All Questions
Tagged with cv.complex-variables reference-request
271 questions
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Description of all biholomorphic maps from annulus [duplicate]
Consider the collection $\mathcal{C}$ of all maps $f \colon B_1 -\overline{B}_\beta \to \mathbb{C}$ such that $f$ is biholomorphic onto its image. Is this collection $\mathcal{C}$ path-connected?
In ...
- 11
3
votes
2
answers
116
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Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
-3
votes
1
answer
193
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Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
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1
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118
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Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
2
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0
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37
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Theta series of well-rounded lattices
I've started looking into well-rounded Euclidean lattices and I was interested in learning whether their theta series have any interesting properties, but haven't found much in terms of bibliography ...
- 223
3
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1
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326
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Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
2
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2
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361
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Size of $\zeta'(s)$ at its zeros
How large can the derivative of the Riemann zeta function be at its zeros?
More specifically, let $\rho$ be a zero of the zeta function with $\Im(\rho)\in (0,T]$. What can we say about $|\zeta'(\rho)|...
user536681
3
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1
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178
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Analytic continuation to the Mittag-Leffler star using Mittag-Leffler summation
This is a reference request for a theorem I thought I had read in a book by Steven Krantz, but I can no longer find it.
Searching for Mittag-Leffler star, I can find references to the following result....
- 1,084
1
vote
1
answer
64
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Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
4
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168
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Explicit bounds on gaps between zeros of $\zeta^\prime(s)$
In $\S$9.1 of "Theory of the Riemann Zeta Function", Titchmarsh uses Borel-Carathéodory and Hadamard Three Circles to show that every circle of radius 6 and center $3+iT$ contains a zeros of ...
- 11.1k
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1
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SOT and WOT convergence of Toeplitz operators
For the Hardy space $H^2$, every $\phi \in L^\infty (\mathbb T)$ induces a bounded Toeplitz operator $T_\phi$ on the Hardy space and $\lVert T_\phi \rVert = \lVert \phi \rVert _{\infty}$. Consequently,...
- 151
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1
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84
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Reference Request: Continuous extension of conformal maps
currently I am trying to find some references on the continuous extension of conformal maps between two simply connected domains of the Riemann sphere $\hat{\mathbb C}$. Let $\gamma_1,\gamma_2$ be two ...
5
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1
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172
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Properties of Poisson Integral and Traces of functions in Sobolev space $W^{1,2} (\mathbb D)$
Let $W^{1,2}(\mathbb D)$ be the complex valued Sobolev space on $\mathbb D$ where $\mathbb D $ is the open unit disk of the complex plane. By definition, $W^{1,2} (\mathbb D)$ is the set of all ...
- 151
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219
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Schwartz's theorem without English language reference
I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor,
Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
- 421
3
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1
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249
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Is Toeplitz operator on the Bergman space bounded iff its symbol is bounded?
It is fairly well known that if $T_\varphi$ is a Toeplitz operator on the Hardy-Hilbert space, then $\lVert T_\varphi \rVert = \lVert \varphi \rVert _{\infty}$.
Now, if $\varphi \in L^\infty (\mathbb ...
- 151
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1
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290
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Analytic maps on Banach spaces: analyticity upgrade
Consider the following problem.
Let $E,F,G$ be real or complex Banach spaces, such that $F\subset G$ with continuous embedding. Let $U\subset E$ an open set and
$$ f:U\to G $$
an analytic map, such ...
- 1,065
6
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0
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200
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
- 1,286
3
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1
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458
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Limit of an infinite series with quadratic arguments
I have encountered a limiting process on some infinite series. So, I would like to ask:
QUESTION. Assume $n$ is an even positive integer. Is this true?
$$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
- 43.1k
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109
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Affine scheme over ring of meromorphic functions with finite poles on unit circle
I am looking into the set $S$ of meromorphic functions with a finite number of poles on the unit circle (i.e., rational functions with poles on the unit circle). I assume that any $h\in S$ has the ...
- 91
3
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89
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Transformation of Julia set sequence emerging from meromorphic function
I consider a sequence of meromorphic functions on the Riemann sphere $f_k:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$ for $k\in\mathbb{N}$ of the form
$$f_k(z)=\sum_{j=1}^{n_k}\dfrac{1}{(z-p_j)^{c_j}}$$
...
- 91
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1
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488
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On a paper by Dimitrie Pompéiu and on one (in two parts) by Edmund Landau
To celebrate the new year and the future of mathematics (or the mathematics of future), I see no better way to ask a question stemming from my researches on power series.
The two papers the title ...
- 6,357
4
votes
2
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373
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Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
5
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0
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321
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Approximating $\zeta^{(r)}(s)$ by a sum
Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
- 20.1k
0
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1
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128
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Rational approximation for continuous function on curve $\Gamma$
Let $\Gamma \in C^{1,\lambda}$ be an oriented Jordan curve in complex plane $\mathbb{C} $, $\mathrm{R}(\Gamma)$ the set of all rational functions without poles on $\Gamma $. "$\mathrm{R}(\Gamma)$...
- 269
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127
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an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.1k
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218
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Analytic continuation of Dixon's identity
Many well-known combinatorial identities has an analytic version. For example, the following identities
$$
2^n = \sum_{k=0}^n \binom{n}{k}
$$
$$
\binom{2n}{n} = \sum_{k=1}^n \binom{n}{k}^2
$$
can be ...
- 1,608
1
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0
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111
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Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
- 11
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90
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Computing a complex integral with many poles
For an integer $k\geq 1$, let $f:\mathbb{C}^k\to\mathbb{C}$ be such that $f$ is analytic in the region $\text{Re}(u_i) > -1$ (say) for each $1\leq i \leq k$, and decays rapidly on vertical lines (i....
