This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \int_0^{2\pi}\mathrm{d} t\,P_r\left(\theta-t\right)\varphi\left(\mathrm{e}^{\mathrm{i} t}\right)$$where the Poisson kernel is$$P_r\left(\theta-t\right)\equiv\frac{1-r^2}{1-2r\cos\left(t-\theta\right)+r^2}.$$Extensive search on the internet has not been very successful. Perhaps because this is trivial, but I don't see it. Of course, a fairly self-contained...Read more

Consider the following optimization problem $$\min \| \textbf{Ax-B}\| $$$$s.t.|x_i|=1,i=1,...,n$$where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th element of $\textbf{x}$, $\textbf{A}\in \mathbb{C}^{m \times n}$ and $\textbf{B}\in \mathbb{C}^{m}$ are constant.I want to find a algorithm to solve a stationary point of the problem. When I replace $x_i$ with theta, some search algorithms seem to be extremely difficult to solve. Maybe there are other methods to transform the problem to familiar one....Read more

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius transformations in both domain and range. For degree 1 and 2, there is only one equivalence class. For degree 3, there is a well-understood one-complex-parameter family, so the real challenge is for higher degrees. Given a set of points to be the critical values [in the range], along with a covering space of the complement homeomorphic to a punctured spher...Read more

Let $S$ be a Riemann surface of genus $g \geq 0$ with $n$ punctures, i.e., with $n$ distict points removed. Let $f: S \rightarrow R$ be a quasi-conformal map. Then $R$ is also Riemann surface of genus $g$ with $n$ punctures.Assume $S'$ and $R'$ are surfaces arising from $S$ and $R$ by filling in the $n$ punctures.Is it always possible to extend $f$ quasi-conformally to $f':S' \rightarrow R'$ such that the filled in points are fixed but not necessarily pointwise.The question is true for replacing punctures by boundary components. Since then idea...Read more

Is there a simple and explicit continuous function $f\colon[0,\infty)^2\to\mathbb C$ such that $f$ is analytic on $(0,\infty)^2$ and $|f(x+iy)|/(x+y)\to1$ as $x+y\to\infty$, where $(x,y)\in[0,\infty)^2$? It seems not too hard to construct such a function $f$ as the sum of a series of polynomials, as is done in the proof of the Carleman approximation theorem; cf. [1], [2]. However, that construction is far from explicit....Read more

If we view the j-invariant of a lattice as a map from the upper-half plane to the complexes by $\tau\mapsto j([1,\tau])$, then it is surjective, holomorphic, and has quite a number of other wonderful properties (see the third part of Cox's Primes of the Form for a great introductory reference). My question is: does $j$ have any fixed points? If so, do we know what any/all of them are? I'm in particular curious what goes into the proof. Specifically, whether the answer is immediate from some complex analysis, or whether you need to have a go...Read more

In a recent Math Stack Exchange question I asked about the function $$f(z)=\sum_{n=0}^\infty z^{2^n},$$ and was informed of its status is a canonical example of a lacunary series with natural boundary at $|z|=1$. A phenomenon observed by the accepted answer was that this function has a multitude of zeroes within the unit disk; it was speculated but not proven that that this set is in fact infinite.That raises the following questions, for which I've not been able to find appropriate literature:Does $f(x)$ have an infinitude of zeros within the u...Read more

Suppose f(z) is a power series with positive integer coefficients centered at zero and positive radius of convergence. What is the likelihood that f has a dense set of singularities on its circle of convergence (or any larger circle for that matter).For instance, a classical theorem shown in Polya & Szego assumes f has integer coefficients and radius of convergence one. In this case, f is either rational or has a dense set of singularities on the unit circle. There are only countably many of the former and uncountably many of the latter, so...Read more

Note:In this question, a complex number is counted as a vector initiated from the origin.______________________________________________________________- Is there a holomorphic function $B:\mathbb{C}^2 \to \mathbb{C}$ such that for every two non zero complex numbers $z,w$ with $z/w \notin \mathbb{R},$ the vector $B(z,w)$ is a non zero vector indicating to the direction of the bisector of the angle $\angle (z,w) $? Motivation:The initial formula for the "Bisector" of $\angle (z,w) $ is $B'(z,w)=|z|w+|w|z$. But it is not a holomorphic ...Read more

Consider a function $F(x, y)$ of two complex variables. For $\Re(y)>0$, we know the analytic structure of the function. In that case, the function is meromorphic, with simple poles in $x$ at locations $x=w_i$, for $i=0, 1, ...$ . The locations of the poles do not depend on $y$, however the residues do depend on $y$.Furthermore, we know that $F(x, y)$ is symmetric in $x$ and $y$. So, we also know its analytic structure for $\Re(x)>0$. In that case, the function has poles in $y$ at $y=w_i$. The problem is to know the analytic structure of t...Read more

The question I'm considering is the following: given an 1-cocycle in of the modular group in Hom$(H;\textrm{GL}_{r}(C))$ call it $f$ when does it induce a vector bundle structure on the corresponding curve $X(1)$ (or more genrally $X(\Gamma)$)?The reason why I'm confused over this is because I have two contradictory views on why it should hold in general/not: This first is baased around a similar intuition to filling in punctured Riemann surface - namely we can clearly define a vector bundle over $X(\Gamma)$ without the cusps & points of no...Read more

Can i get the answer to the following problem. I am having a proof, i feel there is something wrong here..Can you please point out!Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma: [0,1]\to D$, be a smooth embedding. Given a continuous one form $\phi$ along $\gamma$ and $\epsilon >0$, Does there exists a holomorphic function $h$ on some open neighborhood $U$ of $\gamma$, $U\subset D$ such that $|dh-\phi|<\epsilon$. Suggested Proof:Without loss of generality we can assume that $0\notin D$. We can write $\phi= \phi_1 ...Read more

[Changed title as a plea to re-open the question.]If one is to motivate a course in complex variables, what specific analytic (holomorphic/meromorphic) function of one variable would you cite as an example that comes up in other areas of mathematics? Particularly if one can comment on its natural domain (either a subset of $\mathbb C$ or an extension, i.e. a Riemann surface)—perhaps with special mention of any pole structure, conjectural or not—without getting into too much details of that particular field. Number theory of course offers a whol...Read more

Let $n$ be a given even positive integer. We have the following integral\begin{align}\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots dy_n&=\\\int_0^{\infty}\cdots\int_0^{\infty}e^{-(y_1+\cdots+y_n)}\left(\int_0^{\infty}e^{-x}\prod\limits_{j=1}^n(x-y_j)dx\right)^ndy_1\cdots dy_n&>0.\end{align}Let's consider a similar integral:$$\int_0^{\infty}\cdots\int_0^{\infty}e^{-(x_1+\cdots+x_n+y_1+\cdots+y_n)}\prod\limits_{i=1}^n\prod\limits_{j=1}^...Read more

I'm an engineering student but I self-study pure mathematics. I am looking for a Complex Variables Introduction book (to study before complex analysis). I have the Brown and Churchill book but I was told that's for engineers and physicist mostly, not for mathematicians. I also looked for Fisher and Flanigan, but they don't seem to have as many topics as Brown. I wonder which book is best for the subject or if one of the two previously mentioned will do to master most of the topics of complex variables as a mathematician. Thanks....Read more