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Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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Holomorphic interpolation with restricted exponential type

Given a sequence $(a_n)_n$ with no zeros, and if necessary $a_n = O(n)$, we want to find $f \in H(\mathbb{C^+})$ such that $f(n) = a_n$, where we denote $\mathbb{C^+} = \{Re(z) > 0\}$, and $H(\...
user1274777's user avatar
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Abel summation on smooth piecewise function

After reading this article on non-analytic smooth functions, I wondered if is possible to apply the Abel Summation Formula to such a function. The respective Wikipedia articles assert that Abel ...
Richard Burke-Ward's user avatar
1 vote
1 answer
95 views

If $f:D\to D$ is analytic, bijective and $f(0)=0$, then $f(z)=cz$ for some $|c|=1$?

Let $D=\{z\in\mathbb{C}:|z|<1\}$ and $f:D\to D$ an analytic, bijective function with $f(0)=0$. Is it true that $f(z)=cz$ for some $c\in\mathbb{C}$ with $|c|=1$? I think it can be proved using ...
Nah's user avatar
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Can an autonomous bounded, analytical system with vanishing derivative not converge?

Suppose I have an autonomous differential equation $x'(t)=f(x(t)) $, with $x(t) \in [0,1]^d$, such that $x'(t)$ vanishes at $+ \infty$ in the following sense : $$‖x'(t)‖_2 \to 0 \text{ when } t \to + ...
cocojar's user avatar
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number of zeros of a power series defined by an absolutely convergent sequence

Let $a = (a_n)_{n \in \mathbb{N}}$ be an absolutely convergent sequence such that each $a_n \in \mathbb{R}$, and define $f_a(x) = \sum_{n \in \mathbb{N}} a_n x^n$ for $x\in\mathbb{R}$. To avoid a ...
hs12's user avatar
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1 answer
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$f(z)=u+iv$ analytic in a region $R$ show that $\frac{\partial(u,v)}{\partial(x,y)}=|f'(z)|^2$

I want to show that if $f(z)=u+iv$ is analytic in a region $R$ then $\frac{\partial(u,v)}{\partial(x,y)}=|f'(z)|^2$ here is my attempt: By definition, $$\begin{align} \frac{\partial(u,v)}{\partial(x,y)...
bill's user avatar
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Conditions for $ \int_{\Gamma}f = \lim_{\epsilon \to 0} \int_{\Gamma \epsilon} f $

Theorem 7.7 in Bak-Neumann's Complex Analysis book states: Suppose f is continuous in an open set D and analytic there except possibly at the points of a line segment L. Then f is analytic throughout ...
giorgio's user avatar
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2 answers
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Finding the natural boundary of $\sum_{n=0}^{\infty}\frac{z^n}{n^k}$ for $k \ge 2$

Let $k\ge 2$, given $$f(z)=\sum_{n=1}^{\infty}\frac{z^n}{n^k}$$ It is easy to see that it converges for $|z|\le 1$, but how can it be analytically continued beyond the unit circle? Hadamard proved ...
kmxzc's user avatar
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Maximizing the radius of convergence around a point for an analytic function

Let $f:\Omega\longrightarrow \mathbb{C}$ be analytic, and let $z_0\in\Omega$ s.t. $$f (z)=\sum_{n\geq 0} a_n(z-z_0)^n,\quad\forall \left|z-z_0\right|<r,$$ for some $r>0$, and some complex-valued ...
virtualcode's user avatar
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What is the definition of "multiple component of germ"?

Recently I read a paper and I am confused with a word "multiple components", but I don't find its definition in this paper. I guess it is about the singularity. Here is a picture. You can ...
cbi's user avatar
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Hörmander: Analyticity and wave front sets of parameter integral in existence proof for parametrix for real principal type operators

I'm trying to understand the proof in Hörmanders Analysis of Differential Operators Vol 1 for Theorem 8.3.7. (Existence of parametrix for real principal type differential operators) (1) Citation of ...
xajas's user avatar
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restricting a function changes its singular points and analyticity?

