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.

If $U$ is a subset of $\mathbb{C}$, $f$ is analytic at $x_0$ if there exists a series $$ \sum_{j=0}^\infty a_j (z-z_0)^j $$ that converges to $f$ at a neighbourhood of $z_0$. In complex analysis, analyticity is equivalent to homomorphy.

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Images of compact subsets in the plane

Let $K$ be an infinite compact subset of $\mathbb{C}$. Is it true that there exists a sequence $(f_n)_{n>0}$ of functions holomorphic in some neighborhood of $K$, such that the images $f_n(K)$ are pairwise non-homeomorphic? (Motivation for this…
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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 there exists a $\delta>0$ such that $f(x)=g(x)$ for…
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Smoothness of $\frac12[W_0(x)+W_{-1}(x)]$ for real $x<0$

The Lambert W-function, i.e. the multivalued inverse of $z=we^w$, has countably many complex-valued branches $W_k(z)$. The relations between the branches are a bit involved and are summarized here. We will consider the behavior of the $k=0,-1$…
Semiclassical
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Fundamental solution to the Poisson equation by Fourier transform

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{cases} \dfrac{1}{(2-d)\Omega_d}|x|^{2-d}\text{ for }…
Brightsun
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Existence and uniqueness of a function generalizing a finite sum of powers of logarithms

I hope to find a proof of the following conjecture: $(1)$ For every $a>0$ there is a convex analytic function $f_a:\mathbb R^+\to\mathbb R$ such that: $f(1)=0$ and $\forall x>1,\ f_a(x)=f_a(x-1)+\ln^ax$ (thus, for $n\in\mathbb N,\…
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Real-analytic $f(z)=f\left(\sqrt z\right) + f\left(-\sqrt z\right)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f\left(\sqrt z\right) + f\left(-\sqrt z\right)$$ is satisfied near the real line? Also can such functions be entire? And/Or can they be periodic with a real period $p>0$? Does…
mick
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Paley-Weiner theorem and the Fourier transform of a non-analytic smooth function

Many Paley-Weiner theorems are variations on the theme "the faster a function $f(x)$ falls off as $x \rightarrow \infty$, the smoother its Fourier transform $\tilde{f}(k)$ is as $k \rightarrow 0$." In particular, we know that if $f(x)$ decays…
tparker
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Conjecture on zeros of analytic function

I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful. Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ then $|f(z)|=1$. Is it true that the zeros of $f$ can not…
Alonso Delfín
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If $\,f^{7} $ is holomorphic, then $f$ is also holomorphic.

I need some help with this problem: Let $ \Omega $ be a complex domain, i.e., a connected and open non-empty subset of $ \mathbb{C} $. If $ f: \Omega \to \mathbb{C} $ is a continuous function and $ f^{7} $ is holomorphic on $ \Omega $, then $ f $…
felipeuni
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$\forall x \,\exists k$ s.t. $f^{(k)}(x)=0$, then $f$ is a polynomial

My friend sent me the following problem: Suppose that $f$ is real analytic on $(a,b)$, and that for all $x$ in $(a,b)$ there exists a non-negative integer $k$ such that $f^{(k)}(x)=0$. Show that $f$ is a polynomial. I believe I solved it (you…
Eric Auld
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Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show that $H$ is analytic on $\mathbb{C} \setminus…
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Complex differentiable but not analytic on circle of convergence

I'm trying to get a better handle on behavior of complex power series on the boundary of their maximal disk of convergence. I'm reading Bak-Newman's Complex Analysis, Chapter 18.1. A regular point $z_0$ on the circle bounding the maximal disk of…
bryanj
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Function analytic in each variable does not imply jointly analytic

I have heard that a function $f: \mathbb R^2 \to \mathbb R$ can be analytic in each variable (i.e. $f(x,y_0) = \sum_{n=0}^{\infty} a_n x^n, \forall x \in \mathbb R$, and the same for $y$) without being jointly analytic (i.e. $f(x,y) =…
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Does there necessarily exist such a holomorphic function?

This is an old qual problem I'm working on: Let $f:[0,1]\rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Does there necessarily exist a holomorphic function $g: \mathbb{C}\setminus\{0\}\rightarrow \mathbb{C}$ such that $f(x)-g(x)$ vanishes to…
vgmath
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What happens when I convert a Taylor series into an integral?

Suppose we have the Taylor series of an analytic function as follows: $$f(x) = \sum_{k=0}^\infty \frac{1}{k!} a_k x^k$$ Then I decide to (kind of) turn it into an integral: $$g(x) = \int_0^\infty \frac{1}{\Gamma(k+1)} a(k) x^k \, dk$$ Clearly, $f(x)…
Nick
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