All Questions
3,561 questions
2
votes
0
answers
231
views
Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
3
votes
1
answer
308
views
On convergence of entire functions
Suppose we have a sequence of entire functions $f_n$ such that $$\text{$f_n(z)\to0$ for each natural $z$}\tag{1}$$
(as $n\to\infty$).
Is it possible to give general additional conditions on the ...
- 128k
14
votes
6
answers
6k
views
Russian Equivalent of Big Rudin
Is there any Russian-authored textbook on Analysis equivalent to Big Rudin (Real and Complex Analysis)?
I like Russian math textbooks a lot. I am looking for Russian textbooks (either in English or ...
- 149
22
votes
13
answers
8k
views
Category theory sans (much) motivation?
So I have a friend (no, really) who's taking algebra and is struggling to gain intuition for it. My story is as follows: I used to hate abstract algebra, with pretty much a burning passion, until I ...
2
votes
1
answer
215
views
A 2 dimensional integral in polar coordinate [closed]
Recently I got stuck on a 2 dimensional integral in polar coordinate,
the expression is the following:
$I(x)=\lim_{\xi\rightarrow0^+}\int_0^\infty dr\int_{-\pi/2}^{\pi/2}dt\frac{2\xi ^{2-2 x}r^{2x+1} \...
- 71
3
votes
1
answer
342
views
Smooth functions with vanishing normal derivatives
Let $B_1 := \{z ∈ C : |z| \le 1\}$, and let $C_0(B_1,\mathbb C)$ be the space of continuous complex-valued functions on $B_1$ equipped with the uniform convergence topology.
How to show that, the set ...
- 1,269
2
votes
0
answers
134
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?
Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{...
- 1,805
1
vote
1
answer
149
views
Do all closed positive currents lift to a resolution?
Consider a normal complex analytic space $X$ and a projective birational resolution $f:Y\rightarrow X$. Let $T$ be a closed positive current of bi-dimension $(p,p)$ on $X$. Is there always a closed ...
- 329
5
votes
0
answers
197
views
Zeros of functions of the form $F(z) = \int_I g(t-z) f(t) \, dt$ with $g$ entire and $f \in L^1(I)$
Let $I \subset \mathbb R$ be a compact interval, $f \in L^1(I)$ and $g : \mathbb C \to \mathbb C$ an entire function. Define an entire function $F : \mathbb C \to \mathbb C$ via
$$
F(z) = \int_I g(t-z)...
- 173
0
votes
0
answers
74
views
When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?
Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...
- 623
1
vote
1
answer
165
views
Upper bound on double series
We consider the sum
$$ \sum_{m \in \mathbb Z^2} \frac{1}{(3 m_1^2+3m_2^2+3(m_1+m_1m_2+m_2)+1)^2}. $$
Numerically, it is not particularly hard to see that the value of this series is well below $4$, ...
- 73
3
votes
1
answer
129
views
Component wise convergence of a sequence of complex harmonic functions
It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...
- 165
4
votes
1
answer
392
views
Existence of periodic solution to ODE
We shall consider the matrix-valued differential operator
$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$
This is ...
- 192
1
vote
1
answer
219
views
What is the cohomology of the $C^{\infty}$-Koszul complex on $\mathbb{C}^2$?
First we consider the holomorphic Koszul complex on $\mathbb{C}^2$:
$$
0\to \mathcal{O}(\mathbb{C}^2)\overset{\begin{pmatrix}-z_2\\z_1\end{pmatrix}}{\to} \mathcal{O}(\mathbb{C}^2)^{\oplus 2}\overset{(...
- 9,019
3
votes
1
answer
281
views
Modular form not meromorphic at $\infty$
Is there a function $f$ with the following properties
$f$ meromorphic at the upper half plane $\mathfrak h$,
$f$ is of weight $k$ under a congruence subgroup of $\operatorname{SL}_2(\mathbb Z)$,
$f$ ...
- 2,375
3
votes
2
answers
256
views
Points attracting to 0 are dense in $\mathbb C$
I know that the following proposition is true, but at the moment I can't see how to prove it.
Define $f(z)=e^z-1$ for all $z\in \mathbb C$. Then $A:=\{z\in \mathbb C:f^n(z)\to 0\}$ is dense in $\...
