# Tagged Questions

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### 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, ...
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### 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 ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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 \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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### 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 ...