Intereting Posts

Stability of periodic solution
Enumerate certain special configurations – combinatorics.
What is a conjugacy class of reflection?
Totally disconnected space
$i'=i^{-1} \bmod p$, prove or disprove that $\lim_{p\to \infty}\dfrac{1}{p^3}\sum_{i=1}^{p-1}ii'=\frac{1}4$
Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$
The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$
Pólya and Szegő, Part I, Ch. 4, 174.
How to apply Borel-Cantelli Lemma here?
No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$
How would one be able to prove mathematically that $1+1 = 2$?
Understanding the concept behind the Lagrangian multiplier
What resources are there for learning Russian math terminology?
Braid invariants resource
Calculate the series expansion at $x=0$ of the integral $\int \frac{xy\arctan(xy)}{1-xy}dx$

Does there exist an infinitely differentiable function $f:U\to\mathbb{R}$, where $U$ is open subset of $\mathbb{R}$, such that

- the Taylor series of $f$ at $x=x_0\in U$ has radius of convergence $R>0$
- $f$ equals its Taylor series only on the subinterval $(x_0-r,x_0+r)$, where $\color{red}{0<}r<R$

The customary examples of smooth real functions that fail to be analytic, e.g. $e^{-1/x}$ or $e^{-1/x^2}$ at $x=0$, have $R=\infty$ but $r=0$. The substance of the question is whether we can find a less extreme example for which analyticity at $x=x_0$ gives out only at a *nonzero* radius smaller than the radius of convergence of the Taylor series.

Note: I don’t really know complex analysis, but I know that the easiest path to whatever the truth is here is probably through the complex domain.

- Function analytic in each variable does not imply jointly analytic
- Expressing the area of the image of a holomorphic function by the coefficients of its expansion
- what is the relation of smooth compact supported funtions and real analytic function?
- Fibonorial of a fractional or complex argument
- What is the Riemann surface of $y=\sqrt{z+z^2+z^4+\cdots +z^{2^n}+\cdots}$?
- Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

- Can it be proved that a Meromorphic function only has a countable number of poles?
- If a holomorphic function $f$ has modulus $1$ on the unit circle, why does $f(z_0)=0$ for some $z_0$ in the disk?
- Image of complex circle under polynomial
- Existence of an holomorphic function
- Holomorphic functions: is it true that $f(\bar{z})=\overline {f(z)}$?
- Infinite products - reference needed!
- What word means “the property of being holomorphic”?
- Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?
- Evaluate the integral $\int_0^{2 \pi} {\cos^2 \theta \over a + b \cos \theta}\; d\theta$
- mapping properties of $(1−z)^i$

$\newcommand{\Reals}{\mathbf{R}}$Yes: Fix $r > 0$. The function $f:\Reals \to \Reals$ defined by

$$

f(x) = \begin{cases}

0 & |x| \leq r, \\

e^{-1/(|x| – r)^{2}} & |x| > r,

\end{cases}

$$

has Taylor series equal to $0$ (radius $\infty$), but agrees with its Taylor series only on $(-r, r)$.

If you want finite radius instead, add your favorite analytic function with radius $R > r$, e.g.,

$$

g(x) = \frac{1}{x^{2} + R^{2}}.

$$

- Why is $a$ and $b$ coprime if $a\equiv 1 \pmod{b}$?
- Compute $\int_0^{\pi/2}\frac{\cos{x}}{2-\sin{2x}}dx$
- Discriminant of a binary quadratic form and Jacobi symbol
- Why are quotient metric spaces defined this way?
- Quotient space and continuous linear operator.
- An inequality on the root of matrix products (part 2 – the reverse case)
- Derivative of a monotone function that has a finite limit as x goes to infinity
- Is the matrix $A$ diagonalizable if $A^2=I$
- definition of winding number, have doubt in definition.
- Disjoint $AC$ equivalent to $AC$
- Can it happen that the image of a functor is not a category?
- Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?
- Using the LRT statistic to test $H_0$ vs $H_1$
- Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $
- Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface