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0
votes
2answers
459 views

Diametrically opposite points go to diametrically opposite points under stereographic projection

I asked this question in MSE here: http://math.stackexchange.com/questions/184524/diametrically-opposite-points-go-to-diametrically-opposite-points-under-stereogr but I didn’t get an answer. I really ...
3
votes
0answers
336 views

confusion about rational maps and Fatou components

Dear fellows, I have come to another conclusion which must be wrong. Let $f$ be a rational map and let $U$ be a connected but not simply connected open subset of the Fatou set such that $f(U)$ is ...
4
votes
1answer
938 views

Complex Analysis - Analytic Continuation and Residual Integration

At first an example which I know how to treat. Let's say we have the following integral $\int dx \frac{1}{x^2+a^2}$ Now we do an analytic continuation of the constant $a$ to the complex numbers: $a ...
0
votes
1answer
176 views

parabolic immediate basins always simply connected?

Edit: So, my original question (stated below) was to find an error in my "proof" that immediate parabolic basins for rational maps are always simply connected. Since I have not received any answers as ...
3
votes
2answers
201 views

roots with negative real parts

Under what constraints on the parameters a,b, and c does the transcendental equation $$x+a+be^{-x}+ce^{-kx}=0$$ ($k$ is a constant) have ALL its roots with negative real parts? Alternatively, any ...
2
votes
3answers
556 views

Do approximately the same polynomials have approximately the same roots? [closed]

"If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to prove ...
0
votes
1answer
452 views

The zeros of the digamma function

I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)
2
votes
2answers
218 views

Computability of finding roots in holomorphic functions.

Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a ...
12
votes
2answers
768 views

Determining rational functions by their critical points

Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, ...
0
votes
1answer
892 views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
4
votes
2answers
415 views

Zeros of incomplete exponential functions

Let $$ f_N(z) = \sum_{n=N}^\infty \frac{z^n}{n!},$$ where $N$ is a positive integer. Where are the (complex) zeros of those functions located? It would be sufficient for me to know what the ...
13
votes
0answers
626 views

What is the appropriate setting for Cauchy's Integral Formula?

For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula: $$ f(\zeta) = ...
0
votes
0answers
107 views

a question on holomorphic mappings

Could any one give me a hint how to solve this one? Let $\Omega$ be a smoothly bounded pseudoconvex domain with nonconstant automorphism group $G$, Does $G$ contains $\mathbb{R}$?
1
vote
0answers
99 views

Sum of univalent functions

Let $f,g$ be a pair of univalent functions on a proper subdomain $\Omega$ of $\mathbb C$. Does their sum $f+g$ necessarily omit any complex value? Similarly, can all holomorphic function be written as ...
2
votes
2answers
432 views

Positive Fourier coefficients

Hi all, Is there any general way to construct a smooth 2pi periodic function which vanishes on an interval and has positive Fourier coefficients? And if that was too specific I can make this more ...
0
votes
0answers
194 views

Limit of an inverse Mellin transform

In Edwards' very nice book ``Riemann's zeta function'' the following integral comes up in section 1.14. Suppose $\beta = \sigma + i\tau$ with $\sigma > 0$. Suppose $x > 1$. Fix some real number ...
0
votes
1answer
307 views

Uniform convergence of a series to exponent [closed]

I'm trying to prove that in the complex plane $\left(1+\frac{z}{n}\right)^n$ converges uniformly to $e^z$ in every closed disc $|z|\leq c$. I thought about showing the sequence as a logarithm of ...
1
vote
1answer
185 views

( finite ) Blaschke product in higher dimensions ?

Hello, as we know, the (finite) Blaschke product $P$ in $\mathbb{C}$ or in $ \mathbb{R}^2 $ is defined by $\prod_{j=1}^{k} \frac{z-a_j}{1-\bar{a_j}z}, a_j \in \mathbb{D}$. I was wondering whether ...
13
votes
1answer
887 views

Certain functional equations for the Riemann Zeta function?

Referring to this question I asked on math.SE. I am posting a more generalized question here, for answers and further inquiry. For the Riemann zeta function, we know of the standard functional ...
0
votes
1answer
188 views

About an integral transform or generalized Laurent series

We start with a little of context. I needed that a function from $\mathbb{R}^+$ to $\mathbb{R}$ could be represented in the following form, not necessarily uniquely: $$ ...
5
votes
1answer
177 views

How does pseudoconvexity restrict the topology?

A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...
11
votes
4answers
597 views

non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as $$ \zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s} $$ where $Re(s)>1$. Let $\omega$ be either ...
1
vote
1answer
394 views

How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$ between holomorphic varieties $V$ and $W$ such that $f$ is a finite branched covering (in the topological sense) There is a ...
0
votes
1answer
315 views

An asymptotic series for the digamma function

As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number. $$ \psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}} $$ $B_n$ is the first Bernoulli numbers. How ...
2
votes
0answers
237 views

How well do continuously differentiable functions behave from R^2 to R^2 ?

The behaviour of complex smooth vs 1-dimensional real smooth functions is discussed in a previous question. In "Complex Analysis as Catalyst" by Steven G. Krantz, the Cauchy integral formula is ...
2
votes
2answers
594 views

sparsity of QR decomposition

Hi, everyone! I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...
8
votes
3answers
550 views

j-invariant fixed point?

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 ...
2
votes
1answer
305 views

Fatou Coordinate for function with rationally indifferent fixed point, and repelling fixed point

Lets say I have f(z)=z^2+c, with c=0.35676274578 + 0.32858194507i. Then f(z) has a fixed point=0.15450849719 + 0.47552825815i, which is rationally indifferent with a period=5. The "Complex Dynamics" ...
-1
votes
1answer
618 views

When are conformal maps holomorphic?

