All Questions
3,560 questions
17
votes
5
answers
3k
views
Teaching prime number theorem in a complex analysis class for physicists
This is a question about pedagogy.
I want to sketch the proof of the prime number theorem or any other application of complex analysis to number theory in a single lecture, in a complex analysis ...
- 253
3
votes
1
answer
468
views
Relationship between two kinds of classifications of Riemann surfaces
There are two kinds of classifications of Riemann surfaces.
Classification 1: Let $M$ be a Riemann surface. We will call $M$:
elliptic iff $M$ is compact (= closed);
parabolic iff $M$ is not compact ...
- 438
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
7
votes
0
answers
306
views
Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
- 1,730
7
votes
2
answers
484
views
Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free
I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...
- 585
1
vote
1
answer
349
views
Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?
It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus.
Does the general formula for the $n$th derivative of the power-exponential ...
- 1,101
29
votes
2
answers
561
views
A strange infinite fraction, and a functional equation
The following curious-looking fraction, with numerical value approximately $1.7302267782385217$, appears in this Reddit question:
$$1+\cfrac{2+\cfrac{4+\cfrac{8+\cdots}{9+\cdots}}{5+\cfrac{10+\cdots}{...
- 721
24
votes
15
answers
5k
views
Applications of connectedness
In an «advanced calculus» course, I am talking tomorrow about connectedness (in the context of metric spaces, including notably the real line).
What are nice examples of applications of the idea of ...
32
votes
7
answers
8k
views
Interpreting the Famous Five equation [closed]
$$e^{\pi i} + 1 = 0$$
I have been searching for a convincing interpretation of this. I understand how it comes about but what is it that it is telling us?
Best that I can figure out is that it just ...
- 425
5
votes
1
answer
458
views
How to define a current on a complex analytic space
I'm reading a paper in which the author use $(p,q)$-forms and currents on a complex analytic space.
My question is how to define $(p,q)$-current on complex space? Does it have similar properties like ...
- 361
4
votes
0
answers
160
views
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
13
votes
1
answer
929
views
Sendov's conjecture
It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
- 559
23
votes
2
answers
975
views
Does Rademacher's convergent series for p(n) define an analytic function?
Let $p(n)$ be the number of partitions of $n\geq 0$. We can let $n$ be
any complex number in Rademacher's convergent infinite series for
$p(n)$. (See e.g. equation (24) here.)
For what $n$ does it ...
- 50.8k
1
vote
2
answers
307
views
A characterization of plurisubharmonic functions
Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function.
Recall that $u$ is called plurisubharmonic (psh) if its restriction ...
- 21.8k
2
votes
0
answers
89
views
Finding a branch cut or a branch point [closed]
Is there a way to find a branch cut or a branch point, through which a curve over a complex function goes, or in general in some region of complex function, say $\ln(f(z))$, using numerical methods or ...
- 67
7
votes
2
answers
1k
views
How to treat Puiseux series as functions?
I have been reading about Puiseux series in the context of the Newton–Puiseux algorithm for resolution of singularities of algebraic curves in $\mathbb{C}^2$. Given a curve $f(x,y)=0$ with $f$ a ...
- 306
6
votes
2
answers
1k
views
$\log |f|$ is subharmonic
It is known that the logarithm of the modulus of an analytic function $f: D \subset \mathbb C \rightarrow \mathbb C$ ($D$ is a domain) is subharmonic. I have two questions:
(1) Are there some weaker ...
- 285
3
votes
1
answer
712
views
Analytic continuation over boundaries
In D.J Newman's paper
A simple analytic proof of the prime number theorem
there is the following theorem:
Suppose $|a_n|<1$ and form the Dirichlet series $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ ...
- 2,902
9
votes
1
answer
853
views
Moments of the Riemann zeta function
Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$
- 150
1
vote
1
answer
210
views
On a property of complex exponentials
Does there exist a simple smooth closed curve $\gamma:S^1\to \mathbb C$ such that
$$ \int_{0}^{2\pi} e^{\gamma(e^{it})} \, |\gamma'(e^{it} )|\,dt =0?$$
- 4,115
5
votes
0
answers
260
views
What is the winding behavior of the Riemann zeta function around zero along the line $s=1+it$?
Let $\phi: \mathbb R \setminus \{0\} \to S^1 \subset \mathbb C$ be defined by
$$\phi(t)= \zeta(1+it)/|\zeta(1+it)|$$
(the nonvavishing of the denominator being a bit weaker than the prime number ...
- 64k
27
votes
5
answers
6k
views
The Matrix-Tree Theorem without the matrix
I'm teaching an introductory graph theory course in the Fall, which I'm excited about because it gives me the chance to improve my understanding of graphs (my work is in topology). A highlight for me ...
- 22.2k
22
votes
1
answer
3k
views
Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?
Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
- 393
16
votes
2
answers
2k
views
An analogue of the exponential function by replacing infinite series with improper integral
For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$
where $t!=\Gamma(t+1)$. This is motivated by classical exponential function.
Is this function well defined (...
- 366
1
vote
1
answer
66
views
Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequence of smooth bounded domains from below
Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining ...
