Questions tagged [cv.complex-variables]
Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.
3,452 questions
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Finding a condition for an inequality
Let $\mathbb{D}\subset \mathbb{C}$ denote the unit disk.
I would like to seek sufficient—and, if possible, necessary—conditions on a weight function $$\mu: \mathbb{D} \rightarrow (0,+\infty),\quad \mu ...
- 11
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102
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A mistake (???) in Bieberbach's book "Conformal mapping"
Bieberbach gives a proof (reproduced below) of the classification of annuli (up to conformal equivalence) on pages 209-211 of his book Conformal mapping (New York: Chelsea Co. pp. VI+234 (1953), ...
- 30k
3
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1
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Fourier transform of compactly supported functions
Let $F:\mathbb C^n\longrightarrow \mathbb C$, be an entire function of exponential type, that is $F$ is holomorphic on $\mathbb C^n$ and there exist constants $C, A$ such that
$$
\vert F(z)\vert\le C ...
- 16.7k
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2
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Asymptotics of $\sum_n^\infty \left(\log\left|1 - \frac z{a_n}\right| + \Re\left(\frac z{a_n}\right)\right)$
TL;DR: I would like to find an asymptotic for
$$
\sum_{n=2}^\infty \left(\log\left|1 - \frac z{a_n}\right| + \Re\left(\frac z{a_n}\right)\right), \tag{$*$}
$$
as $z\to \infty e^{i\theta}$, where $a_n =...
- 245
2
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0
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227
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On Ozsváth's and Szabó's description of Abel-Jacobi theorem
On p. 1035 of Peter Ozsváth's and Zoltán Szabó's "Holomorphic disks and topological invariants for closed three-manifolds", Annals of Mathematics (2) 159, No. 3, 1027-1158 (2004), MR2113019, ...
- 21
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124
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How to construct elliptic functions with predescribled zeros and poles by means of Weierstrass ℘-function and its derivatives?
Given a lattice $\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ in $\mathbb{C}$. It's well known that given $n_i\in\mathbb{Z}, z_i\in \mathbb{C}$ satisfying $\sum n_i z\in\Lambda$ and $\sum n_i=0$, ...
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0
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202
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Zeros of the partial sums $\sum_{k=0}^n (-1)^k/(z-k)$
Let
$$
D_n(z)=\sum_{k=0}^n \frac{(-1)^k}{z-k}
=\frac{P_n(z)}{Q_n(z)}, \qquad
Q_n(z) = \prod_{k=0}^n (z-k).
$$
We have $\deg P_n = n$ ($n$ even), and $\deg P_n = n-1$ ($n$ odd). Moreover, using the ...
- 703
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173
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Picard-Fuchs equation, Schwarzian derivative and Bers embedding of Teichmuller Space
Let $X$ be the elliptic curve
$$y^2=x^3-g_2x+g_3$$
The $j$ invariant of $X$ is
$$j=\frac{g_2^3}{g_2^3-27g_3^2}$$
I came across the formula of Dedekind
$S(\tau)(j)=\frac{1-\frac{1}{2^2}}{(1-j)^2}+\frac{...
- 515
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0
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144
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Convexity principle in several complex variables
I would like a reference for an analogue of the Phragmen-Lindelof in several complex variables. Specifically, if f is analytic in a region $A \subset \mathbb C^n$ and, by Bochner's theorem it extends ...
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0
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99
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Upper bound for maximum of reciprocal of zeta
The following appears in this paper:
Lemma Let $H=T^{1/3}$. Then we have
$$\min_{T\le t\le T+H}\max_{1/2\le\sigma\le 2}\frac{1}{|\zeta(\sigma+it)|}<\exp(C(\log\log T)^2)$$
where $C$ is an absolute ...
user574181
2
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1
answer
180
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Explicit convexity bound for Dirichlet $L$-functions
The following is a well-known convexity bound for Dirichlet $L$-functions.
Theorem Let $\chi$ be a primitive character to the modulus $q$. Then, for any fixed $\varepsilon>0$ and any $k\in\mathbb{...
user574181
10
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1
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No real roots of $\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}}$
Prove or disprove that $$\frac{1}{2^{1+it}} + \frac{1}{3^{1+it}} + \frac{1}{5^{1+it}} \neq 0$$ for any real number $t$.
