**1**

vote

**0**answers

19 views

### Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f(x)$ satisfying
$$ \int_0^\infty f(x)x^{s-1}dx=e^{-p\psi(s)} $$
where $\psi(s)$ is the usual digamma function and $p>0$. The inverse Mellin formula is
$$ ...

**1**

vote

**0**answers

44 views

### To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of ...

**20**

votes

**2**answers

384 views

### Are there irreducible polynomials with all zeros on two concentric circles?

This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...

**0**

votes

**0**answers

60 views

### Modulus of continuity of analytic functions [on hold]

Let $H$ be the class of all analytic functions of the unit disk onto itself. For $r\in (0,2)$ let $$h(r)=\sup\{|f(z)-f(w)|: f\in H, |z-w|\le r\}.$$ How to determine $h$ explicitly?
Schwarz lemma ...

**7**

votes

**0**answers

86 views

### Angular distribution of zero sets of sparse polynomials

Consider a sequence of complex polynomials $f \in \mathbb{C}[z]$, $f(0) \neq 0$, that are composed of a negligible fraction $o(\deg{f})$ of monomials. Are the zeros of such polynomials necessarily ...

**1**

vote

**0**answers

120 views

### Constructing special holomorphic functions

I would appreciate any help with this question as I am not sure how I should approach it.
Suppose $ D$ is the unit disk and that $A(x)$ is a positive continuous function on $D$.
Does there exist a ...

**7**

votes

**0**answers

295 views

### rings of modular functions on the upper half plane

Let $\Gamma_1\le SL_2(\mathbb{Z})$ be a noncongruence subgroup of finite index.
Let $\Gamma_2\le SL_2(\mathbb{Z})$ be another subgroup of finite index.
Let $M_0(\Gamma_i)$ denote the ring of modular ...

**3**

votes

**1**answer

61 views

### compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21:
Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...

**0**

votes

**0**answers

122 views

### Asymptotics to Taylor expansions?

I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
...

**12**

votes

**0**answers

258 views

### Aligned roots of irreducible polynomials

It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...

**1**

vote

**0**answers

42 views

### Zeros of functions constituting a Riesz-basis for the Paley-Wiener space

I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...

**12**

votes

**3**answers

416 views

### How bad can a circle domain get?

Let $X$ be a domain in the Riemann sphere $\widehat{\mathbb{C}}$. We say that $X$ is a circle domain if every connected component of its boundary is either a circle or a point.
It was conjectured by ...

**2**

votes

**1**answer

210 views

### Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...

**2**

votes

**0**answers

110 views

### Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...

**7**

votes

**1**answer

316 views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**0**

votes

**0**answers

86 views

### Does the fundamental theorem of calculus require continuity of the function being integrated? [migrated]

The (first) fundamental theorem of calculus is typically stated as follows, assuming continuity of the given function:
Suppose that f is continuous on the closed interval [a,b] and F
is defined by ...

**1**

vote

**2**answers

65 views

### classification of rational map with exactly only one Fatou component

We know that there exists a polynomial the Fatou set $F(P)$ is connected, which is just an attracting basin for infinity.
I have a question: Given a rational function $R$
such that $F(R)$ is ...

**2**

votes

**2**answers

77 views

### Is there a non-Shih analog for holomorphic functions of the Intermediate Value Theorem?

Let $C$ be a simple closed curve in the complex plane, and let $f$ be holomorphic on an open set containing $C$. Is there a condition on the signs of Im $f$ and Re $f$ on $C$ that guarantees the ...

**4**

votes

**0**answers

86 views

### Functional equation or analytic continuation of certain approximations to $\zeta^z(s)$?

Let $z$ be a complex number and $\omega(n)$ denote the number of distinct prime factors of the natural number $n$. I am considering the arithmetic functions $|\mu(n)|z^{\omega(n)}$ and their ...

**2**

votes

**0**answers

49 views

### Questions about holomorphy and zeros of the symmetric power $L$-function

Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions ...

**2**

votes

**0**answers

104 views

### Lie algebra of holomorphic vector fields

It's well known that the holomorphic vector fields on a complex manifold form a Lie algebra. In simplest situations, this Lie algebra can be described explicitly.
For example, take $X=\mathbb{P}^n$, ...

**0**

votes

**0**answers

51 views

### Conditions for Mellin inversion

Under which conditions is the function
$$
g(s)=a^{c(s-1)}\Gamma(s),\qquad a>0,c\in \mathbb{R}
$$
the Mellin transform of a probability density function $f$? If $c=-1$, then $f$ is the exponential ...

**0**

votes

**0**answers

43 views

### Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane

We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...

**3**

votes

**1**answer

175 views

### On the search for an explicit form of a particular integral

Let $f$ be integrable over the interval $(0, 1)$, and
$$I_n = \int_0^{1} x^n f(x) \, \mathrm{d}x.$$
Suppose $f(x) = f(1-x)$; we can then show that
$$I_n = \sum_{k=0}^{n} \binom{n}{k} (-1)^k \, ...

**2**

votes

**0**answers

127 views

### $\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is ...

**0**

votes

**0**answers

73 views

### What is the Beltrami differential?

Let $R,S$ be Riemann surfaces and $f: R \to S$ an orientation preserving diffeomorphism. Then $f$ determines what is called a Beltrami differential denoted by $\mu \frac{d\bar{z}}{dz}$.
Local ...