- 3,317
5
votes
1
answer
855
views
$L\log L$ and Hardy space on the upper half plane
Set $\mathbb{T}$ the unit circle, $dm$ the Lebesgue measure on $\mathbb{T}$ and $\mathbb{C}^+=\left\{z\in \mathbb{C},s.t.\,\Im(z)>0 \right\}$ the upper half plane.
It is well-known that the Cauchy ...
- 51
4
votes
1
answer
425
views
Reverse residue theorem without using Serre's duality
In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text):
Let $\{a_1, \dots,a_n\}$ be a set of points in ...
1
vote
0
answers
47
views
Holomorphic "quasi-interpolation" of a function sequence
I am interested in some sort of analytic interpolation. A toy version of my problem is as follows. Let $V \subset \mathbb{C}$ be a complex neighborhood of $[0,1]$. Assume there is some bounded ...
- 981
7
votes
1
answer
299
views
Functions $f \geq 0$ on $\mathbb{R}$ which are of the form $f = |g|^2$ for some entire function $g$
I think the answer to this question must be well known. Is it possible to characterize those functions $f \colon \mathbb{R} \to \mathbb{R}_+$ which are of the form $f(x) = |g(x)|^2, x \in \mathbb{R},$ ...
- 2,101
5
votes
1
answer
521
views
Branched covers of the sphere branched over few points
Let $X$ be a compact Riemann surface of genus $g\geq 2$. By the Riemann-Roch theorem, $X$ is a branched cover of the sphere, branched over finitely many branched values. What is the smallest number of ...
- 6,548
10
votes
1
answer
803
views
A natural residue formula
A residue formula
I have strong evindence to believe that the following identity holds:
$$
\frac{n!}{2\pi i}\oint_{|z-1|=\epsilon} \frac{z^{a-1} \mathrm{d}z}{(z^d-1)^{n+1}} = d^{-n-1}\prod_{j=1}^{n}\...
- 833
8
votes
2
answers
693
views
Seeking a combinatorial proof for a binomial identity
Let $n\geq m\geq0$ be two integers. The below binomial identity is provable by other means:
$$\sum_{j=0}^m(-1)^j\binom{n+1}j2^{m-j}
=\sum_{j=0}^m(-1)^j\binom{n-m+j}j.$$
QUESTION. Can you provide a ...
- 43.2k
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$
where $y\in(0,\infty)$, $u\in(0,1)$, and
$$f(t):=t+\pi (1-t) t \cot (\pi t).$$
Here are the graphs of $f$ (black), ...
- 128k
10
votes
3
answers
931
views
Complex manifold with boundary
My question is of local nature.
Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative.
Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...
- 43.2k
50
votes
4
answers
6k
views
The maximum of a polynomial on the unit circle
Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself.
Suppose we are given $n$ ...
- 23k
2
votes
1
answer
271
views
Local extension of holomorphic vector fields
Let $X$ be an open complex manifold, e.g., the complement of a simple normal crossing divisor $D$ in a (smooth) projective manifold $M$. Let $T^{1,0}X$ be the holomorphic tangent bundle of $X$. Let $K ...
user105074
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
129
votes
2
answers
16k
views
What are the shapes of rational functions?
I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
- 25.1k
12
votes
2
answers
1k
views
Has there been further work on Bender-Brody-Müller approach to RH?
Earlier this year (April 4, 2017), a seemingly tantalizing approach of the Riemann Hypothesis based on ideas dating back to Hilbert and Pólya by Bender, Brody and Müller was made publicly available. I ...
- 6,982
15
votes
1
answer
505
views
Partial sums of $\sum_0^\infty z^n$
Let $z$ be a complex number with $|z|<1$. For every subset $A\subset\mathbb N$, the series $\sum_{m\in A}z^m$ is convergent. Denote $S(A)\in\mathbb{C}$ its sum and $\Sigma_z$ the set of all numbers ...
- 52.4k
1
vote
0
answers
94
views
Less strict holomorphy
Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....
- 117
3
votes
1
answer
203
views
Imaginary only values of the Lambert W function
For which values of $x \in \mathbb{R}$ is the principal branch $W_0$ of Lambert's $W$ function a purely imaginary value, i.e. for which $x$'s is $Re(W_0(x))=0$? For instance, $W_0(2 / \pi) = -i \pi / ...
