All Questions
3,561 questions
13
votes
17
answers
3k
views
Short Course Suggestions For High School Students
I am planning to teach a course for talented high school students at a summer camp and I need suggestions for possible topics. The students usually have different backgrounds but most of them are ...
10
votes
0
answers
303
views
the (non-existent) group of conformal transformations
In physics intros to 2d conformal field theory, people often talk about the "group of conformal transformations". Of course, that's not a group but rather a pseudo-group... that's not what ...
- 43.2k
3
votes
0
answers
128
views
Laplace transform of power of zeta function
Let $s$ is the complex variable. I would like to figure out the region of absolutely convergency of the following integral
$$
e^{\frac{is}{2}}\int\limits_{\frac{1}{2}-i\infty}^{\frac{1}{2}+i\infty}\...
- 51
1
vote
1
answer
159
views
Holomorphic map to Möbius group
$\DeclareMathOperator\PSL{PSL}$Let $U\subset\mathbb C^2$ be an open set, $f:U\to \PSL(2,\mathbb C)$ a holomorphic map. If the image of $f$ is contained in $\operatorname{PSU}(2,\mathbb C)$, I guess ...
- 307
10
votes
1
answer
488
views
Complex plane minus Cantor set admits non-constant bounded harmonic function
Let $K\subset [0,1]$ denote the usual 1/3 Cantor set. I know that $\mathbb{C}\backslash K$ has no non-constant bounded analytic function, since the singularity $K$ can be removed. However, a statement ...
- 103
3
votes
0
answers
185
views
Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
- 11.2k
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
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 60.6k
12
votes
1
answer
523
views
Yet another real-rooted polynomial
In this entry I asked for the real-rootedness of a polynomial, and two very interesting answers were given: one using Malo's theorem and the other a clever rewriting of the expression using Jacobi ...
- 1,889
1
vote
1
answer
984
views
Inverse Laplace Transform using contour integration
So math stack exchange isn't really helping much with this.
So initially, I'm proving the inverse laplace transform using contour integration.
This is a good starting point for my research when I ...
- 19
6
votes
1
answer
379
views
Complex-doubly periodic function in two variables?
I am looking for a function $f:\mathbb C^2 \rightarrow \mathbb C^2$ that satisfies the two equations
$$\partial_{z_2}f_1(z_1,z_2) + \partial_{z_1} f_2(z_1,z_2)=0 \text{ and }$$
$$\partial_{\bar z_1}...
- 536
13
votes
3
answers
1k
views
Teaching polarisation formula
When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\...
- 52.4k
2
votes
1
answer
348
views
What is a generic pencil?
In Voisin book "Hodge theory and Complex Algebraic Geometry 2". There is the following corollary
Corollary 2.10. If $X\subset \mathbb{P}^N$ is a smooth projective complex variety, then a ...
- 519
0
votes
1
answer
578
views
Existence of an inverse to the Schwarz-Christoffel mapping [closed]
As an elementary result in complex analysis, one can use the argument principle to show that the Schwarz-Christoffel transform is injective on the interior of the polygon to which it maps. Could this ...
- 547
4
votes
1
answer
696
views
An asymptotic expansion of a infinite sum
I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula ...
- 193
2
votes
1
answer
124
views
Complex manifolds making Liouville fail
Let us consider $g\colon X\to Y$ holomorphic, where $X$ is a complex manifold and $Y$ is a Stein manifold.
I am searching for all the pairs $(X,Y)$ such that we can find some non constant $g$ with ...
- 779
3
votes
1
answer
440
views
Origin of the Liouville theorem for harmonic functions
What is the paper where the Liouville theorem for harmonic function was first stated? Did it come before or after (or in the same paper) as the Liouville theorem in complex analysis?
user140746
30
votes
1
answer
2k
views
Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?
One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list:
Which of the FTA proofs can, even in ...
- 118k
6
votes
1
answer
240
views
Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$
I have the following function
$$
\int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need ...
- 93
5
votes
1
answer
169
views
Efficients method for finding a zero of a multilinear complex polynomial in an specified region
Let P be a given multilinear polynomial in $\mathbb{C}[z_1,\dots,z_n]$ and $D\subset \mathbb{C}$ be a given disc in the complex plane. Does there exist an efficient method for checking that $P$ has a ...
- 4,484
0
votes
0
answers
112
views
How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
6
votes
2
answers
240
views
Continuity of a differential of a Banach-valued holomorphic map
Originally posted on MSE.
Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
- 5,529
2
votes
0
answers
101
views
Determine whether $\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$ is positive rational (given $x$)
Denote the complete elliptic integral of the first kind by
$$K(x)=\int_0^{\pi /2}\frac{d\varphi}{\sqrt{1-x^2\sin^2\varphi}}$$
and
$$f(x)=\frac{K\left(\sqrt{1-x^2}\right)^2}{K(x)^2}$$
Question:
Given a ...
- 121
5
votes
1
answer
208
views
Seven Bridges of Königsberg for hypergraphs
I am teaching a course involving hypergraphs. I would like to have a physical analogy/motivating problem for hypergraphs similarly to how the Seven Bridges of Königsberg motivate graphs. Can you help ...
- 51
2
votes
1
answer
168
views
Existence of a global analytic solution to a linear first order PDE
Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let
$f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following
first order, linear PDE:
$$f_1\...
- 21
6
votes
1
answer
182
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,115
8
votes
1
answer
1k
views
The inverse of the digamma function
The gamma function is increases on the interval $(x_0, \infty),$ where $x_0$ denotes the unique zero of the digamma function on the positive half line.