Let define $f(z) = \frac{1}{z-2}$ for $z\in\mathbb{C}\setminus\{2\}$. Then it is clear that, $f(z)$ has singular point at $z=2$ (Namely pole of order 1 at $z=2$). However, if I update the definition ...
General Mathematics's user avatar
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1 answer
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The dual of the Bergman space is the Bloch space

The Bergman space $\mathcal{A}^1$ is the space of holomorphic functions on the disk such that, $$ \|f\|_{\mathcal{A}^1} = \int_{\mathbb{D}} | f(z)| \, dA(z) < \infty. $$ I have seen in several ...
Scottish Questions's user avatar
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Minimums of 2-D function form a continuous function itself. [closed]

Let $B_{\delta}\subseteq \Bbb{R}^2$ be a closed ball of radius $\delta < 1$ centered at $(0,0)$. Let $f:B_{\delta}\to \Bbb{R}_{\geq 0}$ be real-analytic and have only one zero in $B_{\delta}$, ...
Doofenshmert's user avatar
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Boundary of sublevel of function

Suppose we are given a function $$ h \colon \mathbb{R}^n \to \mathbb{R}. $$ I am trying to find minimal hypotheses on the regularity of $h$ under which the boundary of $$ C = \{h(x) \leq 0 \} $$ is $$ ...
P. Tolo's user avatar
6 votes
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66 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
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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
3 votes
2 answers
437 views

A question on Gamma function

This might be basic but I have difficulty understanding what exactly goes wrong in the following logic: Consider the Gammma function $$\Gamma(z) = \int_0^{\infty} t^{z-1} \, e^{-t}\,dt \quad \textrm{...
Ali's user avatar
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How to handle questions of the form "Find all entire functions such that... "

I apologize if this is not the right forum to ask this question. Could you please recommend an online lecture, a textbook note to refer to understand how to answer questions of the form Find all ...
Eureka's user avatar
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Analytic continuation of $\mathbb{R}$-analytic function

Let $A$ be a non-empty open set of $\mathbb{R}$ and $f: A \to \mathbb{R}$ an $\mathbb{R}$-analytic function ($Re(f)$; $Im(f)$ analytic functions). Now, my goal is to find an open set $U \subset \...
J P's user avatar
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2 votes
1 answer
33 views

Existence of uniformly convergent sequence of polynomials converging to an analytic function

Let $\Delta_1,\Delta_2,\dotsc,\Delta_n$ be mutually disjoint open discs in the complex plane. Question part 1: Does there exists a sequence of polynomials $(p_n)_{n\in \mathbb N}$ converging uniformly ...
MakeOperatorAlgebrasGreatAgain's user avatar
1 vote
0 answers
43 views

Every $f\in C_c^0$ is analytic?! Where is the mistake in the proof?

Let $f\in C_c^0(\mathbb{R})$ (continuous with compact support) and denote by $\rho_a$ the Gaussian density with standard deviation a and mean 0 (i.e. $\rho_a(x)=\frac{1}{\sqrt{2\pi a^2}} \exp({-\frac{...
XXXHaraldXXX's user avatar
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22 views

Proof of analytic continuation on manifold

I was reading the theorem below in this article. My question concerns the passages in bold. Specifically, why would Z be open? Is it because we can use germs to define a topology where Z would be the ...
yumika's user avatar
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1 vote
1 answer
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Applying the Identity Theorem to Analytic Functions Agreeing on 1D Curves

I'm working through a complex analysis problem and have encountered a problem that I'm struggling to understand. The context involves two meromorphic functions, g and h, which are given on the unit ...
zich's user avatar
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Entire function $f$ such that $f(z_0 +z) =f(z_1 +z) =f(z)$ is constant [duplicate]

Take $z_0,z_1$ $\in \mathbb{C}$ to be $\mathbb{R}$ linearly independent. The exercise is to prove that if $f:\mathbb{C} \rightarrow \mathbb{C}$ is entire such that $f(z_0 +z) = f(z)$ and $f(z_1 +z) = ...
Lucas G's user avatar
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0 votes
1 answer
124 views

How is $|z|^2$ not analytic?