3
votes
0
answers
177
views
A question on the proof of Bedford-Taylor theorem in Demailly's book
I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions.
I am reading a proof in the ...
- 21.8k
5
votes
2
answers
338
views
Approximation of analytic function by a fixed number of monomials
This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...
4
votes
1
answer
396
views
Limit at infinity of infinite series
Let
$$
f(x) = \sum_{n=0}^{\infty} a_n x^n
$$
and suppose that the radius of convergence of this series is infinite. Is there a general method to know whether $\lim_{x \rightarrow \infty} f(x)$ exists ...
- 1,608
0
votes
0
answers
253
views
Singularity of inverse exponential integral function
The exponential integral function is defined by
$$ Ei(z) = \int_{-\infty}^z dw \frac{e^w}{w} \,.$$
Away from the negative real axis the exponential integral function has a Taylor series about $z=0$:
$$...
- 81
6
votes
0
answers
755
views
Discriminant of $\alpha P(u) + (z-u) P'(u)$
I'm trying to find a “closed form” of $\textrm{Discriminant}_u(f(u))$, where $f(u) := \alpha P(u) + (z-u) P'(u)$.
Here $P(u)$ is a monic polynomial of degree $d > 1$ with $u\in\mathbb{C}$, $\alpha$ ...
- 41
12
votes
1
answer
895
views
Newman's proof of the prime number theorem
I am teaching a graduate course in Complex Analysis and I am covering Newman's proof of the prime number theorem. I have been using the simplified version in the papers of
Zagier and Korevaar. However,...
- 411
20
votes
2
answers
9k
views
Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,906
0
votes
1
answer
114
views
Limiting behaviour of Cauchy integral near boundary
Let $D \subseteq \mathbb{C} $ be bounded and simply connected, $\Gamma:= \partial D \in C^2 $, $\phi, \psi \in C^{0,\alpha}(\Gamma)$,
$$
f(z):= \frac{1}{2\pi i} \int_{\Gamma} \frac{\phi(\zeta)}{\zeta -...
- 269
0
votes
1
answer
524
views
To integrate elliptic integral, we glue two Riemann surface to make torus
To deal with elliptic integral, we often cut riemann surface and glue them together, and gain a torus. We do this in order to avoid indeterminacy of integral, in other word, to avoid the condition ...
- 1,541
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
4
votes
1
answer
471
views
Beta function, harmonic numbers, and integral values
A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:
$$
I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k
$$
where $\beta_x( -1 - ...
- 255
10
votes
4
answers
2k
views
Why the unreasonable applicability of complex numbers in physics/engineering? [duplicate]
After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the ...
- 217
1
vote
0
answers
31
views
Helgason's support theorem type result in 2 dimensions
I had posted this question in math stackexchange here. Let $\Omega \subset \mathbb{R}^2$ be an open domain with smooth boundary. Identifying $\mathbb{R}^2$ with $\mathbb{C},$ consider the following ...
8
votes
0
answers
308
views
Is an entire function $\mathbb{C}^n\to\mathbb{C}$ a composition of polynomials, univariate entire functions and integrals?
Let $S$ be a set of entire functions $\mathbb{C}^n\to\mathbb{C}$.
To enlarge it we can take polynomial combinations of its elements, compose them with entire functions $\mathbb{C}\to\mathbb{C}$ and ...
user380251
4
votes
0
answers
760
views
What is a holomorphic foliation?
For a smooth foliation $F$, there are three equivalent definitions:
the leaves of $F$ are tangent to a smooth vector field;
the foliation chart $\phi:U\to \mathbb R^k\times \mathbb R^{n-k}$ is ...
- 307
1
vote
0
answers
80
views
Positive integration on P^1
Let $u: \mathbb{P}^1(\mathbb{C}) \longrightarrow \mathbb{R}$ be a smooth function s.t. $u$ is invariant under complex conjugation and $\displaystyle \int_{\mathbb{P}^1(\mathbb{C})}u \; \omega_{\mathrm{...
- 171
7
votes
2
answers
375
views
Multiplication in Deligne cohomology: explicit formula for $p=q=1$
[This is a double of my question of math.stackexchange https://math.stackexchange.com/questions/3214962/multiplication-in-deligne-cohomology-explicit-formula-for-p-q-1]
In the very beginning of [1] ...
- 1,170