I find it surprising that so simple looking equations involving complex numbers ...
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1
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1
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184
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Critical zeros and Levinson's method
In this paper it was shown that one can prove at least one third of the non-trivial zeros of the Riemann zeta function lie on the critical line. This method relies on the twisted second moment of $\...
user574181
4
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0
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94
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When does the positive part of an entire function being in $L^2$ imply that the entire function is in $L^2$?
Let $f$ be an entire function of exponential type, real on the real axis, and $f^+$ its positive part, i.e.
$$
f^+(x) = \begin{cases}
f(x) & \text{if} f(x) \geq 0,\\
0 & \text{otherwise}.
\end{...
- 167
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1
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189
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Is phase $S(T)$ of Riemann Zeta function jumping maximum by one for small increase of $T$?
The number of non-trivial zeros of the $\zeta$ function is strongly coupled to
the hypothetical number of zeros outside of the critical line that are
counter-examples for the Riemann Hypothesis. Hence,...
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126
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Necessary condition for a partial sum in the limit
For $s\in\mathbb{C}$, let $S(s)$ be a series that checks for the Hurwitz Theorem,
If $S(s)=0$, then by Hurwitz theorem, there exists a sequence $s_n$ so that $s_n \xrightarrow[n\to \infty]{} s$ ...
- 17
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0
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69
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Is the Minkowski sum of two strictly pseudo-convex bounded domains pseudo-convex?
Let me just recall that the Minkowski sum of two sets is defined by
$$A+B=\{a+b|\, a\in A, b\in B\}.$$
- 23.1k
3
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1
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336
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Computing Pic with the exponential exact sequence for singular Varieties
For a smooth complex projective variety $X$, the exponential exact sequence is $0\to \mathbb{Z} \to O \to O^*\to 0$, and gives rise to a LES of cohomology. Here, $O$ is the sheaf of holomorphic ...
- 759
6
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370
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Composition of two functions is holomorphic and second is holomorphic then first is holomorphic
Let $f, g: \mathbb{C}^n \rightarrow \mathbb{C}^n$, $g$ is surjective, $f \circ g$ is holomorphic and $g$ is holomorphic. Is $f$ holomorphic? I found this is true for 1-dimensional case but is it such ...
- 121
34
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2
answers
3k
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Best formulation of Riemann hypothesis for a general audience
There are many equivalent ways to state the Riemann Hypothesis. I'm looking for a statement that is mathematically precise and yet at the same time as accessible as possible to a general audience. The ...
- 399
3
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1
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351
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Barriers to a fixed-width zero-free region for zeta
The classical zero-free region of the Riemann zeta function $\zeta(s)$ says there is a constant $A>0$ such that there are no zeta zeros $$\rho=\sigma+iT$$
with $\sigma>1-\frac{A}{\log T}$. ...
user574181
0
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0
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104
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Why is it hard to get a zero-free half-plane $\Re s>\tfrac12$ for $\zeta(s)$ via Rouché using the Maclaurin (Euler–Maclaurin) formula?
I start from the classical “Maclaurin” form of Euler–Maclaurin for every integer $n\ge2$:
$$
\zeta(s)
=\frac{1}{s-1}+\frac12+\sum_{k=2}^{n} B_k\,\frac{s(s+1)\cdots(s+k-2)}{k!}
-\frac{s(s+1)\cdots(s+n-...
- 481
1
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1
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256
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Average of $\Lambda(n)^2$
Let $\Lambda$ be the von Mangoldt function. I am interested in understanding the average $$\sum_{n=1}^x \Lambda(n)^2.$$ By partial summation and the prime number theorem one can prove that this is $$ ...
- 3,092
9
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1
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393
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Contour integration with poles that coincide with branch points
I would like to compute the following integral:
$$\int_{-\infty}^{\infty}\frac{dx}{2\pi} \frac{e^{-2d\sqrt{-x^2+\alpha^2+i\epsilon}}}{(-x^2+\alpha^2+i\epsilon)^2}$$
where $d, \alpha, \epsilon > 0$ ...