- 63
5
votes
1
answer
169
views
Hadamard factorization of a function in the Fock space
An entire function $F: \mathbb C \to \mathbb C$ belongs to the Fock space $\mathcal F^2$ if
$$
\int_{\mathbb C} |F(z)|^2e^{-|z|^2} \, dA(z) < \infty.
$$
It is well-known that every $F \in \mathcal ...
- 190
2
votes
0
answers
116
views
Spectrum of 'complexified' Laplace operator
Let $(M^n,g)$ be a closed Riemannian manifold. Let $\Delta$ be the Laplace–Beltrami operator acting on scalar functions defined on $M$, and let
$\lambda_1 < \lambda_2 \leq \cdots$ be its spectrum.
...
- 5,048
49
votes
4
answers
6k
views
If the Riemann Hypothesis fails, must it fail infinitely often?
That is must there either be no non-trivial zeros off the critical line or
infinitely many?
I'm sure that no one believes otherwise, but I've never seen a theorem in the
literature addressing this. ...
- 17.6k
6
votes
4
answers
630
views
Generalizing contour integration to quaternions and bicomplex numbers
I am interested in the possibility of generalizing the notion of contour integration to the quaternions or bicomplex numbers. I am aware that the Frobenius theorem prevents the construction of a true ...
- 547
3
votes
1
answer
314
views
Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?
Say we have a power series of two variables, with an associated function $f$ defined as
$$
\begin{split}
f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\
& a_{n,m} \geq 0 \quad \forall n, m \in\...
- 63
16
votes
2
answers
956
views
Affine (or Stein) tubular neighbourhood theorem
Fix an embedding $X\subset Y$ of smooth complex affine varieties, or Stein manifolds.
I would guess that in general there is no analytic neighbourhood $X\subset U\subset Y$ with a holomorphic ...
- 1,191
19
votes
7
answers
3k
views
Is this a rational function?
Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient ...
- 11.3k
4
votes
2
answers
420
views
Borel summation and the Abel function of $e^z-1$
This is a question that has bothered myself and Gottfried Helms a fair amount of late. He has made his case for the following result, but a proof escapes both of us. The question is deceptively simple,...
- 388
3
votes
2
answers
245
views
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
4
votes
2
answers
316
views
Request for references in computational complex analysis
We know complex analysis is one of the most important branches of mathematics connecting myriad areas. It is replete with profound results and theorems and theorems. However, a good number of the ...
- 826
41
votes
3
answers
3k
views
Can the unsolvability of quintics be seen in the geometry of the icosahedron?
Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...
- 151k
4
votes
0
answers
74
views
Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?
Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
3
votes
1
answer
167
views
Entire function with almost periodic boundary condition?
Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...
- 73
1
vote
2
answers
590
views
Inequality between coefficients of a polynomial and its supremum
For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
- 463
1
vote
0
answers
140
views
Does a Borel transform uniquely determine a Borel measure?
It is a known fact that Borel measures are uniquely determined by their Fourier transforms. This is the motivation for the following question.
I came across the concept of a Borel transform of a Borel ...
- 3,535
4
votes
0
answers
450
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
4
votes
0
answers
135
views
Reverse Sobolev inequality for family of holomorphic functions
Denote by $P_m$ the space of polynomials of degree $m$ in a single complex variable $x$. This is a "Reverse Sobolev inequality":
Theorem. Let $U \subset \mathbb{C}^2$ be an open domain and $...
- 981
6
votes
1
answer
452
views
Finite set of numbers whose powers sum up to irrational number
It is well-known that $e/\sqrt{2}$ is irrational.
Indeed, if it was rational, i.e. $p/q$ then $e^2/2 =p^2/q^2.$ Thus, $q^2e^2=2p^2,$ which would imply that $e$ is a root of $q^2x^2=2p^2.$
Now my ...
- 73
26
votes
18
answers
34k
views
Undergraduate differential geometry texts
Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?
(I know a ...
0
votes
0
answers
79
views
Geometry of inner products between the unit vector and several given vectors
Let $\mathcal{S}$ denote the set of all unit complex-valued $d$-dimensional vectors, i.e.,
$$
\mathcal{S} \triangleq \left\{ \mathbf{s}\in \mathbb{C}^{d} \mid \mathbf{s}^{\mathrm{H}}\mathbf{s}=1 \...
- 550
2
votes
0
answers
90
views
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....
- 2,000
4
votes
0
answers
214
views
Cartan–Remmert reduction of an algebraic variety
Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
- 1,170
5
votes
2
answers
408
views
Extended binomial coefficients and the gamma function
For which $(a,b,n) \in \mathbb{Z}^3$ satisfying $a+b=n$ does $\frac{\Gamma(z+1)}{\Gamma(x+1)\Gamma(y+1)}$ approach a limit as $(x,y,z) \rightarrow (a,b,n)$ in $\mathbb{C}^3$, and what is that limit? (...
- 19.7k
0
votes
0
answers
57
views
Integrability of logarithm of a function in Hardy class
Assume that f is in $H^1$ (The Hardy space of holomorphic functions in the unit disk). I need an easy reference to the following fact $ \log |f|$ is in $h^1$, i.e. it belongs to harmonic Hardy space.
- 37