**8**

votes

**0**answers

159 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

59 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 ...

**4**

votes

**0**answers

140 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 $X=\mathbb{P}^n$, ...

**1**

vote

**0**answers

56 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

60 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

210 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 \, I_{k}...

**2**

votes

**0**answers

138 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 $O(\...

**0**

votes

**0**answers

106 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 ...

**11**

votes

**6**answers

701 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 course,...

**2**

votes

**1**answer

94 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

190 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

101 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

315 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

268 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 ...

**6**

votes

**2**answers

134 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

172 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

147 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

147 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 $f$...

**5**

votes

**1**answer

175 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

165 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

571 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

417 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

73 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

765 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

126 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

76 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< \frac{N+...

**1**

vote

**0**answers

154 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

402 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

89 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 $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$.
...

**3**

votes

**1**answer

110 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 ...

**2**

votes

**0**answers

98 views

### When does analytic in the operator norm imply analytic in the trace class norm?

This is a crosspost from MSE. It's been up there for a few weeks now. A 200 rep bounty yielded no results (or even comments). I'm hoping someone here has some helpful ideas. See this post for the ...

**1**

vote

**1**answer

215 views

### Class of functions between $C^{\infty}$ and $C^{\omega}$

I am always curious about that whether there exists a class of function which seems that more smooth than the $C^{\infty}$ class, while it is far from $C^{\omega}$ analytic function .
From my point ...

**5**

votes

**2**answers

111 views

### Equivalence of Definitions of Quasiconformal Surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of Quasiconformal Surface.
Definition: A Quasiconformal surface $S$ is a ...

**1**

vote

**1**answer

354 views

### Help with a difficult integral [closed]

Referring to a previous question, I am trying to do the following integral :
$$\phi(s)=i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log\sqrt{1+ix} \right )^{2}}{\pi ^{2}} \right ]-\log \left[1+\...

**0**

votes

**0**answers

48 views

### Non interacting complex unit

How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...

**0**

votes

**0**answers

46 views

### On sequences of rational functions [duplicate]

Let $\{f_n\}_{n=0}^\infty$ be a sequence of rational functions of the following form: $$ f_n(z) = \sum_{m=1}^\infty \frac{C_{m,n}}{z-m}$$ with $C_{m,n} \in \mathbb{Z}$, $C_{1,n} = 1$, and for each $n \...

**7**

votes

**1**answer

375 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...

**15**

votes

**7**answers

2k views

### Is this a rational function?

Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...

**0**

votes

**1**answer

141 views

### When can two Cauchy transforms intersect?

Given two polynomials $p$ and $q$ over reals and being guaranteed that both have all roots real I want to know if there is any characterization of the solutions of the equation $\frac{p'}{p} = \frac{q'...

**4**

votes

**1**answer

112 views

### Numerical equality testing

I am working on developing an online homework system.
One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe $\sin(2x)$)...

**2**

votes

**0**answers

69 views

### What are “minimal lines” in complex geometry?

Schwarz reflection across a real analytic arc $C$ in $\mathbb{R}^2$ is usually defined analytically. Thinking of the arc as the image of an interval in $\mathbb{R}$ under an invertible holomorphic map ...

**1**

vote

**0**answers

61 views

### Related to derivative of Modified Bessel I function wrt the order

I recently met some problems related to the modified Bessel I funtions. Let $I(\nu,x):=I_\nu(x)$, and $I'_\nu(\nu,x):=\dfrac{\partial}{\partial \nu}I(\nu,x)$.
Using maple, it seems that $Re(\dfrac{I'...

**3**

votes

**2**answers

280 views

### Conformal map and Jordan curve

Here is my question :
Suppose you have a simple (analytic) closed curve $\gamma$ in an open simply connected domain $\Omega \neq \mathbb{C}$. Does there exist a conformal bijection $f : \Omega \...

**1**

vote

**1**answer

146 views

### A Generalization of growth exponents

The growth exponent of a function $f(\sigma + it)$ is defined as
the least nonnegative real number $\psi(\sigma)$ satisfying
$$
f(\sigma + it) \ll |t|^{\psi(\sigma) + \epsilon}
$$
as $t \to \infty$, ...

**1**

vote

**1**answer

68 views

### Oscillation of subharmonic functions of slow growth

Given a sequence of real numbers $c_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z_k\to\infty$ with $|z_k|<k$ such that
$$\displaystyle\...

**3**

votes

**2**answers

331 views

### Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements.
Let $h,k\geq1$ be integer numbers and let $F_{1},\ldots,F_{...

**0**

votes

**0**answers

76 views

### Hurwitz's theorem for a system of functions

First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$.
We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \...

**7**

votes

**0**answers

77 views

### Extremal length of graphs in surfaces

Given a surface $\Sigma$ with conformal structure $\omega$, the extremal length of a homotopy class $\gamma$ of curves in $\Sigma$ is defined to be
$$
\sup_{g \in \omega} \frac{\ell_g(\gamma)^2}{A_g(\...

**0**

votes

**0**answers

73 views

### Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...

**1**

vote

**1**answer

176 views

### Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...