- 91
1
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66
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Inverse branches problem for non-proper holomorphic endomorphism of the unit disk
Let $D(0,1)$ be the unit open disk on $\mathbb{C}$. Let $w=f(z): D(0,1)\to D(0,1)$ be a holomorphic map and continuous at the boundary. We do not assume that $f$ is proper. I want to know whether ...
- 337
2
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0
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225
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Weak Phragmén-Lindelöf for meromorphic functions?
Let $f$ be meromorphic on the half-strip $\Re s\leq 0$, $|\Im s|\leq 1$, and bounded on its boundary. I would like to be able to assert that there is a sequence $0>\sigma_1>\sigma_2>\sigma_3&...
- 22k
8
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1
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775
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What is an application of Goursat's stronger version of Cauchy's theorem?
Many modern undergraduate textbooks give the following path to the Cauchy theory of contour integrals. Assume that $f \colon U \to \mathbb{C}$ is holomorphic: has a complex derivative (not assumed to ...
- 30k
3
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0
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206
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Analytic continuation of algebraic functions
Consider the polynomial $K(z,u)= u^e- z (p_0+p_1 u+...+p_k u^k)$, where: $k,e$ are nonnegative integers and the coefficients $p_i$ are nonnegative reals such that $0< \sum_i p_i \leq 1$ with $p_0&...
- 373
4
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1
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186
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Lower bound on the ratio between length of relative boundary and total boundary in Ahlfors' theory
Sorry I ask the following question about Ahlfors' theory and it seems a little awkward since I'm not an expert in this field.
Ahlfors' theory has the notion "relative boundary" and I feel it ...
- 337
1
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1
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160
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Reference for multidimensional steepest descent with higher order correction terms
I am looking for a reference which proves a multidimensional steepest descent with higher order correction terms.
The integral that I consider is of the form
\begin{equation}
I(\lambda)=\int_\Gamma e^{...
- 39
0
votes
2
answers
204
views
Linear isoperimetric inequality for bounded holomorphic function defined on the unit disk
Let $\mathbb{D}=D(0,1)$ be the open unit disk on the complex plane $\mathbb{C}$. Let $\bar{\mathbb{D}}$ and $ \partial\mathbb{D} $ denote the closure of $\mathbb{D}$ and the boundary of $\mathbb{D}$, ...
- 337
1
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0
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90
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Realizing a Laurent polynomial as a dynamical zeta function or characteristic polynomial from an isolated surface singularity
Given a Laurent polynomial $\Delta(t) \in \mathbb{Z}[t, t^{-1}]$ satisfying:
$\Delta(1) = \pm 1$,
$\Delta(t^{-1}) = t^{\pm k} \Delta(t)$ (symmetry up to a unit),
can $\Delta(t)$ be realized as a ...
- 63
2
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1
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215
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Analytic continuation of the stereographic projection: Riemann surface description
Consider the standard stereographic projection from the real projective line $\mathbb{R}P^1$ to the unit circle $S^1$, which can be described by the map: $$ x \mapsto (\cos(2 \arctan x), \sin(2 \...
- 63
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88
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Solutions of Appell system of type $F_4$
The Appell's hypergeometric function of type $F_4$ is defined as:
$$
F_4 (\alpha,\beta,\gamma,\gamma'; z_1,z_2) = \sum_{m,n = 0}^{\infty}
\frac{(\alpha, m+n)(\beta, m+n)}{(\gamma,m)(\gamma',n)(1,m)(1,...
- 635
6
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1
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349
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Restriction of the Hodge decomposition to Kähler submanifolds
Let $(X, \omega)$ be a compact Kähler manifold with kähler form $\omega$, and let $Y\subset X$ be a Kähler submanifolds with the induced Kähler form. The kähler form $\omega$ induces an isomorphism, ...
- 93
30
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2
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Why do winding numbers replace two-chains in the proof of Cauchy's theorem?
I asked this question over at History of Science and Mathematics three months ago. Since there were no replies there I thought I would try my luck here.
Many modern textbooks on introductory complex ...
- 30k
1
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1
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144
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Kobayashi metric of the tube domain
$\DeclareMathOperator\Kob{Kob}$Assume that $T_\Omega= \Omega+ i R^n\subset C^n$. Is this formula true $\Kob_{T_\Omega}(z,v)=\Kob_\Omega(x,v_x)$,$z=x+ iy\in C^n$ is a point, and $v=v_x+iv_y$ is a ...
