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2
votes
2answers
420 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
187 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
298 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
180 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
876 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
174 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
171 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
585 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
382 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
301 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
231 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
544 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
545 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
293 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
577 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
151 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
1k 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
538 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
257 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
159 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
273 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
312 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
403 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
498 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
337 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
617 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
884 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
790 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
520 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
214 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
446 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
123 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
249 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 ...
5
votes
1answer
188 views

If self-composition of an analytic function on the disk gives a linear fractional map, is the original function linear fractional?

Suppose $f: \mathbb{D} \rightarrow \mathbb{D}$ is analytic. Furthermore, suppose $f \circ f = g$, and $g$ is a linear fractional map. Does this guarantee that $f$ is linear fractional? I know it would ...
30
votes
3answers
1k views

How is the Julia set of $fg$ related to the Julia set of $gf$?

Let $f$ and $g$ be complex rational functions (of degree $\geq 2$ if that helps). What can be said about the relationship between $J(fg)$ and $J(gf)$, the Julia sets of the composite functions $f ...
0
votes
1answer
163 views

graph of the size of a complex function [closed]

Hi Here there are two graphs for two functions from $R^2\mapsto R$. Is there similar graph for the absolute value of a complex variable function $f:C\mapsto C$ that has the same point (like saddle ...
2
votes
1answer
222 views

Little Picard for (open) complex manifolds?

"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The ...
1
vote
0answers
278 views

Residue Cancellation

I am trying to understand how to apply the residue theorem to solve $\frac{1}{2\pi j}\int^{\gamma+j\infty}_{\gamma-j\infty}\Gamma(n-s)\Gamma(s)\Gamma(1-s) {}_1F_1(s;b;c) ...
2
votes
2answers
292 views

Hurwitz's automorphisms theorem for infinite genus Riemann surfaces

Hurwitz's automorphisms theorem states that for a compact Riemann surface $X$ the cardinality of $Aut(X)$, the group of holomorphic automorphisms, is bounded above by $84(g(X)-1)$ and is therefore ...
4
votes
0answers
273 views

Laplace Transform: Are there theorems similar to the Bernstein Theorem?

Bernstein's Theorem states, that if a function is completely monotonic, then it is the Laplace transform of an $L^1$-function. (E.g. Widder, "The Laplace Transform", Chapter IV, Theorem 19b) Are ...
2
votes
0answers
318 views

Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
1
vote
1answer
163 views

How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature?

Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the ...
1
vote
0answers
91 views

Bounding a q-expansion on a bounded open subset of the complex upper-half plane

Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $q:\tau\mapsto \exp(\pi i \tau)$ be the nome on $\mathbf{H}$. Suppose that there are integers $a_j$ ...
1
vote
2answers
329 views

Reversed disc algebra?

Take $U=\mathbb{D}_2\setminus \overline{\mathbb{D}_1}$ ($\mathbb{D}_r$ is the open disc centered at 0 with radius $r$) and consider the space $A(U)$ of all functions on $\overline{U}$ which are ...
8
votes
1answer
390 views

Properties of a matrix-valued generalization of the $\Gamma$ function

I am interested in the following function (Mellin transform of matrix exponential): $$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$ Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ ...
1
vote
1answer
602 views

Convergence of Dirichlet series

I have arrived at an elementary-looking "result" via a sketchy argument. Having unsuccessfully searched for the statement and its "proof" in the literature, I would like to hear if anyone knows ...
8
votes
0answers
326 views

$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$

Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...