Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
976
questions with no upvoted or accepted answers
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Finding roots of equation with gamma functions
Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
- 199
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91
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Tauberian theorem converse (Wiener-Ikehara)
Jacob Korevaar provides a nice converse to to Wiener-Ikehara tauberian theorem on p. 125 of his Tauberian Theory book:
For the non-decreasing, locally of bounded variation function $s$, if we have $\...
- 91
4
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72
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Saddle point approximation for multiple contour integrals
General Question: Is there a reference where the saddle point approximation is applied to multiple contour integrals?
In particular, say we have the integral
$$ I_N = \frac{1}{(2\pi i)^N} \oint \left[\...
- 241
4
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192
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Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$
I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
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4
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211
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Real part of a holomorphic section of a vector bundle
Let $F\to M$ be a holomorphic vector bundle over a complex manifold $M$ and let $s:M\to F$ be a no-zero section. Let $E$ be the complexification of $F$, and suppose that $E$ admits a holomorphic ...
- 153
4
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294
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Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
4
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144
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Entire function which modulus grows only along the real axis
My research in theoretical physics led to the necessity of constructing an entire function with modulus decaying in the significant part of the complex plane. I wonder whether this is possible because ...
- 41
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132
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Independent inner functions on the unit disk
This problem cropped up in a paper that I am writing and I have thought about it for too long to no avail: let $\mathbb{D}$ be the open unit disk in the complex plane and suppose $\varphi:\mathbb{D}\...
- 41
4
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126
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What is the closed form of this integral?
Consider the Chebyshev first function $\psi(y):=\sum_{p^j \leq y} \log p$, where $p$ is a prime. Define $$F(s, k) = s\int_{1}^{\infty} \psi(x + x^k)x^{-s-1} \mathrm{d}x$$ for $ \Re(s) >$ max $(1, ...
- 41
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72
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Asymptotics of a certain integral in singularity theory
Let $f:\mathbb{C}^2\to \mathbb{C}$ be an isolated plane curve singularity. Consider the versal deformation space $\mathbb{C}^\mu$ parameterizing deformations $f_\lambda$ for $\lambda \in \mathbb C^\mu$...
- 2,790
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99
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Generating $H^{\infty}(X)$
Let $X$ be a domain in $\mathbb{C}^d$ and let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$. Consider the Banach algebra $H^{\infty}(X)$ consisting of bounded holomorphic functions on $X$ with ...
- 5,385
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284
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Bers' simultaneous uniformization
I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated.
Let $S$ and $S^{'}$ be two ...
- 221
4
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52
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A relaxed form of quantitative transversality
Let $f$ be a holomorphic function on the unit ball $B_1(0) \subseteq \mathbb{C}^n$. Then given any pair of constants $\delta, \epsilon > 0$, does there exist a smooth function $g: B_1(0) \to \...
- 1,591
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258
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Reference request for some result of de Bruijn on zeros of some holomorphic function
In a video lecture on Youtube, ''Vaporizing and freezing the Riemann zeta function'', Terry Tao states that ''de Bruijn proved that if for some $t_0$ the zeros of $H_{t_0}$ are contained in the strip $...
- 57
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315
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Flat base change in the complex analytic setting
On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension":
Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...
- 41
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115
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Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
- 551
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Is every algebraic curve the critical set of an algebraic function?
Is every algebraic curve in $\mathbb{R}^2$($\mathbb{C}^2$), the set of critical points of a polynomial in $\mathbb{R}[x,y]$($\mathbb{C}[x,y]$)?
In particular what is a real (complex) polynomial whose ...
- 312
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131
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Injective resolution of the ring of entire functions
Let $R$ be the ring of entire functions on $\mathbb{C}$. I heard that the concrete value of the global dimension of the ring depends on continuum hypothesis.
I would think that the injective dimension ...
- 26.2k
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334
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Regular functions vs holomorphic functions
Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...
- 181
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275
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Computing Bohr Radii
The Bohr radius $R$ for $\mathcal{H}(\mathbb{D})$ is defined as the radius $$R = \sup\limits_{0<r<1} \Bigl\{ r\ \Big|\sum\limits_{k=0}^{\infty}|a_k|r^k \leq |f|_\mathbb{D}, \text{ for all }f(z)=\...
- 3,177
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69
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Extension of holomorphic function on family of relatively compact strictly pseudoconvex domains
Let $Y \to M$ be a (proper and locally trivial) family of relatively compact strictly pseudoconvex domains which are smooth (not neccesarily Stein spaces). So $Y$ and $M$ are a complex manifold and ...
- 83
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72
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Is a domain of a holomorphic flow pseudoconvex?
Let $Z$ be a holomorphic vector field on $\mathbb{C}^n$. I would like to know whether (it seems that it is) the domain $D_\phi \subset \mathbb{C} \times \mathbb{C}^n$ of a maximal flow $\phi: D_\phi \...
- 437
4
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101
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Extending a holomorphic map on diffeomorphic affine varieties
Suppose I have two smooth complex affine varieties $X$ and $Y$. Assume that they are each diffeomorphic to $\mathbb{R}^{2n}$ (where $n\geq 3$).
Question: If there exists open dense subsets $U\...
- 8,394
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189
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Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
4
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211
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Analytically continuing Euler's partition function
Author's note: This question might be a little hopeless, but maybe someone has some form of good feedback. It's a long one because I tried to be very thorough. I tried to explained all the odds and ...
user78249
4
votes
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108
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Section of the spinor bundle over $S^{1}$ that extend to sections of the spinor bundle over $D^{2}$
Let $\mathbb{S} \rightarrow S^{1}$ be the spinor bundle associated to the connected double cover $\text{Spin}(S^{1}) \rightarrow S^{1}$. Let $\mathbb{D} \rightarrow D^{2}$ be the spinor bundle ...