- 121
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0
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144
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$f' \in D(\mu)$ implies $f \in D(\mu)$
Let $D(\mu)$ denote the harmonically weighted Dirichlet space introduced by S. Richter. I was wondering if $f' \in D(\mu)$ implies $f\in D(\mu)$.
A result due to Liu, Chacón and Lou, Theorem 3.5 says ...
- 203
4
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1
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361
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Log-free Perron - consequences?
Say you have a form of Perron's formula with an error bound of the form $O(x/T)$. (The companion to this question is Truncated Perron - logarithm-free error term? , where I asked for such a formula; ...
- 22k
4
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0
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224
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How do we know that global analytic functions exist?
Let $(M,m_0)$ be a real analytic manifold. Let $\pi$ be a cotangent at $m_0$. How can one prove that there exists a global analytic function $\theta:M\rightarrow \mathbb{R}$ such that $d\theta(m_0)=\...
- 1,329
2
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0
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394
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What are some verifiable (and reasonably weak?) conditions to guarantee that the zero set of holomorphic functions contains more than one point?
I know very little algebraic geometry, so maybe this is an easy question, but I ran into it and I could use some expert guidance.
Given holomorphic functions $f_1,...,f_m:\mathbb C^n \to \mathbb C$ (...
- 398
4
votes
1
answer
410
views
Generalizing Marden's theorem to quartics
Is there a nice generalization of Marden's theorem which applies to all quartics?
Marden's theorem is a strengthening of the Gauss-Lucas theorem for polynomials over the complex numbers which applies ...
- 8,540
2
votes
1
answer
214
views
How can one prove this convergence theorem in complex analysis?
Reinhold Remmert's "Classical Topics in Complex Analysis"
Reinhold Remmert's Classical topics in complex function theory, (Transl. by Leslie Kay, Graduate Texts in Mathematics, 172, New York,...
- 23
13
votes
1
answer
942
views
Geometric meaning of the Wronskian
Suppose $f:\mathbb C\to \mathbb C^n$ is a holomorphic function. Let
$w(f)=det (\partial^{i+1} f_j)$
be the Wronskian of $f=(f_1,\dots,f_n)$.
Then it is known (Maxime Bocher 1901) that $$w(f)\equiv 0 \...
- 515
4
votes
1
answer
199
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Entire function of order 1 and type $\infty$ with prescribed indicator function $h_f(\theta)$
I would like to know if there exists an entire function $f$ of order 1 such that the indicator
$$
h_f(\theta) := \limsup_{r\to\infty} \frac{\log |f(re^{i\theta})|}{r}
$$
satisfy
$$
h_f(\theta) = \...
- 245
1
vote
0
answers
64
views
Closure of locally closed varieties
Let $K$ be the field of real or complex numbers and I consider the analytic topology on all mentioned spaces.
Suppose that $M$ is a $K$-analytic manifold (not compact or anything) and let $T\subset M$ ...
- 845
3
votes
0
answers
299
views
Singularities of exponential series $\sum a_n \exp(i \sqrt{n} z)$
More generally I'm interested in series of the form $f(z):=\sum a_n \exp(i n^{1/d} z)$ that have analytic continuation, and finding restrictions on where their singularities can be.
The motivation is ...
- 1,173
3
votes
0
answers
700
views
Partial fraction expansion for $\frac{\xi}{\xi'}(s)$
There is a well known partial fraction expansion for the log-derivative of Riemann $\xi(s)$ function. Is there a formula for the reciprocal ?
We move the critical line to the real axis and consider ...
- 2,139
4
votes
0
answers
289
views
On the identity theorem for holomorphic functions of several complex variables
This question was triggered by the recent digitalisation of the almost forgotten but nevertheless important paper by Beppo Levi [3].
Premises
In the field of several complex variables, the name "...
- 6,830
3
votes
1
answer
211
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Bounds on Banach density of zeros of functions of exponential type
It is well known that the upper asymptotic (or natural) density of the set of zeros of a function $f$ of exponential type $\tau$ is constrained. Let $R = \{|z_n|\}$ be the set of absolute values of ...
- 167