**10**

votes

**5**answers

509 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of ...

**2**

votes

**1**answer

78 views

### Higher dimensional analogue of Ahlfors covering surface theory

It is well known that Ahlfors covering surface theory in one dimensional is very powerful in dealing with many problems. I wonder whether there exists some generalization of this theory into higher ...

**6**

votes

**0**answers

171 views

### Criteria for irreducibility using the location of complex roots

I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...

**0**

votes

**0**answers

80 views

### constructing koenigs function

My question is rather simple and I hope someone has some sort of an answer. I am looking for a simple yes or no answer, and a reference if anyone has one.
We have a holomorphic function $f$ defined ...

**3**

votes

**0**answers

223 views

### Why does this example of global residue theorem not work?

This question was previously asked here. I am posting it here also to increase the potential number of people who will see it. I realize that this question might not be entirely in the spirit of ...

**1**

vote

**1**answer

93 views

### Extensions of Real Analytic to Holomorphic Functions in One & Several Variables: References?

A problem I'm working on requires the application of Cauchy's estimate for the modulus of the coefficients of a holomorphic function's power series representation, but the original functions with ...

**7**

votes

**2**answers

121 views

### A Generalization of the Ahlfors function to have varying degrees?

It's a classical result of Ahlfors that, for any sufficiently nice n-connected domain $\Omega \subset \mathbb C$ there is a holomorphic branched covering $f: \Omega \rightarrow \mathbb D$ to the disk ...

**2**

votes

**2**answers

166 views

### Comparision theorem for distance function

Assume that $\rho$ and $\rho'$ are conformal metrics on the unit disk which is a geodesic disk of radius $1$ w.r.t. both metrics $\rho$ and $\rho'$, and assume that $\rho'$ has a constant Gauss ...

**3**

votes

**1**answer

117 views

### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...

**1**

vote

**1**answer

130 views

### A question on $J(f)$ and $J(f')$

I was confused by the following question for a long time:
Does there exists a transcendental entire function $f$ such that
$J(f)\cap J(f')=\emptyset$ ?
where $J(f)$, ($J(f')$) is the Julia set of ...

**5**

votes

**1**answer

160 views

### Which combinations of normality, separability, and paracompactness do complex manifolds possess?

I am interested in what kinds of non-paracompact complex manifolds may exist and which topological properties they may have.
Is there a non-separable complex manifold? Can a non-separable complex ...

**5**

votes

**3**answers

141 views

### Do the bounded isophase lines of a complex polynomial $f$ through the zeroes of $f’$ define a spanning tree?

Let $f: \mathbb{C} \to \mathbb{C}$ be a polynomial and let $\arg(f(z))$ be the phase of $f(z) = | f(z)| \exp(\mathrm{i} \arg(f(z)))$. The zeroes of $f'(z)$ are saddle points of $\arg(f(z))$, i.e. ...

**5**

votes

**1**answer

399 views

### Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...

**6**

votes

**1**answer

405 views

### Analytic Chern classes

I have two questions on Chern classes, following Huybrechts' Complex Geometry.
Are the analytic Chern forms just the elementary symmetric polynomials of the eigenvalues of the curvature?
I googled ...

**0**

votes

**1**answer

52 views

### criterion for a differential of the third kind to be a logarithmic derivative of a function

Let $X$ be a compact Riemann surface of genus $g\geq 1$. If $f$ is a meromorphic function on $X$ then, the meromorphic differential $\omega=\frac{df}{f}$ is a differential
of the third kind with ...

**11**

votes

**2**answers

726 views

### Is Every Holomorphic Near an Entire?

Let $K\subset \mathbb C$ be a closed subset of the complex plane, not necessarily bounded. Let $U$ be the interior of $K$.
Let $f:K\to \mathbb C$ be a continuous bounded function, whose restriction ...

**0**

votes

**1**answer

83 views

### Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together

Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...

**0**

votes

**0**answers

63 views

### solutions of elliptic linear pde depending analytically on a parameter

Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< ...

**2**

votes

**0**answers

112 views

### Steepest descent path and Picard-Lefschetz theory

Assume that an ordinary integral of the form
$$I=\int_{-\infty}^{\infty}dx e^{-f(x)} $$
for some real function $f(x)$ is given where $f(x)$ is well defined over all $\mathbb{R}$ and the integral is ...

**6**

votes

**3**answers

381 views

### Does the proof of Picard's theorem become simpler by increasing the number of points that are not attained?

Let $f$ be an entire analytic function which attains all but $k$ complex numbers $z_1,\ldots,z_k$. Is there any elementary proof, for some $k$, that $f$ is constant?

**2**

votes

**0**answers

70 views

### Tensor product of bounded analytic functions

I asked this question on math.SE, but couldn't get an answer.
Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$.
...

**1**

vote

**0**answers

123 views

### Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...

**2**

votes

**1**answer

101 views

### What is the image of the Ramanujan Delta function?

Consider the Ramanujan $\Delta$ function as a map from the upper half plane to the complex plane. We know that the image of $\Delta$ is unbounded and that it does not contain the point $0$. What else ...

**0**

votes

**0**answers

65 views

### Finding singularities from power series

I am sorry beforehand for the length of my post, but I thought I should give some details. I try to figure out where are the singularities of a rather complicated power series.
This series comes from ...