All Questions
Tagged with cv.complex-variables reference-request
271 questions
2
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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
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.
2
votes
0
answers
90
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Second order estimates for Dirichlet problem for complex Monge-Ampere equation
Let $\Omega\subset \mathbb{C}^n$ be a bounded pseudo-convex domain. Let $0<f\in C^{\infty}(\bar\Omega)$, $\phi\in C^\infty(\partial \Omega)$. Consider the Dirichlet problem for the complex Monge -...
- 21.8k
2
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0
answers
180
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Multiple zeta values related to fractional calculus and an Appell polynomial sequence
There is an Appell sequence of polynomials $p_n(z)$ related to an infinitesimal generator for one rep of the fractional calculus that have coefficients involving the Riemann zeta function values at ...
- 10.5k
2
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0
answers
154
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Algebra of meromorphic functions on a Riemann surface
Let $C$ be compact Riemann surface of high genus. Let $p$ be a general point on $C$ and $z$ a local coordinate around $p$.
Given a meromorphic function on $C$, regular outside $p$, we can look at its ...
- 2,384
2
votes
0
answers
61
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Criteria for a limit to be a proper function
This question is obviously broad; turning this broadness into something sharp is part of the problem.
Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
- 779
2
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0
answers
117
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automorphic form associated with Apollonian Gasket
In /Indra's Pearls/, it's mentioned one can associate automorphic forms with limit sets. Is there an explicit description of the one associated with the Apollonian gasket (up to some appropriate ...
- 975
2
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0
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219
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Integral with product of two infinite sums
I am looking for references and results on integrals with product of two infinite sums:
$$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$
Above integral is ...
- 1,199
2
votes
0
answers
320
views
Solution to algebraic equations over $\mathbb{C}$ and $\mathbb{C}[x]$
$t^n=a$, we get one solution to the equation:
$$t=e^{\frac{1}{n}\int^a_1 \frac{1}{x}}$$ generalizing this result by replacing the exponential with an elliptic modular function and the integral with ...
- 3,084
2
votes
0
answers
59
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explicit conformal map of sinus-shaped region
I wonder weather an explicit conformal map of a sinus-shaped region given by $\Re(z) \in [0, 1+\delta \sin(\Im(z)) ]$ onto say a strip or a ball is known (of course $0<\delta < 1$). Thank you.
- 53
2
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0
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173
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Does this symmetrization operator have a name? Any theory?
Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define
$$f_{\mathrm{symm}}(x_1,\ldots,x_n) =
2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1}
f(\varepsilon_1x_1,\ldots,\...
- 37.7k
2
votes
0
answers
179
views
Questions about transformation or integral transformation
I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
- 3,084
2
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0
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215
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ALE Kähler manifolds are birational to deformations of $\mathbb{C}^n/G$.
I am reading Dominic Joyce's book 'Compact Manifolds with Special Holonomy' and I am struggling to understand a remark he makes at the end of chapter 8. The assertion is (I think) the following:
...
- 21
1
vote
1
answer
655
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Series involving factorials
Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...
- 13.4k
1
vote
1
answer
218
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Generating series of rational$\times \exp($rational$)$
It is known that rational functions $f\in \mathbb C(x)$, $0$ not a pole, are the sum of generating series $\sum_{n\geq 0} a_nx^n$ where $(a_n)_n$ is solution of a linear recurrence with constant ...
- 5,428
1
vote
1
answer
130
views
distance-set along the orbit of $e^{2\pi i\theta}$
Let $z=e^{2\pi i\theta}$ for a fixed real number $\theta$. It's known that if $\theta\not\in\mathbb{Q}$ (is irrational) then the set $S(\theta)=\{z^n: n\in\mathbb{N}\}$ is dense on the unit circle $\...
- 43.2k
1
vote
1
answer
393
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The Dirichlet series of the Hasse–Weil L-function
I have the following question:
Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order.
Thank you in ...
- 1,197
1
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1
answer
179
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One trig "survives" a binomial summation: why?
I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.
In case you wonder where this came from, I was investigating certain $q$-series in ...
- 43.2k
1
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1
answer
241
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Subharmonic function in unbounded regions
The harmonic majorization for a subharmonic function $h$ is well-known for bounded regions $\Omega \subset \mathbb{C}$:
$$h \le 0 \text{ in }\partial \Omega \Longrightarrow h \le 0 \text{ in }\Omega.$$...
- 285
1
vote
2
answers
354
views
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
vote
1
answer
344
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Is there a way to tie up even and "newly suggested odd" Riemann zeta values?
Define the sequence
$$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases}$...
- 43.2k
1
vote
2
answers
698
views
Extension of harmonic function at infinity
Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property ...
