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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
34 views

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
1 vote
0 answers
30 views

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 ...
Rune H's user avatar
  • 11
0 votes
1 answer
33 views

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'...
Reaper's user avatar
  • 1
0 votes
4 answers
74 views

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: ...
Elise9's user avatar
  • 67
0 votes
0 answers
30 views

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
333 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
0 votes
1 answer
32 views

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 $$|\hat{u}_k|\leq K e^{-a|k|}.$$ ...
SoupMath's user avatar
0 votes
0 answers
13 views

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 ...
John Zimmerman's user avatar
0 votes
1 answer
69 views

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_{...
uoiu's user avatar
  • 551
1 vote
0 answers
31 views

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 ...
syphracos's user avatar
  • 464
0 votes
1 answer
112 views

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 ...
disx's user avatar
  • 27
0 votes
0 answers
48 views

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 ...
kavsnzre's user avatar
  • 121
1 vote
1 answer
100 views

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
2 votes
0 answers
33 views

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
  • 941
3 votes
0 answers
100 views

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
1 vote
0 answers
36 views

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
187 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
  • 1,213
1 vote
1 answer
98 views

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 \...
MilesDefis's user avatar
1 vote
0 answers
29 views

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
0 votes
0 answers
29 views

Can someone help me to compare 4 symbolic equations analytically in Python?

I have 4 different profit functions and I would like to compare them analytically and find under which conditions what is that ranks of profit functions. For example, I expect an outcome as follows: ...
mhc's user avatar
  • 1
1 vote
0 answers
53 views

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
0 votes
1 answer
247 views

Solving a Heat Equation with Neumann Boundary Conditions and an initial condition

i have this partial equation (heat equation) with the initial condition and the Neumann boundary conditions Consider the heat equation: ...
servus1991's user avatar
2 votes
2 answers
137 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
  • 3,846
0 votes
1 answer
34 views

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
62 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
  • 318
0 votes
1 answer
59 views

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
0 votes
0 answers
34 views

(Failure of termwise differentiation of $f_n(x)=\frac{e^{inx}}{in}$ for $0 \leq x \leq 2\pi$

It is clear that $f_n \to 0$ uniformly on [0,2π]. But $f^{\prime}_n(x) = e^{inx}$ and this sequence does not converge except at $x = 0$ or $x = 2π$. Consequently, $f_n$ does not converge to 0. ...
James R.'s user avatar
2 votes
2 answers
65 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
  • 392
1 vote
0 answers
56 views

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
  • 21
5 votes
1 answer
79 views

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
0 votes
0 answers
35 views

Cauchy-Riemann equations for $tan(2z)$

Given $w=\tan(2z)$ We can separate $w$ in their real and imaginary parts. $$Re(w) = u(x, y)=\frac{\sin{4y}}{\cos{4x}+\cosh{4y}}$$ $$Im(w) = v(x, y)=\frac{\sinh{4y}}{\cos{4x}+\cosh{4y}}$$ The Cauchy ...
AlexSp3's user avatar
  • 325
0 votes
0 answers
22 views

weaker notion of analytic continuation for functions in quasi analytic classes?

Say for $x>1$ a function is $C^\infty$ and even in a quasi-analytic class, but not real analytic. Then of course there is no analytic continuation for this region. Is there some weaker notion of ...
John Zimmerman's user avatar
1 vote
0 answers
69 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
  • 279
3 votes
0 answers
80 views

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
82 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
0 votes
1 answer
71 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
0 votes
1 answer
33 views

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
339 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: ...
awgya's user avatar
  • 289
1 vote
1 answer
251 views

Suppose $f$ is holomorphic on $\mathbb{C}\setminus\{0\}$ and $f(n)=(-1)^n$ for each positive integer $n$. Prove $\inf_{z\neq 0} |f(z)|=0$.

This is a question from a previous complex analysis qualifying exam that I'm working through to study for my own upcoming exam. Problem: Suppose that $f$ is holomorphic on $\mathbb{C}\setminus\{0\}$ ...
Serafina's user avatar
  • 470
2 votes
1 answer
113 views

What does it mean when the second Wirtinger derivative is sometimes zero?

I have to show that $f(z) =\sin (\bar z)$ is not analytic anywhere. One way is to check the CR equations by letting $\sin (z) = \sin(x+iy)$ and do some algebra. From the CR equations I obtained that $...
Teo LC's user avatar
  • 47
1 vote
0 answers
61 views

Global version of Cauchy-Kowalevskaya theorem

Does there exist a global version of the Cauchy-Kowalevskaya theorem for linear PDEs ? Global in the sense that the solution exist and is analytic on $\mathbb{R}$ if so are the coefficients of the PDE....
perturbation's user avatar
2 votes
0 answers
64 views

Why is the image of a compact complex manifold analytic?

Suppose $f: X \to Y$ is a holomorphic map of connected compact complex manifolds. Is there an easy way to see that $f(X)$ is analytic (i.e. Zariski closed) in $Y$? It is clear that $f(X)$ is closed, ...
red_trumpet's user avatar
  • 7,400
2 votes
1 answer
51 views

Fourier transform $\hat{f}(\gamma)$ analytic in whole complex plane if $f\in L^2(\mathbb{R}^1)$ is compact?

How to show that Fourier transform $\hat{f}(\gamma)$ analytic in whole complex plane if $f\in L^2(\mathbb{R}^1)$ is compact? Attempt: Since $f \in L^2(\mathbb{R}^1)$ we have that $\hat{f}(\gamma)=\...
TOMILO87's user avatar
  • 500
1 vote
1 answer
34 views

Is piece-wise defined function with arccos and arcosh analytic at 1?

Let $f:\Bbb R^{\geqslant-1}\to\Bbb R$ be $$f(x)=\begin{cases} \dfrac{x\sqrt{1-x^2}}{\arccos(x)}, &-1\leqslant x<1\\ 1, &x=1\\ \dfrac{x\sqrt{x^2-1}}{\ln(x+\sqrt{x^2-1})}, &x>1 \end{...
emacs drives me nuts's user avatar
2 votes
1 answer
127 views

Understanding Real analyticity

I'm going to state my assumption of the definition of Real analyticity, and how I understand it based on my current understanding. Please tell me if they are correct or not and please help me ...
Ian Ambrose's user avatar
3 votes
0 answers
69 views

Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.

The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$. I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there ...
aetius's user avatar
  • 53
0 votes
0 answers
78 views

Asymptotic expansion of non analytical function

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function. Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
José Duarte de Azevedo e Cunha's user avatar
1 vote
0 answers
144 views

What is an analytic space?

What is an analytics space? I have seen more definitions, so my question is - what is the version used the most? And is it all the same notion, or can there be confusion regarding more terms with the ...
Tereza Tizkova's user avatar
0 votes
1 answer
51 views

Analytic functions at points other than zero

I am confused about the definition of analytical functions on real line, or, to be more precise, how this definition is used. According to Wikipedia Analytic function is a function that is locally ...
Motoko's user avatar
  • 35
0 votes
1 answer
61 views

A generic complex contour integral

If, let's say, I have an integral such as $$\int_{-\infty}^{\infty}\frac{dt}{(t-a)(t-b)(t-c)(t-d)}$$ where $a$, $b$, $c$ and $d$ are complex numbers such that the poles falls in all the four quadrant ...
Sayan's user avatar
  • 121

1
2 3 4 5
20