The complex-analysis tag has no wiki summary.

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### 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 ...

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### 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 ...

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189 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 ...

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300 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 ...

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182 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 ...

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### 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 ...

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176 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:
$$
...

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### 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 ...

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### 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 ...

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385 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 ...

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307 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 ...

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### 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 ...

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558 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 ...

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### 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 ...

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296 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" ...

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### 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 $f':\mathbb{R}^2\to \mathbb{R}^2$ ...

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### 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{ ...

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### 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(
...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $f(\mathbb{D})$? ...

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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 ...

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### 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 ...

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### 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 .
...

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### 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.
...

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### 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} ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $\dfrac {n} {abc\ldots k}$, ...

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### 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 ...

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### 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 ...

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### 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 :
$E_{\alpha}(z)$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $a_j$ ...

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### 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 ...

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### 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 ...

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### 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 ...