$ be a complex number. Is it true that if $|w| \leq 1$ , $|1-e^w| \leq c|w|$ for some constant $c$?
SunnyMath's user avatar
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2 votes
0 answers
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Application of open mapping theorem in complex analysis?

I recently came along the following question. Let $\Omega\subseteq\mathbb{C}$ be a open connected set and $f\in\mathrm{H}(\Omega)$, i.e., $f$ is holomorphic in $\Omega$. Assume that $f^2(z)=\overline{...
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How to show a complex-valued function defined by a series converges on complex plane

I studied about Weierstrass $P$-function (a.k.a Weierstrass elliptic function). In order to prove the fact that this function converges to a meromorphic function on the whole complex plane, my ...
PKS's user avatar
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Calculation of the contour integral

I'm struggling with the integral $\int_{0}^{2π} \frac{cos(x)}{cos(x)-5/4} \,dx$. The problem itself wants me to solve it using residue theorem,but I have no Idea what to do. My idea was to add and ...
Ігор Ілечко's user avatar
2 votes
0 answers
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A limit related to Poisson kernel in unit disc

I am trying to show $$\lim_{r\to1^-}\int_0^{2\pi}\exp\left(\frac{1-r^2}{1-2r\cos\theta+r^2}\right)d\theta=+\infty.$$ An approach is to apply the theory of Hardy space. Indeed, the function $g(z)=\exp\...
FFGG's user avatar
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1 vote
1 answer
73 views

Fourier transform of incomplete gamma function

Ultimately I am interested in the Fourier transform of $$ e^{-i\zeta}(-i\zeta)^{-2\epsilon}\Gamma(2\epsilon,-i\zeta) $$ in a series expansion around $\epsilon=0$, so to first order in $$ \lim_{\...
Tobias's user avatar
  • 103
-4 votes
0 answers
69 views

How does the Schrödinger equation on compact Riemann surfaces under SL(2, R) action impact? [closed]

In our study, we delve into the closure of orbits and the uniform distribution of points within the context of the SL(2,R) action on the moduli space of compact Riemann surfaces, akin to unipotent ...
amir mohammadi's user avatar
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0 answers
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Lower&upper bound $\alpha_1\|x_1\|_{\infty}+\alpha_2\|x_2\|_{\infty} \leq \|x_1+x_2\|_{\infty}\leq\beta_1\| x_1\|_{\infty}+\beta_2\| x_2\|_{\infty}$?

If $x_1, x_2 \in \mathbb{C}^n$, then is there any lower and upper bound for $\|x_1 + x_2\|_{\infty}$, where $\| x\|_{\infty} := \max_{i=1,\ldots,n} |x_i| $? More specifically, I am wondering, is there ...
learning's user avatar
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0 answers
41 views

Singularities and their nature

What are the singularities of $f(z) = e^{\frac{\sin z}{z}}$ ? It clearly has a removable singularity at $0$. A textbook says that it has essential singularities at $kπ$ for $k \in \mathbb{Z}$. How and ...
Anonymous's user avatar
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1 answer
50 views

A question of complex numbers involving inequality

$\begin {aligned}|a(z_2-z_3)+(z_3-z_1)|&\ge |a|\, |z_2-z_3|+|z_3-z_1|\text{(by triangle inequality)}\\ &\ge 2\sqrt a\cdot \sqrt{|z_2-z_3|\cdot|z_3-z_1|} \text{(by AM-GM inequality)} \end{...
user1318878's user avatar
0 votes
1 answer
65 views

How to show $\log(z) = \log(r) + i \theta$ without implicitly assuming $z = r \exp (i \theta)$ - from Penrose Road to Reality

In Roger Penrose's book Road to Reality - Chapter 5 - he goes to great lengths to arrive at the standard polar expression for a complex number $w = r e^{i \theta}$ via a discussion of complex ...
a_former_scientist's user avatar
3 votes
1 answer
49 views

Radon Nikodym derivative and distribution function

Let $\mathfrak{B}$ be the Borel $\sigma$-algebra over $\mathbb{R}$ and $\beta$ the Borel-Lebesgue measure over $\mathfrak{B}$. Let $\mu$ be a Borel measure over $\mathbb{R}$ s.t. the distribution ...
Kham Bodrogi's user avatar
3 votes
1 answer
95 views

Analytic $f: \mathbb{D} \to \mathbb{D}$, $f(0)=0$, and $f$ has five zeros in $\overline{\frac{1}{2}\mathbb{D}}$

Suppose $f: \mathbb{D} \to \mathbb{D}$ is a holomorphic function and $f(0)=0$. The function $f$ has a total of five zeros (counting multiplicities) in the closed half-disc $\overline{\frac{1}{2}\...
Grigor Hakobyan's user avatar
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1 answer
45 views

