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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 is quite different from the properties of functions over the complex numbers.

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$f^{(k)}(z_0)=0$ for all $k\geq0$ implies that $f=0$

I'm trying understand that given an analytic function in a domain, if the function and all its derivatives vanish at a point, then, the function itself is zero. I was told that this is a direct ...
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Determine if $f(z) = \log(e^z+1)$ is analytic and where

I tried to substitute $z$ with $x+iy$ and then write it down as $f(z) = log(e^xcis(iy)+1) $ but it looks worse than the starting point. on original branch
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Fourier Transform of Exponentially Decaying Function Cannot Have Compact Support

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a measurable function, with $|f(x)| \le e^{-|x|}$ a.e. Then how can we prove that its Fourier transform, $\hat{f}$, cannot have compact support (unless $...
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Definition of analytic function at a point

Suppose $f:I\to \mathbb{R}$, where $I$ is an open subset of $\mathbb{R}$, is a smooth function on $I$, $f\in C^{\infty}(I)$. Let $x_0\in I$. Def. We say that $f$ is analytic on $x_0$ if the Taylor ...
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If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$.

If $f$ is analytic on $(a, b)$ containing at point $x_{0}$ with $f^{(n)}(x_{0}) = 0$ for $n \in \mathbb{N}$, prove $f(x) = 0$ for all $x$. Hi, I need help with the above problem. I'm working ...
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An even function, $f(z)$, analytic near 0 can be written as another analytic function, $h(z^2)$

If f is an even function, $f(z) = f(-z)$, and is analytic near 0, then there exists a function h, also analytic near 0, such that $f(z) = h(z^2)$ I suspect this statement is true because the ...
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Can Analytic Functions on the Circle be Characterized by Sobolev Norms?

In Exercise I.4.4 of Katznelson's book An Introduction to Harmonic Analysis we learn that the Fourier series $$f(z) = \sum_{k=-\infty}^{+\infty} f_k z^k$$ defines an analytic function in the ...
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Bounding the extrema of polynomials from $\frac{d^n}{dx^n} \exp(-1/x)$

As laid out on Wikipedia, the function $$f(x):=\begin{cases} \exp(-1/x) & x>0\\ 0 & x\le 0 \end{cases}$$ has the expression for derivatives at $x>0$, $$ f^{(n)}(x) = \frac{p_n(x)}{x^{...
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Analytic properties of a series

I was wondering about whether it is possible that the analytic properties of a series are different from the ones of the partial sums. For instance suppose we have the series $$S=\sum_{n=0}^{\infty}\...
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How to find a harmonic conjugate or not [duplicate]

Show that $u = \log|z| $ does not have a harmonic conjugate $v$ throughout $\mathbb{C}-\{0\}$, that is, show that there is no function $v$ such that $\log |z| + iv(z)$ is analytic in $\mathbb{C}-\{0\}...
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Show that $h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$ is analytic in a nbh. of zero.

Consider the function $$h(z)=\int_{0}^{1}\frac{t^2}{1-tz}dt$$ whenever the integral exists, $z \in \mathbb{C}$. Show that $h$ is analytic in a neighborhood of the origin and calculate the power ...
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Is a function still analytic if its infinite sequence of subsequent derivatives, at every point of its domain, grow faster than the factorial?

Taylor series have the nth derivative ($f^{(n)}(a)$) in their numerator and n! in their denominator. I was wondering what if the derivatives (for any point $a$) grows faster than the factorial. Does ...
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If $f$ is analytic on $\mathbb{R}$, is it necessary that $f = \sum_{n = 0}^{\infty} a_{n} x^{n}$ converges for all $x \in \mathbb{R}$?

I have two questions regarding analyticity. They are pretty easy, and I think I have them correct, but I just want to make sure. First, regarding the question in the title, I think that it is ...
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If $f$ is an entire function with $|\,f(z)|\le|\operatorname{Re}(z)|$, then $\,f\equiv 0$.

If $f$ is an entire function so that $|\,f(z)|\le|\operatorname{Re}(z)|$ for all $\mathbb{C}$, then $f\equiv0$ on $\mathbb{C}$. There are various ways showing the above property. For example, using ...
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Find a function $f$ analytic at $x_{0} = 0$ so that $f\left(\frac{1}{n}\right) = \frac{n}{n + 1}, n = 1, 2, \ldots$.

I am learning about real analytic functions on my own right now. I've been having trouble with one of the exercises, and it isn't much help that most of the resources online for analytic functions ...
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For $|q|<1$, the function $\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on $|z|<1$.

