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As a matter of fact, real entire functions (that is, entire functions that map the real line into itself, or equivalently, functions represented by a power series centered in 0, with real coefficient …

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 they have a suitable background in linear algebra, I would not omit the interpretation of complex numbers in terms of conformal matrices of order 2 (with nonnegative determinant), translating al …

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 …

Start by the definition of the Bernoulli numbers via their generating function: we have $$\big(e^x-1\big)\sum_{k=0}^\infty B _ k\frac{x^k}{k!}=x\, .$$ Therefore, writing with your notation $ t _ n(x)= …

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). …

I fear this is possibly a bit too abstract for your purposes, but tells that there is such a complete distance topologically equivalent to the uniform distance, and can be made more concrete (check th …

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 …

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 …

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) …

I'm quite sure one could camuflage any complex analysis technique into a real calculus computation, leaving no appeal even to the notion of complex numbers. For instance, the Cauchy's formula on a cir …

Assuming $V:=U=(-\alpha,+\alpha)$, we want the sequence of real numbers $$B_n:=\bigg( \frac{\| f^{(n)} \|_{\infty,U}} {n!M_n} \bigg) ^{1/n}$$ to be bounded (in which case $B:=\sup_{n\in\mathbb{N}}B_n$ …

Consider, for $N=1$, the case where $M$ is the open unit disk $B_1$: diffeomeorphic to $\mathbb{R}^2$, not biholomorphic to $\mathbb{C}$. The compact $K:=\{0\}$ is actually such that $M\setminus K$ …

Yes. The triangle wave $s(x):=\min_{k\in\mathbb{Z}}\big|x-k\big|$ has an absolutely convergent Fourier series $$s(x)=\frac{1}{4}-\frac{2}{\pi^2}\sum_{k=0}^\infty\frac{1}{(2k+1)^2}\cos\big(2\pi (2k+1 …