## New answers tagged cv.complex-variables

1
vote

### What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

Some extended comments:
Investigations of this type appear in the theory of the Mellin transform and, in particular, the use of Ramanujan's master heuristic (theorem, formula) as illustrated in the MO-...

- 8,551

4
votes

### What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

This is also just a comment but large:
It might be worthwhile to try poking around with this formula on functions which cannot be analytically continued to see what arises. Trying to break the formula ...

- 1,183

8
votes

### What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

First of all, to make sense of $f(-n)$ we need some assumptions about $f$. For example,
let $$\sum_{n=0}^\infty f(n)z^n\quad\quad\quad\quad (1)$$ be a series with positive radius of convergence. Then ...

- 81.6k

6
votes

### What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

I'll convert my comment to an answer.
One issue that you're implicitly sweeping under the rug is that we need to be able to make sense of the evaluation of $f$ at negative integers. Given an arbitrary ...

- 19.9k

7
votes

### What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?

This is just a comment but I am not entitled and it will be too long anyway. If we start with a toy example where $f(n)=1$ for all $n$, then $F$ is the function $\frac 1{1-z}$ in the open disc, $...

- 181

7
votes

### What, exactly, has Louis de Branges proved about the Riemann Hypothesis?

In the years since this question was first asked and answered, there have been some new developments (namely, 80+ pages of commentary by Eric Kvaalen, and a large number of new versions of de Branges' ...

- 281

3
votes

Accepted

### Is a local isometry of the hyperbolic plane the restriction of a global isometry?

The hyperbolic plane has this property, as does the Euclidean plane.
If $E$ is any subset of $ \mathbb{H}^2$, and $u : K \to\mathbb{H}^2$ is an isometry, then there is an extension of $u$ which is an ...

- 38.5k

2
votes

### Proper journal for a preprint in complex geometry

As far as I know, if not already peer-reviewed and then published in a journal, the manuscript that you are going to submit for peer-review is still called a preprint (even if it has already been ...

- 313

5
votes

### Holomorphic maps from a Riemann surface of infinite genus

The following is not a real answer but an extensive comment on the OP.
First, recall that an (open) Riemann surface is said to have type $P_{AB}$ (resp. $O_{AB}$) if it admits (resp., does not admit) ...

- 7,631

4
votes

Accepted

### On a lemma of Łojasiewicz in complex analysis of one variable

The assumption of the Lemma is that $f$ has a finite limit as $z\to \zeta_0$. This assumption does not hold in any of the two examples that you mention. In these examples, $f$ has a limit only when $z\...

- 81.6k

3
votes

### Holomorphic maps from a Riemann surface of infinite genus

Edit: as Moishe points out, my answer (just below) is for a different, and easier, question. I will leave the answer up, as it does feel “related”.
The answer is no.
I find it conceptually easier to ...

- 21.1k

3
votes

Accepted

### Quantitative analytic continuation estimate for functions small except on a small set

This conjecture is correct. Take $K=e$, and let $\gamma\leq 1/4$; we will fix $\gamma$ later.
First we give a crude estimate of $c_0$.
Let $g(z)=\sum_{1}^\infty c_nz^n.$ Since $|c_n|\leq e^n$,
we ...

- 81.6k

4
votes

Accepted

### Entire function with almost periodic boundary condition?

The answer seems to be negative. Suppose that an entire function $f$
satisfies $f(z+v_i)=e^{A_iz}f(z)$, where $v_1$ and $v_2$ generate a lattice. Let $\Pi$ be the fundamental parallelogram of this ...

- 81.6k

5
votes

### Quantitative analytic continuation estimate for a function small on a set of positive measure

$\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$As shown in the post by Alexandre Eremenko, the answer to this question is yes if $C=1$ (and hence if $C\le1$).
On the ...

- 85.1k

11
votes

Accepted

### a problem in complex-variable inequality

For a polynomial $Q(x)=\sum_i q_ix^i$, define $N(Q)=\sum_i|q_i|^2$. We need to show that $N(R)\geq 2$, where $R(x)=\prod_i (x-z_i)$.
For a polynomial $Q(x)=\sum_{i=0}^k q_ix^i$, define $Q^*(x)=\sum_{i=...

- 19.8k

7
votes

### Quantitative analytic continuation estimate for a function small on a set of positive measure

The answer depends on $C$. For example, for $C=1$ it is positive. Your estimate $|f^{(m)}(0)|\leq m!$ implies that $|f_n(z)|\leq 1/(1-|z|).$ Take $|z|=1/2$,
you conclude that $|f_n(z)|\leq 2,\; |z|<...

- 81.6k

11
votes

Accepted

### Quantitative analytic continuation estimate for a function small on a set of positive measure

Unfortunately, no, as requested:
Take any sequence $\delta_j\in(0,1)$ decaying to $0$, choose small $\mu_j>0$ such that $\prod_j \delta_j^{\mu_j}=e^{-1}$ and put $f_n(z)=e^n\prod_j B_{\delta_j}(z)^{...

- 54.3k

4
votes

Accepted

### Bounds on zeros of rational function

Let us drop the assumption $x_j\in[1,2]$, it is not needed.
Proving the result by contradiction, denote our function by $f_N$, suppose that $f_N(z_N)=-i$, and $\mathrm{Im}\ z_N= 1/(N^2R_N)$ where $R_N\...

- 81.6k

2
votes

### A question on the use of fractional derivatives in Riemann Hypothesis

(Disclaimer I need to verify if your integral of $\frac{1}{1+e^{iw}}$ trick works but here's an alternative framework which should meet your needs)
So we wish to evaluate
$$(1-2^{1-a} ) \zeta(a) = \...

- 1,183

0
votes

### Riemann uniformization theorem (limit case)

I'll attempt to sketch a proof that this is true. First, it is convenient to apply the map $z\mapsto \log z$, which maps annular regions in question to thin $2\pi$ - periodic vertical strips $S_r$ and ...

- 6,274

6
votes

### An inf-sup estimate for holomorphic functions

This is not true: take $n=1$, $r=1$, $\eta(z)=e^{az},\; a>0,$
then
$$\max_{z\in B(0,r)}|\eta(z)|=e^a,$$
while
$$\min_{z\in B(0,1)}|\eta(z)|=e^{-a}.$$
Since $a>0$ is arbitrary, no $\kappa$ with ...

- 81.6k

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