- 1,990
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2
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354
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Reference request and clarification for Central Limit Theorem for complex random variables
I'm looking for a reference and a proof of the following version (or eventually a more general version) of the Central Limit Theorem for complex random variables.
Theorem. Let $Z_1, Z_2, \dots, Z_n$ ...
- 361
1
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0
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210
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Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform
This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
- 51
6
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3
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536
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A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.1k
29
votes
2
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2k
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Contractibility of the space of Jordan curves
Is the space of Jordan curves in $\textbf{R}^2$ contractible? In other words, is there a canonical or continuous way to deform each Jordan curve to the unit circle $\textbf{S}^1$.
If the curves are ...
- 7,149
4
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0
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279
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Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?
Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
- 1,018
4
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1
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450
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Riemann mapping theorem with smooth boundary
This is closely related to the question here. The setup is that $U\subset\mathbb{C}$ is an open bounded simply connected domain with $C^\infty$ boundary. If $\phi:U\rightarrow\mathbb{D}$ is a ...
- 439
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3
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784
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Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
- 128k
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Groups of conformal isomorphisms of simply connected surfaces
By the uniformization theorem every connected and simply connected surface $M$ is conformally equivalent to one of the following three surfaces:
open disk $D$, complex plane $\mathbb{C}$, or $2$-...
3
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1
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173
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Can we define $\partial\bar{\partial}(\log|z_1|^2)\wedge \partial\bar{\partial}(\log|z_2|^2)$ as a current?
In complex analysis, by Poincare-Lelong theorem, we have
$$
\frac{\sqrt{-1}}{\pi}\partial\bar{\partial}(\log|z|^2)=T_{z=0}
$$
as currents, where
$$
T_{z=0}(\eta)=\int_{z=0}\eta.
$$
Now suppose we have ...
- 9,009
4
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0
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159
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Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
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1
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657
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Is anything known about the power series $\sum x^p$ for $p$ prime?
I'm interested in information about the power series
$$\sum_{\text{$p$ prime}} x^p$$
and the related power series
$$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$
where $p(n)$ is the nth prime.
Immediately, the ...
- 1,730
5
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1
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170
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Analytic continuation for disjoint domains
This question is a question about nomenclature more than anything. I have shown all the math, but I don't know what to search for for similar results. In such a sense, it is more so a reference ...
- 388
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0
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138
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On the solutions of $_2F_1(\alpha, \beta; \gamma, z) = \Lambda$
More General Question
Let
$$F(\alpha,\beta;\gamma;z) = \sum_{n=0}^{+\infty} \frac{(\alpha)_n(\beta)_n}{(\gamma)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)...(x+n-1), \quad (x)_0 = 1$$
be the ...
- 233
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1
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304
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Is it possible to define pseudodifferential operator $p(x,T)$ using Cauchy integral formula?
I was wondering how I can define a pseudodifferential operator using Cauchy integral formula.
Consider a differential operator $p(T)$ ($p$ is a polynomial for instance). $p(T)$ can be defined as:
$$\...
- 350
2
votes
1
answer
112
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References for group of invariance of the Painlevé property
I'm searching for some references about groups of invariance of the Painlevé property other than the book of Robert Conte or more generally birational transformation of Riemann surfaces.
0
votes
1
answer
58
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Analytic continuation of spline wavelets (reference request)
I would like to extend (cubic or higher degrees) spline wavelets to complex domain. First, does this continuation exist? Second, I appreciate it if anyone could point me to some references.
- 350
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1
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206
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Is the Gauss hypergeometric function $_2F_1(a,b;c,z)$ univalent in $\left|z - \frac{1}{2} \right| < \frac{1}{2}$?
Consider the Gauss hypergeometric function
$$_2F_1(a,b;c,z) = \sum_{n=0}^{+\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, \quad |z| < 1, \quad (x)_n = x(x+1)\cdots(x+n-1), \quad (x)_0 = 1$$
The Encylopedia ...
- 233
1
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0
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116
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Converse of transfer theorem : does asymptotic behaviour of coefficients describe the singularity?
I'm interested in Singularity Analysis after reading this Math.Stack post. In the "Analytic Combinatorics" book by P. Flajolet and R. Sedgewick, at p.390, the following transfer theorem (...
- 233
3
votes
2
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243
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Is every planar bounded $C^2$ domain finitely connected?
Let $\Omega \subset \mathbb R^2$ be a bounded $C^2$ domain. Is $\Omega$ then finitely connected? As I learned recently a domain in $\mathbb R^2$ is finitely connected iff “[its] complement has ...
- 313
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votes
1
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129
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Seeking an integral formulation for an algebraic function
While working with a generating function for the Catalan numbers, I came across the integral representation
$$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\...
- 43.1k
3
votes
0
answers
184
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"Circulant-Vandermonde" matrix: in search of a formula
An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form
\begin{align}
\mathbf{X}_n= \begin{bmatrix}
x_1 & x_2 & \cdots & x_{n-1} & x_n \\
x_2 & x_3 & \cdots & x_n&...
- 43.1k
0
votes
1
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163
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Reference(s) on the smallest concave majorant for the sequence of coefficients of a given power series?
This question is based on this Math.SE answer, so let's recall a few concepts dealt with there. If $\{a_n\}_{n\in\Bbb N}$ is the sequence of coefficients of a power series $\sum_{n=0}^\infty a_nz^n$ ...
- 6,357