All Questions
3,560 questions
38
votes
2
answers
2k
views
Residues in several complex variables
I am trying to educate myself about the basics of the theory of residues in several complex variables. As is usually written in the introduction in the textbooks on the topic, the situation is much ...
- 1,190
1
vote
0
answers
126
views
Canonical basis of cycles of Riemann surfaces
Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
2
votes
0
answers
109
views
Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
- 295
32
votes
9
answers
21k
views
Interesting applications of the classical Stokes theorem?
When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y ...
2
votes
1
answer
215
views
An integral transform computation
In Erdelyi, Tables of Integral Transforms, p. 344 Section 7.2.
they note that
\begin{align}
\frac{1}{2 \pi i} \int_{c-i\infty}^{c+i\infty} s^{\nu} e^{\alpha s^2} x^{-s} \, ds
= 2^{-\nu/2} \pi^{-...
- 21
1
vote
2
answers
163
views
Transcendental functions with two prescribed values
Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$.
I would like ...
- 515
4
votes
1
answer
488
views
Linear combinations of roots of unity
Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form
$$ z = \sum_{i=0}^{n-1} x_i \omega^...
- 2,222
7
votes
1
answer
195
views
Are entire functions uniformly bounded from below on a line through the origin?
Let $F : \mathbb C \to \mathbb C$ be an entire function of finite order. Since the zeros of $F$ are countable there exists a constant $c \in \mathbb R$ such that $F$ is zero-free on the line $e^{ic} \...
- 437
52
votes
22
answers
19k
views
Interesting Calculus Questions/Exercises
I am in the process of redesigning the calculus course that I have taught five or six times. What I would like to know is if anyone has some really good examples or exercises that I could either do ...
106
votes
6
answers
19k
views
Why does the Riemann zeta function have non-trivial zeros?
This is a very basic question of course, and exposes my serious ignorance of analytic number theory, but what I am looking for is a good intuitive explanation rather than a formal proof (though a ...
- 29k
6
votes
1
answer
468
views
a problem in complex-variable inequality
Let $n\ge2$ be a given positive integer, and $z_{1},z_{2},\cdots,z_{n}\in \mathbb{C}$,such
$$|z_{1}|^2+|z_{2}|^2+\cdots+|z_{n}|^2\ge n.$$
Prove or disprove
$$f_{n}=\sum_{j=1}^{n}\left|\sum_{I\subseteq ...
- 4,280
2
votes
1
answer
170
views
A specific question on the Griffiths' paper: the reduction of the pole order
If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10?
I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\...
- 73
7
votes
0
answers
219
views
Partitions, weights and polynomials with roots on the unit circle
Let us consider the set $[n]=\{1,\ldots,n\}$ and all of its partitions into exactly $m$ blocks, but let us allow each block to be internally ordered. For example, taking $n=6$ and $m=2$, we will ...
- 1,889
5
votes
3
answers
343
views
Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 2,689
6
votes
2
answers
348
views
Does there exist a framework for determining if a power series is "differentially algebraic"
It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
- 2,689
3
votes
0
answers
146
views
A Hartogs analogue?
Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$.
For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
- 31
34
votes
23
answers
29k
views
Textbook recommendations for undergraduate proof-writing class
I am teaching the proof-writing class (for the 3rd time) in the Fall and plan to buck the party line and use a different text than the default Bond and Keane. My parameters are as follows:
Logic, ...
5
votes
1
answer
291
views
Asymptotics of the Liouville sum at the primes
Let $\lambda$ denote the Liouville function, and let $L(x):=\sum_{n \leq x} \lambda(n)$ be the Liouville sum. Define $c$ to be the supremum of the real parts of the zeros of the Riemann zeta function. ...
- 103
4
votes
2
answers
432
views
Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$?
So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of
$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \...
- 2,689
1
vote
0
answers
210
views
Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform
This question is reposted from Math Stack Exchange (you can see the original post here). The motivation for reposting is that I feel like the question isn't getting much attention in MSE - if there is ...
- 51
6
votes
0
answers
131
views
Complex beta function $\int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\beta-1} \,dx\,dy$
I am interested in showing that the integral
\begin{align}
& \int_{\mathbb{C}} |z|^{2\alpha-2}|1-z|^{2\beta - 2} \,dA(z) \\[8pt]
= {} & \int_{\mathbb{R}^2} (x^2+y^2)^{\alpha-1}((1-x)^2+y^2)^{\...
