All Questions
372 questions
33
votes
20
answers
5k
views
Do names given to math concepts have a role in common mistakes by students?
Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "...
30
votes
1
answer
4k
views
Proof of "Possible new series for $\pi$" without use of physics
Related post: The post Possible new series for $\pi$ is about whether the identity is new, so to avoid confusion I was advised to ask this question separately.
I am looking for a proof of the ...
- 1,459
30
votes
4
answers
3k
views
Distribution of roots of complex polynomials
I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...
- 151k
29
votes
7
answers
7k
views
Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
user100272
28
votes
5
answers
3k
views
Continuous + holomorphic on a dense open => holomorphic?
Let D ⊂ ℂ be the closed unit disc in the complex plane, and let C be a continuously embedded path in D between the points -1 and 1. The curve C splits D into two halfs $D_1$ and $D_2$.
Let ...
- 43.2k
28
votes
2
answers
1k
views
Are there irreducible polynomials with all zeros on two concentric circles?
This is somewhat similar to this recent question, but extending in a different direction.
Let $f(x)$ be an irreducible polynomial of degree $n$ with integer coefficients. Call such $f$ a bicycle ...
- 13.4k
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
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
21
votes
6
answers
1k
views
What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 1,730
21
votes
0
answers
2k
views
Cartan–Oka vanishing in one variable without $\overline{\partial}$?
This is a literature question, about possible proofs of some very basic results in complex analysis.
Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
- 21.3k
21
votes
10
answers
6k
views
Not especially famous, long-open problems which higher mathematics beginners can understand
This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...
21
votes
7
answers
3k
views
What should be taught in a 1st course on Riemann Surfaces?
I am teaching a topics course on Riemann Surfaces/Algebraic Curves next term. The course is aimed at 1st and 2nd year US graduate students who have have taken basic coursework in algebra and manifold ...
- 3,284
20
votes
2
answers
9k
views
Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,906
20
votes
2
answers
1k
views
Belyi functions on non-compact surfaces; or: Building Riemann surfaces from equilateral triangles
Some background on (compact) Belyi surfaces
$\newcommand{\Ch}{\hat{\mathbb{C}}}$
A compact Riemann surface $X$ is called a Belyi surface if there exists a branched covering map $f:X\to \Ch$ such that $...
- 6,548
19
votes
14
answers
4k
views
Excellent uses of induction and recursion
Can you make an example of a great proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen ...
18
votes
2
answers
2k
views
Homotopy types of schemes
Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...
- 15.5k
16
votes
1
answer
2k
views
What are some of the earliest examples of analytic continuation?
I'm wondering how Riemann knew that $\zeta(z)$ could be extended to a larger domain. In particular, who was the first person to explicitly extend the domain of a complex valued function and what was ...
- 3,699
15
votes
5
answers
2k
views
Zeros of the derivative of Riemann's $\xi$-function
The Riemann xi function $\xi(s)$ is defined as
$$
\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).
$$
It is an entire function whose zeros are precisely those of $\zeta(s)$.
Since $\xi$ is real ...
- 11.1k
15
votes
3
answers
4k
views
Elementary Luroth theorem proof?
Hi, everyone!
I'm trying to explain the proof of Luroth theorem (every field $L$, s.t. $K\subset L\subset K(t)$, is isomorphic to $K(t)$) to the high-school audience. I'm not going to use such ...
- 1,422
14
votes
1
answer
1k
views
What is the analytic continuation of $\varphi(s)=\sum_{n \ge 1} e^{-n^s}?$
My research has lead me to the following function that I'm trying to continue. 3 Months ago I posted this question to MSE, and have placed 3 bounties on the question, but haven't received an answer, ...
- 101
13
votes
1
answer
929
views
Sendov's conjecture
It has been more than fifty years for famous Sendov's conjecture which states that if $p(z)$ is a polynomial of degree $n$ having all its zeros in the unit disc $|z|\leq 1$ then each of the n ...
- 559
12
votes
4
answers
2k
views
Seeking a Geometric Proof of a Generalized Alternating Series' Convergence
Let $z \in \mathbb{C} \backslash \lbrace 1 \rbrace$ with $|z| = 1$. We consider the following infinite series, which necessarily converges:
$$S(z) := \sum_{n = 1}^{\infty}\frac{z^n}{n}$$
Note that $S(...
- 7,831
12
votes
6
answers
1k
views
Functions holomorphic on a region minus a Cantor set
Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is ...
- 367
12
votes
2
answers
1k
views
Fourier transform of the critical line of zeta?
This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
- 4,907
11
votes
3
answers
1k
views
Can a metric conformal to a Kahler metric be Kahler?
Let $X$ be a non-compact complex manifold of dimension at least 2 equipped with a Kahler metric $\omega$. Take a smooth positive function $f : X \to \mathbb R$, and define a new hermitian metric on $X$...
- 4,343
10
votes
3
answers
2k
views
Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-...
- 565
10
votes
2
answers
2k
views
Characterize where the Dirichlet Problem for the Laplacian is always solvable
Conway's 1978 textbook Functions of One Complex Variable I gives an unsatisfying characterization of the regions for which the Dirichlet Problem can always be solved, and then comments no cleaner ...