It is a standard fact from elementary complex analysis that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ is a conformal mapping. Now, suppose I have a map $f':\mathbb{R}^2\to \mathbb{R}^2$ ...
2
votes
2answers
154 views

approximation of holomorphic functions on a halfplane.

Let $\mathbb {C} _ + $ denote the right halfplane and $A$ the algebra $$ A = \{ f \in H^\infty({\mathbb C} _ +) \cap C(\overline{{\mathbb C} _ +}): \; |f(z)| \le M (1+|z|)^{-\epsilon} \text{ ...
2
votes
5answers
2k views

Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?

For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ defined as $$ \frac{\partial}{\partial z}= \frac{1}{2}\left( ...
14
votes
3answers
1k views

Perron, Fourier

Perron´s formula is in some sense just Fourier inversion, but I have never seen proven it that way in a textbook. I take this must be because the conditions for the Fourier inversion formula to hold ...
7
votes
1answer
551 views

distinct zero points for polynomial

I met an interesting phenomenon. Suppose $f(z)=\frac{1}{p(z)}$ where p(z) is a polynomial in $\mathbb{C}[z] $. If there exists a $ k \in \mathbb{N} $ and $ k>1 $ such that after you take $k$-th ...
4
votes
0answers
265 views

Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...
3
votes
1answer
166 views

Can $-1/a_2$ belong to the range of a schlicht function $z+a_2z^2+\cdots$? Or is $-1/a_2$ necessarily an omitted value?

Is there an example of a schlicht function $f(z)=z+a_2z^2+a_3z^3+\cdots$, which is analytic and injective on the open unit disk $\mathbb{D}$, such that $-1/a_2$ belongs to the range $f(\mathbb{D})$? ...
1
vote
1answer
288 views

complex convex functions

I call a function f defined and valued on a domain A in the plane convex if it maps convex areas to convex areas. Some obvious example of convex functions. If f is also a bijection, what can we say ...
4
votes
1answer
314 views

Need there be infinitely many Gaussian primes along lines that contain at least one?

Greetings from EuroCG 2012, from which I post via iPod, so apologies for lack of problem motivation, background research and mathematical formatting. Question:Suppose L is a horizontal or vertical ...
5
votes
2answers
413 views

Help with a mellin-type integral

greetings . i've been trying to do this integral for many days now, with no clue on how to attack it . the integral is a mellin inverse of some kind, and appears in analytic number theory . ...
3
votes
1answer
504 views

Algebraic relationships between elliptic functions

Let $f$ and $g$ be tow elliptic function with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of tow variables and constants coefficients. ...
0
votes
0answers
177 views

Asymtotic Complexity Analysis using logarithms and binomial coefficients

On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} ...
4
votes
1answer
339 views

Characterization of the moduli space of the pair of pants in terms of the modules of the extremal ring domains

Hi, I was thinking about the following question ; I will appreciate it if somebody can give me a full or partial answer or can at least cite any reference(s)/ papers etc : By $ \bar{P} $ , we ...
6
votes
2answers
618 views

analytic approximation of a non-negative matrix by a sequence of positive matrices

Let $L \in \{0,1\}^{n \times n}$ be a non-negative matrix whose row sum is 1. ($L$ stands for "limit"). It is known that there exists a unique $r \times r$ principal minor of $L$ that is a permutation ...
15
votes
4answers
957 views

Why are lacunary series so badly behaved?

Hi! I just came across the Ostroski-Hadamard gap theorem, and while I can understand the proofs as well as the principle that the series $\sum_{n=0}^\infty z^{2^n}$ ought to have a singularity at ...
1
vote
0answers
238 views

Are there any known bounds on this function?

For a sequence of functions $f_{k}(z,s)=\frac{1}{k} \sqrt[s]{Li_s(z^k)}$ with $s>2$ and $Li_{s}(s)$ is the Polylogarithm, I am trying to show If $\Re f_{1}(e^{\frac{2\pi i}{3}},s) > \Re ...
-1
votes
1answer
800 views

1895 Math Trip problem on primitive roots of unity

How to prove that if $\theta _1,\theta _2,\theta _3$ be the arguments of the primitive roots of unity, $\sum \cos p\theta = 0$ when $p$ is a positive integer less than $\dfrac {n} {abc\ldots k}$, ...
1
vote
3answers
532 views

Convergence of analytic covering maps to a covering map

Suppose a sequence of analytic maps $f_n: \mathbb{D} \to \mathbb{D}$ from the unit disk to itself, each of which is a topological covering map to its image, converges locally uniformly to an analytic ...
2
votes
1answer
216 views

Are there polynomials (almost) all of whose intersection numbers are divisible by some integer?

I've been playing around with some basic intersection theory, and I've wondered the following: For every two integers $n$ and $m$, and complex numbers $a_1,...,a_n$, are there polynomials ...
3
votes
0answers
451 views

analytic continuation of an integral involving the mittag-leffler function

greetings. we have the following integral : $$I(s)=s\int_{0}^{\infty} \frac{dx}{2x}\left(E_{s/2}((\pi x)^{s/2})-1\right)\omega(x)-\left(E_{s/2}(( x)^{s/2})-1\right)e^{-x}$$ where : $E_{\alpha}(z)$ ...
0
votes
0answers
124 views

Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.

This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...
4
votes
1answer
250 views

Convergence of the summation 1/p^(1+iy) (over all primes p with y a nonzero real number)

For $z\in\mathbb{C}$ with real part greater than $1$ the sum $$\sum_{p}{\frac{1}{p^z}},$$ where the sum is taken over all primes $p$, converges absolutely. It is also well known that the same sum with ...