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Questions tagged [analyticity]

A function is analytic if it has a converging power series expansion. Please also use one of (real-analysis) or (complex-analysis) to specify real or complex analyticity: though the definition is the same, the properties of functions analytic over real numbers are quite different from the properties of functions over the complex numbers.

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Convolution of $\mathcal{C}^\infty$ is analytic

Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as: $$(f*g)(x) = \int_\mathbb{...
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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 ...
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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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In complex number system, sin z and cos z are unbounded and periodic. But they are continuous also. How can that be possible?

I know that a continuous periodic function must be bounded because if a function is continuous and periodic, its graph will have to turn at certain points to reattain the values and hence, it cannot ...
Ria Talwar 's user avatar
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Regarding a Coin Toss Experiment by Neil DeGrasse Tyson, and its validity

In one of his interviews, Clip Link, Neil DeGrasse Tyson discusses a coin toss experiment. It goes something like this: Line up 1000 people, each given a coin, to be flipped simultaneously Ask each ...
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Analyticity of an "Integral" type function.

I found the following exercise in Conway's complex analysis. Determine the region in which $ f(z) = \int_{0}^{1} \frac{1}{{t - z}} \, dt $ is analytic.
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Pointwise limit of sequence of holomorphic functions given constraint on their derivatives at the origin

Consider a sequence of functions $g_n(z)$. $n$ takes values on the natural numbers, and $z$ is a complex variable. For all $n$, $g_n(z)$ is guaranteed to be an analytic function of $z$ within a disk ...
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Analytic Solution to: $ n_t = C_1(n_x^2 + nn_{xx}) \text{ where }n(-L,t)=n(L,t)=0 \text{ and }n(x,0)=e^{-x^2} $

I am working with a nonlinear PDE and am looking for an analytical solution. I'm unsure how to figure out if a PDE has a known solution, so I figured that someone here may know. This is the PDE: $$ \...
diogenes's user avatar
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Why is a polynomial in $z,\bar z$ analytic iff it does not involve monomials $z^i\bar z^j$ with $j\ge1$?

I have encountered some note on complex analysis: In particular, given a polynomial in the real variables x and y , with complex coefficients, these properties tell us that the polynomial is analytic ...
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Spatial analyticity for solutions of linear parabolic PDEs

I believe that the following result is true, but am having a hard time finding a suitable reference to convince myself: Suppose $X:\mathbb{R}^n\to \mathbb{R}^n$ is analytic. If the smooth function $u:(...
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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 ...
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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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Why does Jost function have analytic continuation on the $p$-plane?

I have question in the analyticity of Jost function on the $p$-plane. The chapter 12 of the book Scattering Theory by John R. Taylor states (p.218): The Jost function is defined as $$ f_{l}(p)=1+\frac{...
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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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Find the analyticity of |z|

I have been struggling finding the domain of analyticity in complex analysis, my teacher finds them with a drawing like the one I've attatched (in the pic they were asking for $\sqrt{z-1}=e^{1/2 \log(...
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What does it mean for a distribution to be analytic and how do we analytically continue it?

Let $T \in S'(\mathbb{R}^n$) be a tempered distribution. I've seen a few resources (such as the first sentence of this Wikipedia article) refer to distributions being analytic or how one can ...
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Is $\ln(1+x)$ analytic for values of $-1<x<0$?

It can be shown easily that the taylor series of $\ln(1+x)$ about $x=0$ is,$$\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}$$ and it converges for $-1<x\leq 1$. However to check if $\ln(1+x)$ ...
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If $D$ is a domain symmetric about the real-axis and $f:D \rightarrow \mathbb{C}$ analytic, then $\overline{f(\overline{z})}$ is analytic

My original problem that I encounter was: "Let $D$ be a domain symmetric about the real-axis and containing the real axis. Suppose that $f:D \rightarrow \mathbb{C}$ analytic on $D$ and that when $...
obitobi_tobias's user avatar
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What does the Wirtinger derivative of a non-analytic function (absolute value squared) represent?

Let $z = x + i y$ be a complex number and consider the modulus squared function: $$ f(z) = |z|^2 = z z^* = x^2 + y^2 = u(x, y) $$ where the asterix denotes complex conjugation and $u(x, y)$ is the ...
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Identifying Non-Analytic Regions for the Function $f(z) = \frac{1}{{z^2 + 5iz - 4}}$

I'm working with the complex function $f(z) = \frac{1}{{z^2 + 5iz - 4}}$, and I'm trying to determine where this function is not analytic. I've been trying to compute its domain of analyticity, but I'...
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Let function $f=u +iv$ is analytic on $D$ and for some $a,b,c \in \mathbb R$, $a^2+b^2 \neq 0$ and $au+bv=c$ on $D$. Prove that $f$ is constant on $D$

Let function $f=u +iv$ be analytic on some domain $D$. Let $a,b,c \in \mathbb R$ such that $a^2+b^2 \neq 0$ and $au+bv=c$ on $D$. Prove that $f$ is constant on $D$. What I have done is following: ...
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Real-analytic bijection is linear?