I want to prove that for $|q|<1$, the function $f(z):=\frac{(az;q)_\infty}{(z;q)_\infty}$ is analytic on the set $\{z:|z|<1\}$. My approach: We consider the sequence of functions $\{f_n\}$ ...
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Analytic continuation of representations between $SO(4)$ and $O(1,3)$ Lie algebras

$SO(4)$ transformations and Lorentz transformations are isomorphic in a neighbourhood of their identity element. Can anyone shed light about how this could lead to the analytic continuation of their ...
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Smooth and not analytic

Can someone show me, without reference to Taylor series, why a complex function can be smooth but not analytic? I do not understand it intuitively or visually either. I would like an explanation ...
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Is it impossible that gradient descent converges to strict saddle point?

Suppose $f(x), x \in X \subset R^n$, is a smooth function (or real analytic function) which has only one stationary point $x^* \in X$, ($\nabla f(x^*)=0$), and $x^*$ is a strict saddle point which ...
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Doe a smooth function map positive measure sets to positive measure sets

Suppose $f: X \subset R^n \to R^n$ is a smooth function (for example $C^2$ function), and for each $y \in R^n$, the set $f^{-1}(y)$ is finite. Do we have $f(A)$ is a positive measure set if $A$ has ...
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Understanding why f(z) can be differentiable, but not analytic

I'm working through a practice exam and can only do part of this problem. Let $f(z) = x^3 + iy^3$. Find all points where the Cauchy-Riemann equations are satisfied, and give a brief explanation of ...
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Are conformal maps between Riemannian manifolds real-analytic?

Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there exist real-analytic ...
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Wanted: Simple invertible function with the following properties

The function $$f(x) = x \tanh(\frac{\pi}{2}\sinh{x})$$ has useful properties in numerical calculations as it is an even function that converges extremely quickly to $x$ for large input values. ...
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$f(z)$ is analytic if and only if $f(z)$ cannot be written as function of $\bar z$ ( where $z\in\mathbb C$ and $\bar z$ is conjugate of $z$)

Maybe I understand it wrong. I don't feel relax if I don't ask this question. I study complex analysis myself. In the nptel's video(given below), Prof. V. Balakrishnan says that If a function of $z=...
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Extending the domain of an analytic function to a larger set

I have the following problem: Given some function $f : \mathbb C \rightarrow \mathbb C$ that is analytic on some vertical line segment, there exists some (non-trivial) rectangle around that line ...
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“Nearly-analytic” functions

Call a function $f : \mathbb{R} \rightarrow \mathbb{R}$ nearly analytic iff it is smooth, and for each point $x \in \mathbb{R}$, either $f$ is analytic at $x$, or there exist $a,b > 0$ such that $f$...
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How to show a real valued function of several variables is analytic?

Let $f:\Omega\subset\Bbb{R}^m\to\Bbb{R}^n$ be a given function. In general, how can I show that $f$ is smooth (infinitely differentiable) and analytic. I know this is a bit vaguely stated question, ...
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46 views

Differentiability of $f(x+iy)=|x|+i|y|$

I am trying to find where the function $f(x+iy)=|x|+i|y|$ is differentiable and then determine where it is analytic. I have been using the Cauchy-Riemann equations to show where a function is ...
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Is a real-analytic function which vanishes on a set of positive measure identically zero? [duplicate]

Suppose $n>1$, and let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $f$ be a real-analytic function on $U$. Suppose that $f=0$ on a subset of $U$ of positive measure. Is it true that $...
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What uniquely characterizes the germ of a smooth function?

Let $X$ be the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which are infinitely differentiable at $0$. Let us define an equivalence relation $\sim$ on $X$ by saying that $f\sim g$ if ...
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Analyticity of a Fourier transform

I was reading a physics paper and the author made a statement that I didn't follow. Suppose $p(t)$ is given by the Fourier transform of $\omega(E)$ \begin{equation} p(t) = \int_{-\infty}^{+\infty} ...
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Which one of the following are true?

Consider the function $e^{-z^{-4}}$ for $z≠0$ and $f(0)=0$. Then, (A) $f$ is not analytic. (B)$f$ is not differentiable at $z=0.$ (c)$f$ does not satisfy the C-R(Cauchy-Riemann) ...
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Error bounds for Gauss-Hermite quadrature, for analytic functions

I am working with the class of analytic functions and I want to derive some estimates, using error bounds for Gauss-Hermite quadrature, for analytic functions (based on Bernstein ellipses). The issue ...
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Extensions of real analytic functions with multiple variables

I wonder if my statements below are correct. Let $V$ be an open domain in $\mathbb{R}^d$, and $U$ an open domain in $\mathbb{C}^d$ with $V=\operatorname{Re}U:=\{\operatorname{Re}z:z\in U\}$. I want ...
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Analytic function. Bounding number of zeros with the number of critical points

Let $f$ be a real analytic function on $|x|< R$. Let $N(R,f)$ be the number of zeros of $f$ in the region of analyticity. Can we show taht \begin{align} N(R,f) \le N(R,f^\prime)+2 \end{align} ...
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Infinite sum of analytic functions becomes non-analytic

I am looking for a simple example that a series of analytic functions can become non-analytic. This is in the context of phase transitions, where one considers the analyticity of the partition ...
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Analysis of $\mathbb C^{n\times n} \to \mathbb C^{n\times n}$ functions and integrals of them.

