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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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 $...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 $$ Neumann Boundary condition:
$...
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Solving a Heat Equation with Neumann Boundary Conditions and an initial condition
i have this partial equation (heat equation) with the initial condition and the Neumann boundary conditions
Consider the heat equation: ...
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Can Schwartz class functions be nowhere analytic?
By "Schwartz class functions" I will be referring to the functions of the Schwartz space on $\mathbb{R}$, that is, smooth ($\mathcal{C}^\infty$) functions $f : \mathbb{R} \to \mathbb{R}$ ...
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Find the domain of analyticity of a function $\sum_{n=0}^{+\infty}\frac{(-1)^n}{z-n}_, \ \ \ z\in\mathbb{C}.$
Find the domain of analyticity of a function $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{z-n}_, \ \ \ z\in\mathbb{C}.$$
If there's uniform convergence, there's also analyticity. Let's use Weierstrass's test ...
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Is $f(x+\epsilon i) \approx f(x)+\epsilon i \frac{d}{dx}f(x)$ approximation valid?
My understanding is that, in complex domain, if $f$ is a holomorphic function around $z_0$, it can be locally expressed as its own Taylor series, therefore $f(z_0+\epsilon i) \approx f(z_0)+\epsilon i ...
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How do I find the principal branch?
I'm trying to find the domain on which a function is analytic, specifically, $\text{Log}\left(\frac{1}{z}+i\right)$.
Would I need to find $\text{Log}\left(\frac{1}{z}+i\right)=\ln(r)+i\theta$ to find ...
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(Failure of termwise differentiation of $f_n(x)=\frac{e^{inx}}{in}$ for $0 \leq x \leq 2\pi$
It is clear that $f_n \to 0$ uniformly on [0,2π]. But $f^{\prime}_n(x) = e^{inx}$ and this sequence does not converge except at $x = 0$ or $x = 2π$. Consequently, $f_n$ does not converge to 0.
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Is $z \mapsto \int f(x,z) K_z(x) dx$ analytic if $K_z(x)$ is a distribution analytic in the paramter $z$?
Let $K_z(x) \in \mathcal D'(\mathbb R)$ be a distribution kernel depending analytically on $z \in O \subset \mathbb C$, i.e. for $f \in \mathcal D(\mathbb R)$ fixed,
$$
\int K_z(x) f(x) dx
$$
is ...
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Branch cut of the function log(z + i).
I have a short question. I have the following function $f\left(z\right) = \log\left(z + i\right)$.
The question is, construct a branch cut such that $f\left(z\right)$ is analytic at $f\left(0\right) = ...
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If $f$ is analytic defined on $D:=\{z:|z|<1\}$ and $|f(z)|\le 1$, can $f$ be extended continuously to $\overline D$?
If $f$ is analytic defined on $D:=\{z:|z|<1\}$ and $|f(z)|\le 1$, can $f$ always be extended continuously to $\overline D$? In other words, can all analytic functions $f:D\to\overline D$ be ...
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Cauchy-Riemann equations for $tan(2z)$
Given $w=\tan(2z)$
We can separate $w$ in their real and imaginary parts.
$$Re(w) = u(x, y)=\frac{\sin{4y}}{\cos{4x}+\cosh{4y}}$$
$$Im(w) = v(x, y)=\frac{\sinh{4y}}{\cos{4x}+\cosh{4y}}$$
The Cauchy ...
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weaker notion of analytic continuation for functions in quasi analytic classes?
Say for $x>1$ a function is $C^\infty$ and even in a quasi-analytic class, but not real analytic. Then of course there is no analytic continuation for this region.
Is there some weaker notion of ...
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Why is $f(z) = \frac{1}{1-|z|^2}$ analytic?
I have been trying to find an analytic onto function from $\mathbb{C} \to D$ where $D$ is open unit disc. I had a few candidate functions in mind, but couldn't prove that they are analytic due to a $\...
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Does the zero locus of a real analytic function have a smooth point?
Let $F \colon \mathbb{R}^m \to \mathbb{R}^n$ be a real analytic function (i.e., each of its component functions is real analytic). Does the set
$$Z = \{x \in \mathbb{R}^m : F(x) = 0\}$$
have a smooth ...
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Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus?
Is there a surjective $C^{\omega}$ map from the 2-sphere to the 2-torus?
More explicitly, is there a surjective real-analytic mapping $f \colon S^2 \to T^2$ where
$$S^2 = \{x \in \mathbb{R}^3 : \|x\| =...
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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 - ...
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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 ...
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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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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 $\mathbb{C}\setminus\{0\}$ ...
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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 $...
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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....
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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, ...
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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)=\...
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Is piece-wise defined function with arccos and arcosh analytic at 1?
Let $f:\Bbb R^{\geqslant-1}\to\Bbb R$ be
$$f(x)=\begin{cases}
\dfrac{x\sqrt{1-x^2}}{\arccos(x)}, &-1\leqslant x<1\\
1, &x=1\\
\dfrac{x\sqrt{x^2-1}}{\ln(x+\sqrt{x^2-1})}, &x>1
\end{...
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Understanding Real analyticity
I'm going to state my assumption of the definition of Real analyticity, and how I understand it based on my current understanding. Please tell me if they are correct or not and please help me ...
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Description of complete analytic function for $\sqrt{4z-\sqrt[3]{z}}$.
The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$.
I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$
So, there ...
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Asymptotic expansion of non analytical function
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real smooth (not necessarily analytical) function.
Suppose I tell you that $f(t)$ admits a full asymptotic, at $t\rightarrow0$ expansion up to all orders ...
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What is an analytic space?
What is an analytics space? I have seen more definitions, so my question is - what is the version used the most? And is it all the same notion, or can there be confusion regarding more terms with the ...
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Analytic functions at points other than zero
I am confused about the definition of analytical functions on real line, or, to be more precise, how this definition is used. According to Wikipedia
Analytic function is a function that is locally ...
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A generic complex contour integral
If, let's say, I have an integral such as
$$\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 ...
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