- 1,795
1
vote
1
answer
65
views
Reference dual Dirichlet space $D^1$
Let $\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$ be the unit disk. The Bergman space $A^1 = A^1(\mathbb{D})$ is the Banach space of holomorphic functions on $\mathbb{D}$ such that
$$
\|f\|_{A^1} ...
1
vote
1
answer
151
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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
1
vote
1
answer
61
views
Reference request for value distribution theory of bicomplex meromorphic functions
While there is abundant literature available on value distribution of meromorphic functions, I am interested to know whether the value distribution theory for bicomplex meromorphic functions has been ...
- 165
1
vote
1
answer
518
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Zeros of Multivariate Complex Functions [need reference]
I am looking for a good accessible reference that would summarize properties of zeros of complex analytic functions.
For my purpose, it would be interesting to see a discussion on the following ...
- 671
1
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1
answer
166
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Is the $n$'th super root analytic in a half plane?
This question has been bogging me down lately. I'm not sure how to come up with an approach to tackle the proof exactly. I'm without a proof, butI think the result I'm searching for is true. Similarly,...
user78249
1
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1
answer
252
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Hilbert scheme of an infinitesimal neighborhood of a subvariety
Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...
- 21.8k
1
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1
answer
2k
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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 :)
- 19
1
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1
answer
119
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Estimating two dimensional theta function
My feeling is that this should be written somewhere but I don't know what to search for.
Let $Q(x,y)$ be a binary quadratic form over $\mathbb{C}$, with $\operatorname{Re}(Q)$ positive definite. Then ...
- 2,178
1
vote
1
answer
486
views
Mandelbrot set and logistic map connection
I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
1
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1
answer
177
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Conformal mapping of multiply connected domains
I am studying conformal mappings of multiply connected domains. Most of the results that I can reach concern existence and uniqueness of such mappings, whereas I can not find anything satisfactory ...
- 11
1
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1
answer
380
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Infinite compositions of holomorphic functions, is there literature on the subject?
I've recently become very intrigued by infinite compositions. To get at what I mean by the term, I'll be as explanatory as possible.
Consider a sequence of holomorphic functions $\{\phi_j\}_{j=0}^\...
user78249
1
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1
answer
1k
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A generalization of the Grauert direct image theorem
EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...
- 21.8k
1
vote
1
answer
143
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On the geometry of roots of a sum of complex linear fractions
I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form
$$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$
where the $a_j$'s are nonzero complex ...
- 2,154
1
vote
1
answer
554
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Conformal mapping of C \ D* onto C \ (-1, 1) [closed]
Which is the concrete formula for the conformal mapping (normalized at infinity),
acting from $\mathbb C \backslash D^*$ onto
$\mathbb C\backslash[-1, 1]$?
Here $\mathbb C$ denotes the set of all ...
- 21
1
vote
0
answers
52
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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
1
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0
answers
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 $E_6$).
$\...
- 43.2k
1
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0
answers
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
1
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0
answers
210
views
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
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
1
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0
answers
144
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Zeroes of Mellin transform
There exist a "standard" or canonical way to construct a real valued function whose Mellin transform has a prescribed set of zeroes? Clearly for some set of zeroes this could be impossible ...
- 131
1
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0
answers
79
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Resources/books/articles about estimation of $|f(z)|$ [closed]
I am new to complex analysis and I want to explore the following problem: for a complex valued bounded function $f(z)$ in a domain, I would like to know how we estimate the modulus values in a given ...
- 826
1
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0
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199
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What is decoupling theory means on Tao Blog ? And what is its purpose in mathematics? [closed]
I accrossed on Tao Blog a new theory for me which it is called "Decoupling theory", But I didn't find in the web its definition and its purpose , I find only this article in wiki but this very far ...
user147204
1
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0
answers
49
views
Permutation of eigenvalues induced by a loop
A friend of mine just mention me what I think is a very fun phenomena and I would be very interested to learn more about it:
Let $A,B\in \mathbb{C}^{n\times n}$ two matrices. And let $\lambda_1(z), \...
- 4,361
1
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0
answers
294
views
Can an entire function have every root function?
My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$
$$\...
user78249
1
vote
0
answers
52
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Composing between Schröder functions in complex dynamics
Assume that $f(z)$ is a holomorphic function that sends some open and connected set $G$ to itself. Assume $f$ has a single fixed point $z_0$. Assume $f(f(...(n\,times)...f(z))) = f^{\circ n}(z) \to ...
user78249
1
vote
1
answer
133
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Iterated sums--something like a differsum
So I've been fiddling around with the cauchy product of sequences lately, and am curious about a little identity I've found (which I'm sure is ubiquitous in finite differences, as I can't be the only ...
user78249
1
vote
0
answers
201
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Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions
In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved:
Let $U,V \subseteq \mathbb{C}$ be open subsets. Every $\mathbb{...
- 21
1
vote
0
answers
217
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Homeomorphism of fibers of holomorphic maps
EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...
- 21.8k