Integral of a multivalued function

I need to calculate this integral using complex analysis methods $$ \int_{0}^{1}\frac{\,\sqrt[4]{\,{x^{3}\left(1 - x\right)}\,}\,}{x + 1}\,{\rm d}x $$ I encountered a problem: I received an inadequate ...
  Alina Gabriel's user avatar
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Which of the following function does not have a Laurent series around the point z=0 with a non-zero region of convergence around z=0. [closed]

(i) Which of the following function does not have a Laurent series around the point z=0 with a non-zero region of convergence around z=0. $tan(1z)$ $e^{(-iz)^{-2}}$ $1z(z-1)$ $log(1+z)$ Contour
Miss M's user avatar
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1 vote
0 answers
23 views

Construction of a function subject to conditions

Is it possible to construct $f \in H(B(0,1))$ such that $f(1/n) = z_n$, where: $z_n = (-1)^n$ $z_n = \frac{n}{n+1}$ $z_n = 0$ if $n$ is even and $z_n = \frac{1}{n}$ when $n$ is odd? Is the following ...
A. Random's user avatar
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0 votes
0 answers
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Arithmetic Mean of a Harmonic Function over concentric circles

I'm going through Ahlfors Complex Analysis. I'm trying to understand theorem 20 in the book. Basically, given that in spherical coordinates the conjugate differential is $^*du=r(\partial u/\partial ...
Redcrazyguy's user avatar
1 vote
0 answers
40 views

Singularities of a complex function with exponential components

I'm working on trying to classify all singularities and find their residues in the following function in $\overline{\mathbb{C}}$: \begin{equation} f(z)= \frac{1}{e^{z^2}-e^{4z-4}} \end{equation} I ...
Febrero's user avatar
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6 votes
0 answers
55 views

Anything interesting known about this generalization of even and odd functions?

Let $n \in \mathbb N$. Let's say a complex function $f: U \rightarrow \mathbb C$ is "of type $k \pmod n$" if for one (and hence every) primitive $n$-th root of unity $\omega$, $$f(\omega z) =...
Torsten Schoeneberg's user avatar
1 vote
0 answers
20 views

Using the Singular Inverstion Theorem to count the number of $r$-ary trees

For $r \ge 2$ let $\mathcal{C}_r$ denote the class of $r$-ary trees, i.e. Cayley trees in which every vertex has at most $r$ children. We denote the EGF of $\mathcal{C}_r$ by $C_r(z)$. Show that the ...
3nondatur's user avatar
  • 4,178
3 votes
1 answer
52 views

Using Cauchy's estimate to show a function is constant

If $f$ is entire function satisfying $|f(z)|\leq |z|^{1/2}\log(1+|z|+|z|^2)$. Show that $f(z)=c$ for some $c\in \mathbb{C}$. My approach: Let $p\in \mathbb{C}$ and $\mathbb{D}_r(p)\subset \mathbb{C}$. ...
Remu X's user avatar
  • 1,055
-2 votes
0 answers
33 views

definite integral of square root with negative squared trigonometric identity inside the root. [closed]

How do we integrate $\int_0^{\pi/3}\sqrt{1+(-2sin(x))^2}dx$.
shomikc's user avatar
0 votes
0 answers
23 views

Covering properties of non-constant holomorphic function $f: X \rightarrow \mathbb{C}$

I'm working through a proof that Riemann surfaces are second countable, and one of the main steps is showing that if $X$ is a connected Riemann surface such that there is a non-constant holomorphic ...
Dalop's user avatar
  • 715
0 votes
0 answers
45 views

Solving Dirichlet problem on the unit disc. Is it correct?

I would like to solve the Dirichlet problem in $\Omega = B(i,2)$ and with boundary function $\varphi(x+iy) = x^2y^2$. Attempt I first consider the conformal map $f(z) = \frac{z-i}{2}$, with inverse $f^...
Mths's user avatar
  • 65
-1 votes
0 answers
33 views

Trigonometric nonlinear equation [closed]

I try to solve the following equation: $exp(i\phi(x)) ( sin(x)^4 \int_a^b exp(-i\phi(u))\sqrt{cos(u)} sin(u) (1 + cos(u))du + 4sin(x)^2 \int_a^b exp(-i\phi(u)) \sqrt{cos(u)} sin(u)^3 (1 + cos(u))du + ...
nicolas.bachelard's user avatar
-1 votes
0 answers
77 views

Why using contour integration we consider $\cos x$ as real part of $e^{ix}$ but not the same for $\sin x$ as imaginary part of $e^{ix}$? [closed]

I am from Physics background, I am doing definite integrals involving infinity as limits using complex analysis, where we consider a semi circle in upper half of the real axis, and close the contour, ...
Vivek Panchal 's user avatar
0 votes
0 answers
27 views

Bound on Hyperbolic Cosine Identity

I've worked on this problem for a day now and still can't find the desired bound. I want to show that $$\frac{2\cos(2 \pi x)}{\cosh(2 \pi y)-\cos(2 \pi x)} \leq M, \ M\in \mathbb{R}$$ with the ...
atebake's user avatar
1 vote
0 answers
34 views

