Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

An analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. Besides, not all infinitely differentiable real function are analytic; for instance the fonction $f\colon\mathbb{R}\longrightarrow\mathbb{R}$ defined by $f(x)=\exp\left(-\frac1{x^2}\right)$ if $x\neq0$ and such that $f(0)=0$ is infinitely differentiable, but not analytic. On the other hand, every differentiable function from an open non-empty subset of $\mathbb C$ into $\mathbb C$ is analytic.

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.

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Why is the notion of analytic function so important?

I think I have some understanding of what an analytic function is — it is a function that can be approximated by a Taylor power series. But why is the notion of "analytic function" so important? I guess being analytic entails some more interesting…
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Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot be extended to complex entire function always. I…
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What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which I will drop the notation now). The function $F$…
Jeremy Upsal
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On certain algebraic functions on the interval $[0, 1]$

Let $\mathcal{C}$ be the class of continuous and polynomially bounded functions that map the interval [0, 1] to [0, 1]. A function $f(x)$ is polynomially bounded if both $f$ and $1-f$ are bounded below by min($x^n$, $(1-x)^n$) for some integer $n$…
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Show that $f(z)=\sum_{n=0}^{\infty}z^{2^n}$ can't be analytically continued past the unit disk.

I'm reading the problems of Stein and Shakarchi's Complex Analysis, Chapter 2 Problem 1 asks to show that $$f(z)=\sum_{n=0}^{\infty}z^{2^n}$$ cannot be analytically continued past the unit disk. (Hint: Suppose $\theta =\frac{2\pi p}{2^k}$ for…
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Theorems that give sufficient condition for a $C^{\infty}$ function to be analytic

What are general theorems that give sufficient criteria for a $C^{\infty}$ function to be analytic? The more general/simple the test, the better. I'm trying to understand in a more thorough way what prevents $C^{\infty}$ functions from being…
Vik78
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The Lebesgue measure of zero set of a polynomial function is zero

Suppose $f :\mathbb R^n \to \mathbb R$ be a non zero polynomial(more generally smooth) function.Suppose $Z(f)=\{ x \in \mathbb R^n \mid f(x)=0 \}$. Show that Lebesgue measure of $Z(f)$ is zero. I am trying to use induction on $n$.The result holds…
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An integral involving a smooth function

Let $f : [0,1] \to \mathbb [0,1]$ be a smooth function (class $C^\infty$) that is not necessarily real-analytic. Let $g : (-1, \infty) \to \mathbb R$ be the function defined by $g(x) = \int_0^1 f(t) \, t^x dt$. Is $g$ necessarily a real-analytic…
Vladimir Reshetnikov
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Why does $f(z) = z^n$ have no antiderivative only for $n=-1$?

The complex valued function $f(z) = z^n$ has an analytic antiderivative on $\mathbb{C} \setminus \{0 \}$ for every $n$ except for $n=-1$. What is so special about $-1$? To show why this is such an anomaly, imagine if $z^n$ had an analytic…
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Asymptotic behaviour of $f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!}$ for $\varepsilon\in(0,1)$

Let $\varepsilon \in (0, 1)$ and consider the analytic function $$f(x) = \sum_{n=1}^\infty n^\varepsilon \frac{x^n}{n!}.$$ What is the order of growth of $f(x)$ as $x \to \infty$? From the basic inequality $1 \leqslant n^\varepsilon \leqslant n$ I…
Unit
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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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A series of nonanalytic-smooth functions $f'_n = f_{n+1}$ with finite sum?

Good day, we know, in analytic functions, examples of derivative "ladders", i.e. an infinite set of functions $$\{f_n\}_{n=-\infty}^{n=+\infty}$$ $$\frac{d}{dx}f_n=f_{n+1}$$ which have finite sums $$\sum_{n=-\infty}^{n=+\infty} f_n(x).$$…
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Analytic "Lagrange" interpolation for a countably infinite set of points?

Suppose I have a finite set of points on the real plane, and I want to find the univariate polynomial interpolating all of them. Lagrange interpolation gives me the least-degree polynomial going through all of those. Is there an analogous construct…
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On the function $n \mapsto |a_n|^{\frac 1n}$ for a given power series $\sum_{n} a_n z^n$

I am currently doing research involving power series on the unit disk in $\Bbb C$: precisely I am studying the properties of converging power series of a standard form $$ f(z)= \sum_{n=0}^\infty a_n z^n \label{1}\tag{1} $$ where (assuming…
Daniele Tampieri
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If $f$ is complex analytic on $S=\{x+iy : |x|<1, |y|<1\}$, continuous on $\bar{S}$ and bounded by $1,2,3,4$ on each side, then is $|f(0)|>2$ possible?

I'm a second-year undergraduate taking an introductory course in complex analysis. I am stuck on this problem from one of the previous year's exam: True or False: For a function $f$ analytic on $S = \{ x + iy : x \in \mathbb{R}, y \in \mathbb{R},…
user833460
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