I was wondering if there is a generalization of the integral discussed here to a case like, \begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\ver …

I want to know the source of the following "folkloric" fact about holomorphic functions. It seems well described by the phrase: The three-point characterization of holomorphy …

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 $\mathca …

I am wondering if there is a general explanation for the following phenomenon. The partial sums of the geometric series $\sum_{n\geq 0} x^n$ evaluated at a root of unity $\zeta\ne …

Consider an analytic function $f : U \longrightarrow \mathbb{C}$ where $U$ is an open subset of the complex numbers which contains the closed unit disk. I have $|f(x)| \geq 1$ for …

Suppose I have (roughly speaking) a multivalued meromorphic function $f(z)$ on all of $\mathbb{C}$ that is single-valued and holomorphic on the open unit disc and has some branch p …

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: http://en …

I would like to find inverse Laplace transform of the function: $$F(s)=e^{-a\sqrt{s(s+b)}}$$ which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be …

I'm having some trouble figuring out the properties with respect to analytical continuation of functions defined using an integral kernel. More particularly, I am working with the …

What are the Hermitian metrics in a trivial line bundle on a Riemann surface X? I read that a Hermitian metric in the trivial line bundle is equivalent to a $\mathcal{C}^{\infty}$ …

Let $X$ be a compact Riemann surface and $x\in X$. Is $X - \overline{D(x,r_x)}$ hyperbolic?

A Riemann surface is said to be: -Potential-theoretically hyperbolic if it has a non-constant bounded subharmonic function. -Poincaré hyperbolic if it is covered by the unid disk …

Let $\bf{u}$ be a smooth complex vectorfield defined on the closed unit ball $B_1(0)\subset \mathbb{R}^3$. Let $\phi$ be any smooth complex function defined on $B_1(0)$. My quest …

Let $g(x):\mathbb{R}_{\geq0}\rightarrow\mathbb{R}$ be real analytic s.t. $g(0)\neq 0$ and $g(x)=O(x^{-2})$ as $x\rightarrow\infty$. I think the following integral should be bounded …

In regard to the characteristics of certain "explicit formulae" arising in number theory, I am pondering the connection between the rate of convergence of series and the asymptotic …