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Consider $$\eqalign{f_1(x) &= (x+1)^2-1\cr f_2(x) &= \sqrt{x+1}-1\cr f_3(x) &= f_4(x) = x }$$ $f_1, f_3$ and $f_4$ being polynomials, their radius of convergence is $\infty$, while $f_2(x)$ has a Macl …

Your second version is not true for arbitrary $E \subset [0,1]$. Consider $m=1$, $f(x) = x$ and $E = [0,\epsilon - \epsilon^2] \cup [1-\epsilon^2, 1]$ where $0 < \epsilon < 1$. We have $|E|=\epsilon …

Yes, of course. Since $g(\lfloor z \rfloor)$ is analytic on $k < \text{Re}(z) < k+1$, and agrees with $f(z)$ for $z \in (k,k+1)$, it is the unique analytic continuation of $f$ from $(k,k+1)$ to the …

The continuous or smooth functions from a closed interval to $\mathbb R^3$ form an inner product space, but it is not a Hilbert space because it is not complete. The $L^2$ functions from an interval …

If $\Re(s) > 1/2+\epsilon$ for $\epsilon > 0$ we have $\log(1+p^{-s}) = \sum_{j=1}^\infty p^{-sj}/j$ so $|f_p(s)| \le \sum_{j=2}^\infty p^{-\Re(s)j}/j \le p^{-1-2\epsilon}/(1-p^{-1/2-\epsilon})$. S …

Try $P(z) = z^2 + z + 1$. Its zeros are in the sector $-2\pi/3 \le \theta \le 2\pi/3$, but the zero of $P'$ is $-1/2$ which is not.

Some cases in point: $A_1 = B_1 = \exp(z)$ entire with no zeros, $A_1 \star B_1 = I_0(2 \sqrt{z})$ with infinitely many. $$\eqalign{A_2 &= \sum_{n=0}^\infty \frac{z^n}{n!!} = 1 + z e^{z^2/2} + \sqrt …

$f(x)$ is the characteristic polynomial of its companion matrix $$ A = \pmatrix{0 & 0 & 0 & 0 & c\cr 1 & 0 & 0 & 0 & 1\cr 0 & 1 & 0 & 0 & 1\cr 0 & 0 & 1 & 0 & 1\cr 0 & 0 & 0 & 1 & 1\cr}$$ and $A^5$ ha …

Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple. For the number of real roots, see OEIS sequence A094937 and references there. EDIT: As requested b …

The distinction between "determined" and "represented" is not clear. Consider a function $f$ analytic in domain $U$ containing, say, $0$. The values of $f$ on a sequence $p_n$ of nonzero points with …

EDITED: No, it is not possible if $f_1, \ldots, f_m$ are not all constant. Suppose wlog $f_1$ is not constant. I'll ignore $f_2, \ldots, f_m$ and consider polynomials $p(f_1(z))$. For convenience …

There are separation-of-variables solutions of the form $$\eqalign{u(x,y) &= X(x) + Y(y)\cr \text{where}\cr X'' &= c - (X')^2 \cr Y'' &= -c - (Y')^2\cr}$$ There are radially symm …

I don't know how you want to define "pathological", but some of the corollaries of Runge's theorem give you functions with interesting properties. One of mine: there is a rational function $f$ such …

So you're looking at $F(z) = \sum_{m=1}^M H_m(z) R_m$. According to Wikipedia, a matrix-valued function is positive-real if Each element of $F(z)$ is analytic in the open right half-plane. Each ele …

Consider the case $a_n = 1$ if $n$ or $n-1$ is a power of $3$, $0$ otherwise, and $\xi = -1$. Since $f(z) = (1+z) \sum_{k=0}^\infty z^{3^k}$ and $\sum_{k=0}^\infty z^{3^k}$ is a lacunary series, the …