I am aware that the C.R equations are not satisfied for $|z|^2$. However, I tried to prove using limits. My approach was using the formula for derivative of a complex function: $f'(0) = \lim_{z\to 0}\...
Praneel65's user avatar
1 vote
0 answers
50 views

Reference request: if $f$ is real analytic and $\{x:f(x)=0\}$ has a limit point, then $f=0$

I would like to know a reference for the following theorem. Theorem: If $f$ is a real analytic function on an open set $G$ in $\mathbb{R}^p$ and if $\{x:f(x)=0\}$ has a limit point in $G$, then $f(x)=...
rfloc's user avatar
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0 answers
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Growth of preimages of singular values of finite type entire map

Let $f\colon \mathbb{C} \to \mathbb{C}$ be an entire map having precisely two distinct singular values $w^1$ and $w^2$. If $w^i$ has infinitely many preimages under $f$, we write $(z_n^i)_{n \in \...
A B's user avatar
  • 125
0 votes
1 answer
91 views

Find the analytic function $f(z)$ if $v(x,y)=\frac{x-y}{x^2+y^2}$

I try to solve this in the polar form, then by taking the derivative with respect to $\theta$ and $r$ from the question I obtain $v_r = -\frac{1}{r^2}(\cos \theta - \sin\theta)$ $v_\theta = \frac{1}{r}...
Ocean's user avatar
  • 105
2 votes
1 answer
25 views

Bounding $f'(z)$ with $O(\log(\frac{1}{1-r}))$ for an Analytic Series

I am working with an analytic function defined within the unit disk $|z| < 1$ as follows: $$ f(z) = \sum_{n=1}^{\infty} a_{n} z^{n}, $$ where I have the condition that $\sum_{n=1}^{N} n|a_{n}| = O(\...
El Sh's user avatar
  • 81
1 vote
1 answer
55 views

Are analytic functions with highest convergence radius at a point analytic everywhere? [duplicate]

This seems to be some elementary point and I have red several questions about it, but I could not have a clarifying answer, so I directly ask myself. I say that a real-valued function $f$ on a real ...
Hair80's user avatar
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0 answers
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What's the relationship between these concepts about power series expansion?

A bivariate function $f:\mathbb{R}^2 \to \mathbb{R} $ is called real analytic if it can be expanded into a power series about $x-x_0$ and $y-y_0$ within a neighborhood of $(x_0,y_0)$,i.e. $$f(x,y)=\...
kmxzc's user avatar
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0 answers
28 views

Singularities of $f(z) = \sin(\cot(1/z))$

I am asked to find all singularities for the function $f(z) = \sin(\cot(1/z))$. Here is my attempt, would appreciate any feedback if this is correct or otherwise where I am wrong / not rigorous enough:...
giorgio's user avatar
  • 583
1 vote
0 answers
107 views

How do you solve the differential equation $f'(x) = f(ax)$ [closed]

I was looking at $C^\infty$ functions that are not analytic anywhere. For that I was looking at solutions of $f'(x) = f(ax)$. There is such a function for $a = 2$. But how would you solve that ...
José Tierno's user avatar
1 vote
0 answers
28 views

Clarification on one-to-one mapping of a disk in complex analysis

I have a some difficulties understanding the proof in section 7.2 of "Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics" by Edward Saff and Arnold ...
david's user avatar
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1 vote
0 answers
40 views

Analyticity and Complex Differentiability

I'm seeing two different definitions for analytic functions. A function $\Phi: \mathcal{D}\rightarrow \mathbb C$ is analytic if, for every $z_0 \in \mathcal{D}\subseteq \mathbb C$, there exists a ...
Dee's user avatar
  • 511
2 votes
2 answers
107 views

Question About Differentiability Of Principal Branch Of log(z).

Theorem:1 Let the function $$f (z) = u(r, θ) + iv(r, θ)$$be defined throughout some neighborhood of a nonzero point $z_0 = r_0 \exp(iθ_0)$ and suppose that (a) The first-order partial derivatives of ...
Meet Patel's user avatar
0 votes
3 answers
121 views

$|f(z)|$ is constant but $f(z)$ is not constant. [closed]

According to this post Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$ We have, If $f$ is analytic in a domain $D$ and $|f(z)|...
General Mathematics's user avatar
4 votes
1 answer
41 views

Can we write every $C^1$ complex function on the unit circle as the the difference of two approriate functions?