It is well-known that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ which is injective must be of the form $f(z)=az+b$ for some $a,b\in \mathbb{C}$ with $a\neq 0$. See here for a proof. My ...
TheEmptyFunction's user avatar
3 votes
2 answers
376 views

How to prove this integral function is analytic?

Given $$G(z) = \int_{1+i}^{z}\text{sin} (\theta^2) d\theta$$ Prove that $G(z)$ is an analytic function of $z$. I read that integration preserves analyticity. But why is that true when $z$ is in the ...
s_a94248's user avatar
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Condition for real analyticity of multiple Fourier series

In 1D, the Fourier series $u(x)=\sum_{k\in\mathbb{Z}}\hat{u}_ke^{ikx}$ is analytic on the torus if and only if there exist constants $K>0$ and $a>0$ such that $$|\hat{u}_k|\leq K e^{-a|k|}.$$ ...
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globally extending a real analytic foliation to the entire strictly positive Euclidean plane with the canonical metric.

Consider the manifold $(\Bbb R^2, \mathrm{can})$. I derived a PDE on this manifold: $$r^2 \frac{\partial ^3\Psi(r,s)}{\partial r^3}=s^2 \frac{\partial \Psi(r,s)}{\partial s}$$ and found a particular ...
zeta space's user avatar
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Finding order of pole, why is this argument not valid?

I'm learning about different types of singularities in complex analysis, and stumbled upon a problem with finding what order of pole $f(z)=\frac{\sin(z)}{z^3}$ have at $z_0=0$. My solution is $$\lim_{...
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function with two complex variables

I have a project regarding analyticity of functions with two complex variables. My question is, what would be some interesting/special functions in $\mathbb C^2$? Maybe something like the complex ...
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Is $x+\frac{x}{x^2+y^2}+i\left(y-\frac{y}{x^2+y^2}\right)$ an analytic function? [closed]

How can I find out wheter the function below is an analytical function or not? $$x+\frac{x}{x^2+y^2}+i\left(y-\frac{y}{x^2+y^2}\right)$$ I found a solution with Cauchy-Riemann equations. However, I ...
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A smooth, non-analytic real function which is not flat?

It is possible for the Taylor series around 0 of a smooth real function $f$ to converge pointwise to $f$ only in a neighborhood $(-r,r)$ of 0 such that $0<r<R$, where $R>0$ is the radius of ...
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1 answer
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When can we conclude via the Identity theorem?

Suppose $f:D_1\mapsto \mathbb{C}$ and $g:D_2\mapsto \mathbb{C}$, where $D_1, D_2$ are domains in the usual complex analysis sense, are two holomorphic functions such that the following identity holds ...
harrydiv321's user avatar
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Analytic function with constant finite radius of convergence everywhere

I am trying to construct an example of a function $f : \mathbb{R} \rightarrow \mathbb{R}$ which is analytic everywhere, where the radius of convergence of the Taylor series of $f$ at any $x \in \...
Julius's user avatar
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When is the zero set of a multivariate $p$-adic power series algebraic?

Let $f = f(z_1, \dots, z_n)$ be a power series in $n$ variables with coefficients in the $p$-adic integers $\mathbb{Z}_p$. Let $g(z_1, \dots, z_n) = f(pz_1, \dots, pz_n)$, so that $g$ converges on all ...
Ashvin Swaminathan's user avatar
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0 answers
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Real analytic manifold with curvature

Suppose $(M,g)$ is a real analytic Riemannian manifold (that is with coordinate charts that are real analytic functions with analytic inverses). Can we conclude that the manifold is always flat? The ...
Wreck it Ralph's user avatar
5 votes
1 answer
286 views

Question about the "nowhere analyticity" of the Fabius function

As you could see in Wikipedia the Fabius function is a known "example of an infinitely differentiable function that is nowhere analytic". In the following answer where it is explained how is ...
Joako's user avatar
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Some tricky path integrals without residue theory

I am trying to work out the following path integrals without residue theory and have a really hard time doing so: \begin{align*} \int_{\partial K_2(0)}\frac{e^{iz^2}-1}{z^2}dz \...
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curve selection lemma

$\quad$I am recently reading Proof of the Gradient Conjecture of R. Thom by Kurdyka, Mostowski, and Parusinski and I have a question about how to apply the curve selection lemma (CSL). Curve Selection ...
polyroot's user avatar
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is my analytical solution for Heat Equation correct?

