Mathematics Stack Exchange
Mathematics Stack Exchange

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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Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?

It is well-known that the function $$f(x) = \begin{cases} e^{-1/x^2}, \mbox{if } x \ne 0 \\ 0, \mbox{if } x = 0\end{cases}$$ is smooth everywhere, yet not analytic at $x = 0$. In particular, its Taylor series exists there, but it equals $0 + 0x +…
The_Sympathizer
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Is this function nowhere analytic?

One usually sees $f(x):=\exp\frac{-1}{x^2}$ as an example of a $C^\infty$ function that is not analytic, having one point of non-analyticity (the point $0$). The Fabius function is a canonical example of a $C^\infty$ function that is non-analytic on…
s.harp
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Fibonorial of a fractional or complex argument

Let $F(n)$ denote the $n^{\text{th}}$ Fibonacci number$^{[1]}$$\!^{[2]}$$\!^{[3]}$. The Fibonacci numbers have a natural generalization to an analytic function of a complex argument: $$F(z)=\left(\phi^z - \cos(\pi…
Vladimir Reshetnikov
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How badly-behaved are the derivatives of non-analytic smooth functions?

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a smooth function such that $f^{(n)}(0) = 0$ for all $n \in \mathbb{N}_{\geq 0}$ and that $f$ is not analytic. In particular, we assume that $f$ is not identically $0$ in any neighbourhood of $x=0$. Does it…
MathematicsStudent1122
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The Identity Theorem for real analytic functions

What is the condition for two real analytic functions to be identically equal? We know that there is a nice condition (Identity Theorem) for holomorphic function to check if they are the same. What is its version for real analytic functions?
Yui
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Function $f(x)=\int_0^\infty\left|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\right|\,dt$

Let $$f(x)=\int_0^\infty\Big|\sin(t)\cdot\sin(x\,t)\cdot e^{-t}\Big|\,dt,$$ where $|\dots|$ denotes the absolute value. We are concerned only with positive values of $x$ (i.e. let the domain of the function be $\mathbb{R}^+$). The graph of this…
Vladimir Reshetnikov
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Can a function "grow too fast" to be real analytic?

Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for all real analytic functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$, there exists a real number $y$ such that $\: x < y \:$ and $\:…
user57159
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Proving that a doubly-periodic entire function $f$ is constant.

Let $f: \Bbb C \to \Bbb C$ be an entire (analytic on the whole plane) function such that exists $\omega_1,\omega_2 \in \mathbb{S}^1$, linearly independent over $\Bbb R$ such that: $$f(z+\omega_1)=f(z)=f(z+\omega_2), \quad \forall\,z\in \Bbb…
Ivo Terek
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What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?

The following appears as the second-to-last problem of Stewart's Complex Analysis: Describe the Riemann surface of the function $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$. This problem intimidated me when I first saw it as an undergrad, as the…
Semiclassical
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composition of power series

Does anyone know how to derive a formula for the coefficients. That is if, $f(x)=\sum _{n=0}^{\infty } a_nx^n$ and $g(x)=\sum _{n=0}^{\infty } b_nx^n$ suppose the compostion is an analytic function, $h(x)=f(g(x))=\sum _{n=0}^{\infty } c_nx^n$ Is…
aukie
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Analytic continuation of a real function

I know that for $U \subset _{open} \mathbb{C}$, if a function $f$ is analytic on $U$ and if $f$ can be extended to the whole complex plane, this extension is unique. Now i am wondering if this is true for real functions. I mean, if $f: \mathbb{R}…
ThePortakal
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The distinction between infinitely differentiable function and real analytic function

I have known that all the real analytic functions are infinitely differentiable. On the other hand, I know that there exists a function that is infinitely differentiable but not real analytic. For example, $$f(x) = \begin{cases} \exp(-1/x), &…
Juntao Huang
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Hilbert's 19th problem: Why do we care?

Hilbert's 19th problem asks: Are the solutions of regular problems in the calculus of variations always necessarily analytic? This was proven to be true (through the work of Sergei Bernstein, Ennio de Giorgi, John Nash, among others). My question…
user13888
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How many smooth functions are non-analytic?

We know from example that not all smooth (infinitely differentiable) functions are analytic (equal to their Taylor expansion at all points). However, the examples on the linked page seem rather contrived, and most smooth functions that I've…
tba
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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…
Math Lover
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