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You can transform (1) into the equation of the flow of a 2-dimensional real vector field with time $\omega$ (just differentiate it with respect to $\omega$). Then you plug the resulting system into an …

Mohammad, I do believe your integral is divergent. Taking for granted the computations that led you to your last comment, I can assure you that the integral $\int_{1}^{+\infty}\sin\left(\frac{\ln{x}} …

This is a "no-answer" answer. If for some $n$ the quantity $M_n:=\sup_{r\to\infty}\frac{\log_n(r,F)}{\log r}$ is finite then $M_{n+k}=0$ for all $k>0$. Therefore you are asking for entire functions wi …

If the degree of $P$ is greater than $1$ then the Julia set $J$ of $P$ is a nonempty perfect compact set of $\mathbb C$, completely invariant by $P$. Obviously $f$ is constant on any orbit $(P^{\circ …

At some point I needed to prove that some formal (iterative) construction yielded an actual convergent power series. To do so I was led to prove the following lemma, where $\mathcal{B}(\Delta)$ is the …

Edit: the answer is given by user1952009 in the comments when $\infty$ is locally accessible from $\mathbb{C}\setminus{S}$, but I'll leave my original incomplete answer here for illustration. Note th …

I posted this question on MSE a few time ago, but it did not receive much attention. I thought there might be an elementary answer so didn't want to post it directly on MO. My apologies if this questi …

In what follows we assume $\Re(a)>0$ and $\Re(b)>0$. Begin with the case $a+b=k\in\mathbb N$. Using Pochhammer contour $P$, one can relate what's going on on $[0,1]$ to what is going on on a circle …

There is a paper of Breuer and Simon, "Natural Boundaries and Spectral Theory" (some slides here ). They give, among other things, the definition of "strong natural boundary". This concept relates to …

EDIT: at the time of this answer the OP did not specify that the zero should be real. The order (as an entire function) of $E_{\alpha,1}$ is $\frac{1}{\alpha}$. It so happens that entire functions wi …

The equation $\Phi(w,z)=0$ can be solved using Puiseux series. If $\frac{\partial{\Phi}}{\partial{w}}\not\equiv 0$ then there exist finitely many formal series $f(z)=\sum_{n\geq0}a_nz^{n/p}$ such t …

I know that's been a while now that the question has been asked, but as I'm looking more or less into this topic, I think I should share some of my discoveries in the literature. I'm somewhat amazed a …

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path …

It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant coeff …

I would like to point out the following very nice reference (unfortunately I cannot plug into MathSciNet right now and only have a preprint version on my computer, but you should be able to track it d …