- 546
4
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207
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Gap Between Abscissae of Conditional Convergence and Holomorphicity for Dirichlet Series
For a Dirichlet series, $D = \sum_n a_n n^{-s}$ we may define the abscissae, in (non-strictly) increasing order
$\sigma_c(D) = \inf\{\sigma : D \text{ converges in } \mathrm{Re} s > \sigma \}$, ...
- 576
4
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202
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A statement in Siegel's paper on Riemann-Siegel Formula
Siegel left the following comment in the last paragraph of his famous "Über Riemanns Nachlaß zur analytischen Zahlentheorie", Section 1:
Das Integral $\Phi(u)$ ist ein Spezialfall des Integrales $$\...
- 3,317
4
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261
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Is the following integral positive or not?
Let $n$ be a given even positive integer. We have the following integral
\begin{eqnarray}
&&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
- 1,987
4
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243
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Expansion of Jacobi theta function at $p$-torsion
I am aware of the formula $$\Theta(z,q)=z\exp\left( -2\sum_{k\geq 1} \frac{z^{2k}}{(2k)!}E_{2k}(q)\right)$$ for the Jacobi theta function at the origin $z=0$. The definition I am using for the theta ...
- 113
4
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85
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Action of the monodromy on the cycle made of the real points
Let $f : \Bbb C^n \to \Bbb C$ be a polynomial function with real coefficients.
Let $X_t = f^{-1}(t)$ denote the fiber above some $t \in \Bbb C$. Let assume that the set of real points of $X_t$, for $t ...
- 1,044
4
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148
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Modulus of an annulus with a cut
Let $A_r$ be a complex annulus of modulus $r>0$ obtained from a $1\times r$ rectangle in $\mathbb C$ with vertices $A=0$, $B=r$, $C=r+i$, $D=i$, by identifying isomterically $AB$ with $DC$.
Let us ...
- 14k
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157
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Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14k
4
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128
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Topological invariance of the relative multiplicities
Let $f:(\mathbb{C}^n,0)\to (\mathbb{C},0)$ be a reduced complex analytic function. We write
$$f=f_m+f_{m+1}+\cdots+f_k+\cdots$$
where each $f_k$ is a homogeneous polynomial of degree $k$ and $f_m\neq ...
- 41
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144
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Universal cover of Kodaira surface
From an earlier question, the universal cover of a Kodaira fibered surface is a bounded domain in $\mathbb{C}^2$. It is also not the polydisk or the ball. Can we say more about the structure of the ...
- 1,149
4
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120
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Linearizing an operator
This question is more about a curious identity I have come across, than to do with explicit research. The question is somewhat advanced so I'm posting it here rather than on math stackexchange. It ...
user78249
4
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122
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Complex L^1 spaces; reference request
I have been doing a fair amount of research into a complex analytic modified version of the Mellin transform. I have hit a few roadblocks, and am hoping there may already be literature on the subject. ...
user78249
4
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160
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Automorphism groups of elliptic bundles
This is a question in complex geometry, but even for algebraic varieties I don't know the answer:
Let $S$ be a smooth compact Kähler surface (for example a smooth complex projective surface) that is ...
- 41
4
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207
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classification of homogenous complex manifolds
Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?
- 1,091
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677
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Can one integrate around a branch-cut?
How meaningful is it to try to integrate around the branch-cut of a function?
For example lets say I have the function $\log(z^2+a^2)$ for $a>0$ and I choose my branch-cuts to be starting at $\pm ...
- 1,883
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242
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Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?
Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...
- 15.3k
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676
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What is the status on questions related to Bhargava's factorial function?
In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times l!$...
- 7,646
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664
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A difficult integral which the Risch algorithm shows is not elementary
For reasons which aren't conceptually related to the problem a few of my colleagues and I are in need of finding an expression for the following integral in terms of $a$ and $\delta$:
$$\int_{\delta}^...
- 1,401
4
votes
0
answers
279
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Is there a coordinate free proof of the Morrey--Kohn--Hormander identity?
The Morrey--Kohn--Hormander identity is the key to proving vanishing/existence results on bounded pseudoconvex domains in $\mathbb{C}^n$, or more generally, Stein domains. See, for instance, the ...
- 18.3k
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85
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Question concerning Mellin transforms
I've recently come across a result I've been trying to generalize.
Say that $\phi(\sigma \pm iy) < Ce^{\frac{\pi}{2}|y|}$ in the strip $a < \sigma < b$
then then the following integral is ...
- 181
4
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193
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Proving convergence of certain Mellin transforms
I am wondering about the following modified mellin transforms and if they are absolutely converging or not.
Let $\phi$ be some holomorphic function such that for all $y \in \mathbb{R}$ we have $\phi(\...
- 181
4
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111
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status of Invariant subspace problem on Krein Space
What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
- 2,076
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363
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generalizations of the interplay between Cauchy Riemann equations and complex-linearity
A function $f: \mathbb{C} \rightarrow \mathbb{C}$ is naturally viewed as mapping $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Suppose $f = (u,v)$ is continuously differentiable on $D \subseteq \mathbb{...
- 453
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484
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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 ...
- 93
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435
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Why is Mellin-inverse of Gamma periodic?
Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...
- 1,243