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We can say a bit more: if a Neumann series converges point-wise, it also converges absolutely and in $\mathbb C[[z]]$, and the limit is an entire function. Let $(a_n)_{n\ge0}$ be a sequence of complex …

edit: This was meant to recall a first elementary but relevant fact that was not mentioned at all, that is orthogonality. Bessel functions $\{J_n\}_{n\in\mathbb N_+}$ are orthogonal on $\mathbb R$ w.r …

As for the sum $$\sum_{k\geq1}(-1)^k\binom{k}{\frac23}^{-1}$$ one can evaluate it by means of the Beta function integral, like in this recent computation. $$\sum_{k\geq1}(-1)^k\binom{k}{\frac23}^{-1}= …

I do not have a reference, but I’d say it is a plain instance of the Implicit Function Theorem for formal power series. I add the computation below, in case you had a different one. We choose our inde …

An elementary argument based on the identity principle for power series. If $\text{deg}(P)>1$, it has a fixed point $z_0$. We can assume $z_0=0$ (we can replace $f$ with $F(z):=f(z_0+z)$ and $p$ wit …

To make it a bit quantitative, let $\omega$ be a modulus of continuity of $u$. Extend $u\in C^0(B_1)$ to $u_0(x):=u\big(\frac x{ \|x\|\vee1 }\big)$, which incidentally has modulus of continuity $ \om …

Alternatively, wlog the rectangle $R$ is $[0,1]\times[0,1]$, and denoting $L_1$, $L_2$ the corresponding parametrization curves, one may also consider the map $$[0,1]\times[0,1]\ni (t,s)\mapsto L_1(t) …

This has indeed a quick proof for Jordan arcs, if you include the usual property of separation of Jordan curves. To avoid complications due to $L_1$ and $L_2$ possibly having other points in the boun …

Let's prove the equivalent claim: The only complex solutions to $2z=z^{n+1} +1$ with $|z|\le 1$ are $z={1\over\alpha}$ and $z=1$. Remark: The number $\beta:={1\over\alpha}$ is the minimum fixed …

"Objection, the question assumes facts not in evidence!" Talking about the general question as in the title, I wonder in what measure can we say that lacunary series are particularly badly behaved. M …

I think it should be true in the stronger assumption $q>p+2$, but I suspect it could be false for integer $p\ge1$ and $q=p+2$, and an even $k$. Let me explain why -it could be possibly useful to make …

For $n\ge 3$, let's take $z_j$, for $1\le j\le n$, be the $n$-th roots of unity, and $w_j:=z_j+4\bar{z_j}$, so that the assumption are satisfied by $$f(z):=z+4z^{n-1}$$ $$g(z):=4z+z^{n-1}.$$ Howev …

Here's a proof that also uses Bernstein's representation theorem for totally monotone functions (I wondered if that result was needed, and Alexander Eremenko's answer now makes me think, that it is). …

We may assume w.l.o.g $0<\beta<1$. Write $\beta^{-1}$ in the $ \beta^{-2}$ expansion as: $$\beta^{-1}=\sum_{k=0}^\infty d_k\beta^{2k}$$ with integer digits $0\le d_k<\beta^{-2}$, and define $$f(x):=-1 …

If you consider the conjugate of $I:=\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{k=1}^n\prod\limits_{j=1}^n(x_k^{\frac{1}{2}}-iy_j^{\frac{1}{2}})^2dx_1\cdots …