- 1,601
9
votes
3
answers
797
views
An integral identity
$\newcommand\la\lambda\newcommand\w{\mathfrak w}\newcommand\R{\mathbb R}$Numerical calculations and other considerations (The min of the mean of iid exponential variables) suggest that
$$\int_\R \frac{...
- 128k
9
votes
3
answers
929
views
Are real-analytic functions in $\mathbb{R}^2$ holomorphic after suitable change of coordinates?
I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes.
Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always ...
- 5,590
8
votes
2
answers
2k
views
Higher dimensional residues in complex analysis
Consider a function $f:\mathbb{CP}^1\times\mathbb{CP}^1\to \mathbb{CP}^1 $ defined by $f([x_1,x_2],[y_1,y_2])=[x_1y_1,x_2y_2]$. This function is well defined except at $([0,1],[1,0])$ or vice versa (...
- 1,143
6
votes
1
answer
241
views
Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
- 1,730
6
votes
1
answer
900
views
What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the RH?
Let $\zeta$ denote the Riemann zeta function and let $\rho$ denote one of its complex zeros. What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming that all zeros are simple (SZC), ...
- 149
5
votes
1
answer
192
views
Topology of the space of univalent functions
Let $D\subset CP^1$ be a domain (a nonempty open connected subset) and let $S_D$ denote the space of conformal embeddings $D\to CP^1$ equipped with topology of uniform convergence on compacts. Is it ...
- 12.3k
5
votes
3
answers
2k
views
Continuous change of basis (and on the definition of determinant) [closed]
Let $(u_1, \ldots, u_n)$ and $(v_1, \ldots, v_n)$ be two ordered bases of $\mathbb R^n$. The orientation of the first basis is defined as the sign of the determinant of $[u_1 \cdots u_n]$, and ...
- 1,174
4
votes
1
answer
704
views
Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)
Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x ...
- 10.5k
4
votes
2
answers
701
views
Basic questions on the Hilbert scheme/ Douady space
Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More precisely,...
- 21.8k
4
votes
2
answers
986
views
Reference request: Oldest complex analysis books with (unsolved) exercises?
Per the title, what are some of the oldest complex analysis books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am aware of the ...
4
votes
1
answer
476
views
Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$
I am struggling trying to understand an statement in a paper I am reading:
Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose ...
- 601
3
votes
1
answer
385
views
Explicit form for hermitian structure $h$ with respect to $\omega$
Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$...
user21574
3
votes
1
answer
396
views
Conformal map onto a circle, from an identification space composed of five squares
I am looking to derive a conformal map for the problem illustrated in this image. I've read a bit about how to map a square onto a circle, but I'm struggling to extend the concepts for the domain at ...
- 167
3
votes
0
answers
226
views
On an exact expression for the squares of the distances of the critical points to a given zero of a polynomial
Let $p(z) = \prod_{j=1}^{l+1} (z - z_j)^{M_j}$ be a complex polynomial of degree $n$, where the $z_j$ are distinct for $1, \ldots, l+1$. The first $l$ entries in the list $\{z'_1, \ldots, z'_{n-1} \}$ ...
- 429
3
votes
0
answers
646
views
On properties on a certain functional
Consider the following function:
$$F(z) = \omega(z)\sin^2\left(\frac{c\Gamma(z)}{z}\right)$$
Here, $\omega(z)$ is a weight we have to construct and $c$ is a constant.
The following three conditions ...
- 375
2
votes
2
answers
2k
views
What is a simplified intuitive explanation of conformal invariance? [closed]
Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...
- 157
2
votes
1
answer
614
views
Half spaces free of roots of a given polynomial
I thank Loic Teyssier and Emil Jerabek who helped me to revise the two previous version
This question is motivated by the following fact in complex variable:(I learned this fact from the book of ...
- 366
2
votes
2
answers
860
views
$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear
I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. Show that $$ 2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$
Furthermore, there is ...
- 125
1
vote
1
answer
759
views
meromorphic extension of a function
Let $\Lambda\in \mathbf{C}$ be a discrete subset. We assume that $\mathrm{Re}(\lambda)<0$ for all the $\lambda\in \Lambda$. For $i\in \mathbf{N}$, $\lambda\in \Lambda$, let $m_{i,\lambda}\in \...
- 1,111
1
vote
0
answers
443
views
Multivariate solution to Lambert W / product-log function
Consider solving the following system for $x$
\begin{align*}
a - b e^{\theta x} - cx = 0
\end{align*}
According to your favorite computer algebra program, one possible (and the simplest) is
\begin{...
- 229
0
votes
2
answers
1k
views
Question on Hartogs's Extension Theorem
Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)?
For Hartogs's Extension Theorem see here:
http://en.wikipedia.org/wiki/...
- 53
158
votes
8
answers
7k
views
Resources for mathematics advising.
This question is possibly ill-advised. (If it is not right for this site I will delete it.)
I, suddenly, have students.
It is very clear to me that there is nothing in my education that has ...