If $g$ is $C^1$ on the unit circle $C(0,1)$. Then there is a function $f^+$ holomorphic on $B(0,1)$ and continuous on $\bar B(0,1)$, a function $f^-$ holomorphic on $\mathbb{C}\backslash\bar B(0,1)$ ...
Derewsnanu's user avatar
1 vote
0 answers
71 views

Smooth Riemannian metric is locally real analytic?

Let $U$ be an open subset of $\mathbb{R}^n$ and $g$ be a $C^\infty$ Riemannian metric on $U$. Given a point $x_0\in U$, does there exist a local neighborhood $x_0\in V\subset U$ and new coordinates ...
crimsonmist's user avatar
1 vote
0 answers
46 views

Decomposition of analytic functions on the upper half plane

Let $$h:\mathbb{H}:=\{z\in \mathbb{C}:\text{Im}(z)>0\}\to \mathbb{H}\cup \mathbb{R}$$ be an analytic function. We decompose $h$, using partial fraction decomposition, into two parts $h_1$ and $h_2$,...
Jonas Müller's user avatar
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0 answers
34 views

Real analytic function on $X$ is subanalytic on compact subsets $Q\subseteq X$.

$\textbf{Background.}$ I'm trying to apply Lojasiewicz's inequality to a specific function, $f$, and the Euclidean distance function $d$. Hence, I need to prove that $f$ and $d$ are subanalytic. ...
Doofenshmert's user avatar
-1 votes
1 answer
139 views

Proving that $f$ is rational if it is known that $f$ is analytic in $\mathbb{D}$ and satisfies $|f(z)| \to 1$ as $|z| \to 1$

This question was already asked here, I wouldn’t ask it again however I do not think that the question gets the point across that it’s trying to get across, and I can’t find anything else related to ...
no lemon no melon's user avatar
1 vote
1 answer
81 views

If an analytic function is identically zero in a domain, can it be always divided by another analytic function in that domain?

Let's say I have two functions $f$ and $g$, both analytic on the same domain (open and connected set) $D$, and suppose also I was able to prove that $f = 0$ on the entire domain. Question. Is it ...
giorgio's user avatar
  • 583
0 votes
0 answers
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Radius of convergence of a power series defined in terms of co-efficients around another point

I'm trying to answer the following question, having a hard time figuring out where to begin. Any help appreciated: Let f be analytic everywhere it is defined, and for $|z| < 2$ the following holds: ...
giorgio's user avatar
  • 583
3 votes
0 answers
40 views

$f=\phi(v)+\textbf i v$ is analytic, prove that $f$ is constant

Statement: Complex function $f=u+\textbf i v$ is analytic in domain $D\subset \mathbb C$ and everywhere in $D$ $u=\phi(v)$, where $\phi=\phi(t)\in C^{1}(\mathbb R\to\mathbb R)$ and $\phi$ is strictly ...
Egor Ivanov's user avatar
2 votes
0 answers
43 views

Analytically continuing a function of two complex variables.

I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex ...
mathphy24's user avatar
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0 answers
16 views

Characterizing the unimodular functions from the closed disk $\mathbb{C}$ to $\mathbb{C}$ with constraints

It is well known that if $f:\mathbb{D}\to\mathbb{C}$ is analytic, continuous on the boundary, and is unimodular (say with a finite number of zeros) then $f$ is a finite Blaschke product up to some ...
Math101's user avatar
  • 4,653
1 vote
1 answer
94 views

Meaning of "$f$ has a power series expansion around $p$"

In Complex Analysis by Donald Marshall (page 29), there is an exercise problem that starts with "Suppose $f$ has a power series expansion at $0$ which converges in all of $\mathbb{C}$. " ...
Koda's user avatar
  • 1,268
3 votes
2 answers
107 views

Computing the domain of analyticity of $f(z)=\sqrt{z^2-1}$

In this question, it is said that the domain of analyticity of the function $f(z)=\sqrt{z^2-1}$ over the branch $(0,2\pi)$ is $\mathbb{C} \setminus ((-\infty,-1) \cup (1,\infty))$. My question: I ...
Math's user avatar
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