I want to see if this series converges or not: $$ \sum_{n=1}^\infty n^{-1/2}\sin(n)\sin(n^2). $$ I tried comparison tests but nothing. I saw that integral criteria works but I don't know how to show that. Thank you

I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit. 0. Introduction The starting point of this question is this important article by Mutesa et al. where a hyper...

My question refers to some not very well known (and unpublished) fragments of Gauss that treat the problem of finding a conformal mapping (angle-preserving mapping) in the complex plane from the interior of the ellipse to the interior of the unit circle. These fragments date from 1839, much later...

By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of circles and radii around the center of the disk, like on this image taken from this site: I am ...

I need to answer the following question, hopefully in the negative. Question: Does there exist a conformal map $f$ of degree $1$ from the annulus $\{1<|z|<R\}$ to the punctured disk $\{0<|z|<r\}$, such that $f$ extends to a continuous map $\{1\leq|z|<R\}\rightarrow\{|z|<r\}$ sending the inner...

In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli are conformally equivalent iff the ratio of the outer radius and the inner radius is the same for ...

Remove the closure of simply connected region from the interior of a simply connected region. Is it true that the resulting domain can be mapped conformally to some annulus?

Let in the complex plane be a bounded Jordan region T (that is a bounded and simply connected set with the boundary a Jordan curve), containing the origin, with its Riemann mapping onto the open unit disk, having in its Taylor expansion all the coefficients as real numbers. Question : There exis...

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the compact-open topology) have the structure of a manifold in any sense? b) Is there even a notion of ...

Consider two doubly-connected open subsets $A$ and $A'$ of the Riemann sphere. We assume these two domains to be of same modulus (the moduli space being one real parameter), i.e. we assume that there exists a holomorphic bijection $\phi:A\rightarrow A'$. Note that the map $\phi$ is then unique up...

I am looking for some results on the boundary behavior of conformal maps between simply connected domains. In particular I am interested in conformal maps between $\mathbb{C}-\Delta$, where $\Delta$ is an interval (in general, $\mathbb{C}-\Gamma$, where $\Gamma$ is a Jordan arc) onto the exterior...

Now that MathOverflow has moved to 2.0, there is a potential solution to the long-standing tension between people wanting to make minor improvements to old posts, and others not wanting the 'active questions' list being cluttered by these minor edits. A "minor edit" feature has been proposed pre...

Minor edits See this post of Scott Morrison : << Can we have a "minor edit" checkbox in the edit interface, along with the parenthetical text "minor edits do not bump posts on the list of active questions, but are subject to review but another user"? >>

Every edit, no matter how minor, bumps a question to the frontpage of an SE site. This behaviour is important to allow the community to review edits, but it also creates significant problems when a lot of edits are performed at once. What I propose is to allow minor edits that are not bumped to t...