The conjugate of a cosine series is a sine series

In Katznelson's An Introduction to Harmonic Analysis the author defines the conjugate $\widetilde{S}$ of a trigonometric series $$S \sim \sum_{n = \infty}^\infty a_n e^{inx}$$ by $$\widetilde{S} \sim \...
approximate-identity's user avatar
1 vote
0 answers
27 views

Simple Connectivity on the Complex Plane

There is a ton of equivalent definitions of a simply connected set on the plane, reading Ullrich's Complex Made Simple I have some of these equivalences, my question is about (iii) and (iv) ...
underfilho's user avatar
0 votes
0 answers
21 views

Show two functions are identical on a domain

Let $U$ be a simply connected region, such that $U\subset \mathbb{C}$. Let $p,q\in U$ such that $p \neq q$. And $f,g$ be biholomorphic map from $U$ to $U$. Prove if f(p) = g(p) and f(q) = g(q), then f(...
Remu X's user avatar
  • 1,055
32 votes
4 answers
830 views

Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$

Show that $$\{z \in \mathbb{C}: |z^2+1|\le 2 \} \subseteq \{z \in \mathbb{C} : |z^3+3z+2|\le 6 \} \tag{*}$$ (In other words: Let $z\in \mathbb{C}$ satisfy $|z^2+1|\le 2$. Prove that $|z^3+3z+2|\le 6.$)...
Sam's user avatar
  • 3,072
-1 votes
0 answers
48 views

Proof of the class number formula?

I'm looking for the proof of the Analytic Class Number Formula, but unfortunately am having trouble finding it would love any help with it! For clarity, this the result I'm trying to find the proof ...
Sandro's user avatar
  • 31
0 votes
0 answers
35 views

Show the function is constant on its domain

Suppose $f$ is holomorphic in the unit disc such that $|f(z^2)| \geq |f(z)|$ for all $z$ in the disc. Prove that $f$ must be a constant in the disc. My approach: The only thing I could think of is the ...
Remu X's user avatar
  • 1,055
0 votes
1 answer
24 views

Application of Roche's theorem

Let $a>1$ be real number. Prove that $z\exp(a-z)=1$ has a single solution in $\mathbb{D}_1(0)$, which is real and positive. approach: Take $f(z)=z\exp(a-z)-1$ and $g(z)= z\exp(a-z)$ then $$|z\exp(a-...
Remu X's user avatar
  • 1,055
0 votes
1 answer
62 views

Complex valued integral

I'm trying to evaluate the integral $$\int\limits_{-R}^R\frac{e^{ipt}}{1+t^2}dt$$ where $R\in\mathbb{R}$ and $p\in\mathbb{C}$. My actual goal is to show that for $R\to\infty$ this integral evaluates ...
John Doe's user avatar
  • 700
0 votes
0 answers
33 views

Find a sequence polynomials that converges to $e^{x-y}$ on $S^1$ but diverge anywhere else

I'm trying to do Exercise 3.3.1 in scv.pdf Let $z=x+i y$ as usual in $\mathbb{C}$. Find a sequence of polynomials in $x$ and $y$ that converge uniformly to $e^{x-y}$ on $S^1$, but diverge everywhere ...
hbghlyj's user avatar
  • 2,770
3 votes
2 answers
53 views

Show that there does not exist any holomorphic function on the open unit disk and continuous on the closed unit disk with the given property.

Let $\mathbb D : = \left \{z \in \mathbb C\ :\ \left \lvert z \right \rvert < 1 \right \}.$ Prove that there is no continuous function $f : \overline {\mathbb D} \longrightarrow \mathbb C$ such ...
Anacardium's user avatar
  • 2,450
0 votes
0 answers
24 views

Border of riemann surface given by quotient of fuchsian group

Let $\Gamma \subset PSL(2,\mathbb{R})$ be discrete, and consider the Riemann surface $\mathbb{H} / \Gamma$ with the unique complex structure for which the quotient map $\pi : \mathbb{H} \rightarrow \...
porridgemathematics's user avatar
2 votes
0 answers
59 views

Independence of z and z* in Wirtinger derivatives

In a lecture about complex analysis, I was introduced to the Wirtinger derivatives $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$. They can be expressed as $\partial_z = \frac{1}...
theta_phi's user avatar
  • 115
1 vote
0 answers
42 views

Proving $\frac{\sin\pi z}{z}$ is bounded in the punctured unit disk.

I want to prove that $f(z)=\frac{\sin \pi z}{z}$ is bounded in the unit disk. It has a removable singularity in $z=0$. This is because $\lim_{z\rightarrow 0}\frac{\sin \pi z}{z}=\frac{1}{\pi}$. From ...
muhammed gunes's user avatar

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