- 177
8
votes
1
answer
722
views
A robust version of "a holomorphic function is determined by its modulus"
It is well known that if $f(z)$ and $g(z)$ are both holomorphic on a (path-)connected open set $C$ and $\lvert f(z)\rvert=\lvert g(z)\rvert$ on $C$ then $f(z)=cg(z)$ on $C$ for some constant $c$. Do ...
- 1,551
2
votes
1
answer
185
views
Inverse of Bochner–Martinelli formula
Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that
$f(z) = \int_{\...
- 661
1
vote
1
answer
88
views
Mean values of polynomial and holomorphic matrices
Lemma. Assume $H: \mathbb{R} \to \mathbb{R}^{d \times d}$ is a polynomial of degree $m$, such that for all $x \in \mathbb{R}$, $H(x)$ is a symmetric semidefinite matrix. For all $n \geq 0$ and real ...
- 981
-1
votes
1
answer
175
views
On the bound for $\int_{x}^{x+i\infty} (\cot(\pi z)+ i)z^{-s} \, \mathrm{d}z$
I'm reading Titchmarsh's "The theory of the Riemann zeta function", and on p.81 it is claimed that
$$ \int_{x}^{x+i\infty} (\cot(\pi z)+i)z^{-s} \, \mathrm{d}z \ll \frac{x^{-\sigma}}{2(n+1)\...
- 1,019
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 "...
8
votes
2
answers
381
views
Real analytic subvariety in complex manifold which is complex outside of its singular set
Let $M$ be a complex manifold, and $Z \subset M$ a closed real analytic subvariety. Suppose that the set of smooth points in $Z$ is complex analytic in $M$. Will it follow that $Z$ is complex analytic?...
- 9,237
5
votes
1
answer
834
views
A statement on complex polynomials
I have a feeling the following is true.
Assume that there are $n$ mutually disjoint closed disks $D_i$ in the complex plane and $n$ complex polynomials $p_i(z)$ of degree $n - 1$, with both types of ...
- 5,215
3
votes
1
answer
261
views
Where does the Weierstrass expansion of $\operatorname{sn}$ come from?
In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$')
$$\operatorname{sn}u=\frac{B}{A}$$
where $...
- 41
2
votes
1
answer
261
views
Uniqueness results for holomorphic functions bounded in a strip with a certain decay to zero
Let $A=\{ z \in \mathbb C : |\mathrm{Im}(z)| < a \}$ for some $a >0$. Further let $F:A \to \mathbb C$ be a holomorphic function on $A$ with the following properties:
$F$ is bounded on $A$
For ...
- 173
1
vote
0
answers
111
views
Residues of analytic operators
Suppose we have analytic operators $P_{z}: C^1[0,1]\to C^1[0,1]$, where $z \in \mathbb{C}$, and the spectrum of $P_{z_0}$ possesses an isolated eigenvalue $1$ (assuming multiplicity is 1 and $I-P_z$ ...
- 11
2
votes
0
answers
70
views
Zero sets of integral power series that converge on disks
Fix a radius $r \leq 1$. I'm interested in any necessary conditions, or any sufficient conditions, for a subset $S$ of $B(0,r)$, the origin-centered open disk of radius $r$, for $S$ to be the set of ...
- 17.6k
1
vote
1
answer
95
views
Common holomorphic forms for two distinct complex structures
Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
- 363
0
votes
1
answer
431
views
Approximation of the product $(\bar{z} - a)^{-1} \cdot (z - b)^{-1}$
$\def\zbar{\smash{\overline z}\vphantom z}$I would like to construct an approximation of the product
\begin{equation}
f(z) = \frac{1}{\zbar-a} \frac{1}{z-b},
\end{equation}
where $a, b \in \mathbb{C}$,...
- 11
1
vote
2
answers
418
views
Global theory of holomorphic functions [closed]
I am trying to develop a theory explaining analytic continuation of a holomorphic function $f(z)$ defined on an open set $D \subset \mathbb{C}$. Recently, I was looking at the last chapter of Lars ...