Could you please check my analytical solution for accuracy and correctness? Consider the heat equation: $$ u_t(x, t)=u_{x x}(x, t) $$ where $$ 0<x<1, \quad t>0 $$ Neumann Boundary condition: $...
servus1991's user avatar
2 votes
2 answers
228 views

Can Schwartz class functions be nowhere analytic?

By "Schwartz class functions" I will be referring to the functions of the Schwartz space on $\mathbb{R}$, that is, smooth ($\mathcal{C}^\infty$) functions $f : \mathbb{R} \to \mathbb{R}$ ...
Bruno B's user avatar
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Find the domain of analyticity of a function $\sum_{n=0}^{+\infty}\frac{(-1)^n}{z-n}_, \ \ \ z\in\mathbb{C}.$

Find the domain of analyticity of a function $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{z-n}_, \ \ \ z\in\mathbb{C}.$$ If there's uniform convergence, there's also analyticity. Let's use Weierstrass's test ...
fragileradius's user avatar
1 vote
1 answer
63 views

Is $f(x+\epsilon i) \approx f(x)+\epsilon i \frac{d}{dx}f(x)$ approximation valid?

My understanding is that, in complex domain, if $f$ is a holomorphic function around $z_0$, it can be locally expressed as its own Taylor series, therefore $f(z_0+\epsilon i) \approx f(z_0)+\epsilon i ...
Mirar's user avatar
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1 answer
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How do I find the principal branch?

I'm trying to find the domain on which a function is analytic, specifically, $\text{Log}\left(\frac{1}{z}+i\right)$. Would I need to find $\text{Log}\left(\frac{1}{z}+i\right)=\ln(r)+i\theta$ to find ...
Flloyd56's user avatar
2 votes
2 answers
67 views

Is $z \mapsto \int f(x,z) K_z(x) dx$ analytic if $K_z(x)$ is a distribution analytic in the paramter $z$?

Let $K_z(x) \in \mathcal D'(\mathbb R)$ be a distribution kernel depending analytically on $z \in O \subset \mathbb C$, i.e. for $f \in \mathcal D(\mathbb R)$ fixed, $$ \int K_z(x) f(x) dx $$ is ...
Cream's user avatar
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Branch cut of the function log(z + i).

I have a short question. I have the following function $f\left(z\right) = \log\left(z + i\right)$. The question is, construct a branch cut such that $f\left(z\right)$ is analytic at $f\left(0\right) = ...
Oliver4's user avatar
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5 votes
1 answer
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If $f$ is analytic defined on $D:=\{z:|z|<1\}$ and $|f(z)|\le 1$, can $f$ be extended continuously to $\overline D$?

If $f$ is analytic defined on $D:=\{z:|z|<1\}$ and $|f(z)|\le 1$, can $f$ always be extended continuously to $\overline D$? In other words, can all analytic functions $f:D\to\overline D$ be ...
Akiva Weinberger's user avatar
1 vote
0 answers
130 views

Why is $f(z) = \frac{1}{1-|z|^2}$ analytic?

I have been trying to find an analytic onto function from $\mathbb{C} \to D$ where $D$ is open unit disc. I had a few candidate functions in mind, but couldn't prove that they are analytic due to a $\...
Axo's user avatar
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Does the zero locus of a real analytic function have a smooth point?

Let $F \colon \mathbb{R}^m \to \mathbb{R}^n$ be a real analytic function (i.e., each of its component functions is real analytic). Does the set $$Z = \{x \in \mathbb{R}^m : F(x) = 0\}$$ have a smooth ...
ccriscitiello's user avatar
4 votes
1 answer
116 views

Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus?

Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus? More explicitly, is there a surjective real-analytic mapping $f \colon S^2 \to T^2$ where $$S^2 = \{x \in \mathbb{R}^3 : \|x\| =...
ccriscitiello's user avatar
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1 answer
106 views

If $f$ is analytic in $\Omega\subset\mathbb{C}$, then why is $f$ bounded in a neighbourhood of $z_0 \in \Omega$?

The notion of analyticity I am working with is that a complex function $f:\Omega\subset\mathbb{C}\to\mathbb{C}$ is analytic in $\Omega$ if for any $z_0 \in \Omega$ $f(z) = \sum_{n=0}^\infty a_n(z - ...
Cartesian Bear's user avatar
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1 answer
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To prove a function is analytic on the upper half plane.

The map $z\mapsto \int_0^\infty e^{itz}dt$ is well defined on the upper half plane $\mathbb{H}=\{z\in\mathbb{C}: \Im{z}>0\}.$ Is it also analytic on $\mathbb{H}?$ I tried to prove using Morera's ...
Mathemajician's user avatar
3 votes
2 answers
734 views

Existence of Taylor series and Analyticity

I have realised my understanding of Taylor series is not as complete as I would like it to be and therefore have formulated some questions which I am struggling to find answers that I can understand: ...
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