Due to a previous question being too sloppily written by me I here try to be a bit more clear on what I wondered. For functions defined $$\mathbb C^{n\times n} \to \mathbb C^{n\times n} : A+Bi \to f(...
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Finding region where $f(z)=e^{-x}\cos(y) -ie^{-x}\sin(y)$ is analytic

The text I'm using asked me to find on what region $f(z)=e^{-x}\cos(y) -ie^{-x}\sin(y)$ is analytic. So solving the appropriate Cauchy-Riemann Equations. \begin{align} \frac{\partial u}{\partial x}=-e^...
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Condition implying that the argmax of a function is piecewise continuous.

Let $f_i$ be a smooth function mapping $[0,1)$ to $\Re$, for $i = 1, ... n$, and let $\bar f=\max f_i$. Define $T_i = \{ t \in [ 0 , 1 ) s.t. f_i(t) = \bar f(t) \}$. I'm looking for a condition on ...
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51 views

Does there exist complex function which is analytic in all $\Bbb{C}$ except for 5 points?

Does there exist complex function which is analytic in all $\Bbb{C}$ except for 5 points? I would say it exist. $$f(z)=\frac{1}{z(z-1)(z-2)(z-3)(z-4)}$$ But how should I prove that it is analytic ...
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Why are all analytic functions equivalent to their Taylor series?

"A function is analytic if and only if its Taylor series about $x_0$ converges to the function in some neighborhood for every $x_0$ in its domain." Clearly if its Taylor series converges to $f$ then ...
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A Question concerning the definition of Analytic functions

A power series is an infinite series of the form $$\sum_{n=1}^∞ a_n(x-c)^n.$$ And a function $f$ is analytic if it is locally given by a power series. Yet the formal definition of an analytic ...
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Why are these two functions different for negative numbers?

Looking at some straightforward generalizations of the Sophomore's Dream identity I derived the following identity which works for $\Re(b) > 0$: $$ \int_0^1 x^{a x^b} dx = \sum_{n=0}^{\infty} \frac{...
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Is there an analytically continued function of $z^p$ at zero?

For a rational number $p > 1$. We know that the function $z^p$ is holomorphic on $\mathbb{C} \setminus \mathbb{R}^-$ (excluding $z = 0$). Is there an analytic continuation of the function $z^p$ at ...
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extending convergence radius using one boundary point

Show that if $f(z)=\sum\limits_{n\geq0}a_nz^n$ is analytic in $\{z\in\mathbb{C}:|z|<1\}\cup\{1\}$ and $\forall n\geq0:a_n\geq0$ then the radius of convergence of the power series is strictly larger ...
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If $f(z) = u(z)+iv(z)$ is analytic, are both of $u(z)$ and $v(z)$ analytic? [closed]

I think the title says it all, if $f(z) = u(z)+iv(z)$ is analytic, are both of $u(z)$ and $v(z)$ analytic? I betting that this is not the case and that there are some obvious counterexamples, but I ...
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24 views

Proving that a function is analytical without knowing $f$ holomorphic $\Leftrightarrow$ $f$ analytic

I know that $$f:\mathbb C^*\to\mathbb C, \\ z\mapsto\frac{1}{2}\Big(z+\frac{1}{z}\Big)$$ is an analytic function since it is holomorphic. Is there any "basic" way to see that this function is analytic ...
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The continuity of linear functionals with respect to uniform convergence of entire functions on balls

Let $X$ be a Banach space and $H_b (X)$ be the algebra of complex-valued entire functions on $X$ which are bounded on bounded sets, with the topology of uniform convergence on bounded sets. Let $\...
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Understanding why this step in the proof of analyticity of holomorphic functions is necessary?

In this wikipedia article, a proof that every holomorphic function is analytic is given based on Cauchy's integral formula. I understand how they got to the following line: $$f(z)=\frac{1}{2\pi i}\...
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Give the largest open set where the function $f(z)=\sum_{n=1}^{\infty} ne^{nz}$

Give the largest open set where the function $f(z)=\sum_{n=1}^{\infty} ne^{nz}$ is analytic. Here are the steps: For the series to converge it is necessary that $ne^{nz} \rightarrow 0$. But $|e^{nz}...