- 3,699
2
votes
2
answers
260
views
Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,730
4
votes
0
answers
281
views
Order of growth of $\left|\frac{1}{\zeta’(\rho)}\right|$ as $\Im(\rho)\rightarrow\infty$?
Let $\zeta$ denote the Riemann zeta function, and let $\rho\in\mathbb{C}$ be a variable that takes its values among the zeros of the zeta function, so that $\zeta(\rho)=0$, and write $\rho=\sigma+it$. ...
- 1,018
5
votes
1
answer
353
views
Family of functions with prescribed derivatives
Suppose $f: \mathbb C \times (-1,1) \to \mathbb C$ is a smooth function that satisfies $f(0,t)=1$ for all $t\in (-1,1)$. Assume that for any $k\in \mathbb N$, any $z \in \mathbb C$ and any $t \in (-1,...
- 4,115
5
votes
2
answers
289
views
Direct proof of the global submean property for $\log |f|$
Given an entire function $f : \mathbb{C} \to \mathbb{C}$, $\log |f|$ is subharmonic. Globally, this means that for any disk $D_r(c)$ we have the submean property
$$\log |f(c)| \le \frac{1}{\mu(D_r(c))...
- 1,808
1
vote
2
answers
151
views
Location of the negative real roots of certain integer-valued polynomials
The following question on polynomials arose as a potentially helpful intermediate step on a proof of a Theorem that I want to demonstrate. Its statement is quite elementary, and I can think of a ...
- 1,889
6
votes
1
answer
241
views
Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
- 1,730
4
votes
1
answer
203
views
Finding a real-analytic diffeomorphism
Let $U_1\subset \mathbb R^3$ be a simply connected bounded open set with a smooth boundary and let $U_2$ be a neighborhood of $U_1$. Does there exist a real-analytic diffeomorphism $\psi: U_2 \to W_2$ ...
- 4,115
2
votes
2
answers
281
views
Most general reverse Hölder inequality for polynomials
Theorem. Let $m$ be an integer and $P_m$ the vector space of degree $m$ polynomials in one real variable. There is a constant $C$ such that, for all $a<b$ and $p \in P_m$,
$$\|p\|_{L^\infty(a,b)} \...
- 981
3
votes
0
answers
124
views
An open problem of Hardy and Littlewood on $p$-integral means
In Duren's book "Theory of $H^p$ spaces" (MSN) in the comment section after Section 4, it is mentioned that Littlewood and Hardy proved in Some properties of conjugate functions that if $u$ ...
2
votes
1
answer
475
views
Value of $\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}$
I wonder if any of you knows how to find the value
of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$
This function shows up while solving a magnetostatic problem
with complex-valued ...
- 29
3
votes
0
answers
179
views
Topology of level sets for meromorphic function
Let $F$ be a meromorphic function on $\mathbb{C}$.
I consider the "level set" $$E_\varepsilon=\{z:|F(z)|\leq\varepsilon\}.$$ My objective is to find conditions under which $E_\varepsilon$ ...
- 1,352
0
votes
2
answers
285
views
When I know self convolution of the complex function can I recover function itself or its modulus?
I have a function $A : \mathbb{R} \to \mathbb{C}$.
I know there exists unknown function $u: \mathbb{R} \to \mathbb{C}$, such that $A$ is convolution of $u$ and its complex conjugate $A = u * u^*$.
I ...
- 151
10
votes
1
answer
705
views
On entire functions with polynomial Schwarzian derivative
The Schwarzian derivative of an entire holomorphic function $f$ is defined as
$$Sf:=\left(\frac{f^{''}}{f'}\right)'-\frac{1}{2}\left(\frac{f^{''}}{f'}\right)^2.$$
In the following, we only consider ...
- 1,350
58
votes
4
answers
5k
views
Advice for PhD Supervisors
My first PhD student is having his viva tomorrow. Hence, I began contemplating a bit about the whole process of supervising. One thing I realized is that while there seems to be plenty of advice for ...
24
votes
9
answers
9k
views
How to motivate and present epsilon-delta proofs to undergraduates?
This would seem to be a common question, but I am surprised not to see it already asked and answered on MO!
I am teaching an undergraduate course, and I want to teach them to construct basic epsilon-...