The inverse function of gamma function defined ...
- 551
0
votes
0
answers
103
views
Computing $\int^{4b}_0 {e^{-tx}\biggl(\frac{\sqrt{4bx-x^2}}{(2b-2c)^2+4cx}\biggr) dx}$
I am a PhD student working on complex analysis. After integrating over a keyhole contour to obtain the inverse of a particular Laplace transform, I ended up with the following integral:
$$\frac{1}{\pi}...
- 9
1
vote
0
answers
79
views
Resources/books/articles about estimation of $|f(z)|$ [closed]
I am new to complex analysis and I want to explore the following problem: for a complex valued bounded function $f(z)$ in a domain, I would like to know how we estimate the modulus values in a given ...
- 826
2
votes
0
answers
126
views
Sub Banach spaces (Banach algebras) of the disc algebra which are invariant under the differentiation operator
In this question the disc algebra $\mathcal{A}(\mathbb{D})$ is the Banach algebra of all holomorphic functions on the unit open disc $\mathbb{D} \subset \mathbb{C}$ which have a ...
- 366
5
votes
0
answers
139
views
Liouville property in the Bost theorem on foliations
Let $X$ be a smooth algebraic variety over a number field $K$, and let $\mathcal{F}$ be an involutive coherent subsheaf of the tangent bundle. After a pullback to $\mathbb{C}$, $\mathcal{F}_\mathbb{C}$...
- 485
7
votes
1
answer
749
views
Factorization in formal power series versus in convergent power series over the complexes
Let $R=\mathbb C\{x_1,...,x_n\}\subset S=\mathbb C [[x_1,...,x_n]]$ denote the ring of convergent, respectively formal, power series over $\mathbb C$.
Suppose $f\in R$ is irreducible in $R$. Does it ...
- 417
0
votes
1
answer
117
views
Is there any non-normal family $\mathcal{F}$ of meromorphic functions on $|z|<1$ whose each zero has multiplicity $2$ but $\mathcal{F'}$ is normal
It is well known that if a family of meromorphic functions is not normal (a family is said to be normal if each sequence of functions in the family has a subsequence which converges locally uniformly ...
- 165
10
votes
1
answer
419
views
Convex Julia sets
Consider the classical Julia set $J_f$ associated with $f(z)=z^2+c$.
Since $J_c$ is completely invariant,
we know that $f^{-1}(J_f) \subseteq J_f$.
Now, let $H_f$ be the convex hull of $J_f$.
Is it ...
- 15.8k
2
votes
2
answers
269
views
Ratios of polynomials and derivatives under a certain functional
Let $p(x)$ be a polynomial of degree $n>2$, with roots $x_1,x_2,\dots,x_n$ (including multiplicities). Let $m$ be a positive even integer. Define the following mapping
$$V_m(p)=\sum_{1\leq i<j\...
- 43.2k
4
votes
1
answer
159
views
Real part of tail of logarithm
Given a positive integer $n$, consider $f_n = -\min_{|z|=1} \Re \sum_{i>n} \frac{z^i e^{-i/n}}{i}$. What can be said about the growth of $f_n$? How large can it get?
Taking maximum instead of ...
- 14.7k
3
votes
0
answers
196
views
Coefficients for Expansions of $1-\zeta_p$
Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that
$$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$
So ...
- 31
4
votes
1
answer
772
views
Understanding Remmert-Stein extension theorem
I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (...
- 675
1
vote
1
answer
342
views
A mysterious expression from a discriminant
I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of ...
- 41
3
votes
1
answer
387
views
A question about Lelong number
If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the Lelong number of $f$ at $0$ is defined by $$\nu_{f}(0) = \liminf_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$
My ...
- 33
3
votes
2
answers
312
views
Complex Hermite polynomial orthogonality on weighted space
Consider the "probabilist's" Hermite polynomials given by $$H_n(x)=(-1)^ne^{\frac{x^2}{2}}\partial_x^ne^{-\frac{x^2}{2}}.$$
These polynomials trivially extend to functions of $w\in\mathbb{C}$...
- 854
16
votes
2
answers
2k
views
There are two points on the Earth's surface that ... ?
At every moment in time, there are two points on the Earth's surface that have the same $\lbrace x, y, z, ... \rbrace$...?
What is the strongest, most impressive statement one can make here? The ...
- 151k
11
votes
2
answers
1k
views
Algebraic vs analytic normality
Let $X$ be a complex algebraic variety. We can ask if $X$ is normal as an algebraic variety, but also, if its analytification is normal as a complex analytic space. Is there a relationship between the ...
- 113
10
votes
1
answer
1k
views
Is there a holomorphic function on open unit disc with this property?
Let $D=\{z\in \mathbb{C}\mid |z|<1\}$. Is there a holomorphic function $f:D\to \mathbb{C}$ such that for every $n\in \mathbb{N} \cup \{0\},\;f^{(n)}$ has a continuous extension to $\bar D$ but $f$ ...
- 366
1
vote
0
answers
80
views
A constant for univalent functions
Assume that $f:D\to C$ is univalent holomorphic from the unit disk into the complex plane so that $f(0)=f'(0)-1=0$. What is the precise constant $r$ so that $f(D)$ does not contain the disk $D(0,r)$. ...
- 333
0
votes
1
answer
122
views
Existence of an eigenpair for d-bar operator in the unit disck
Let $\overline{\partial}=\frac{1}{2}(\partial_{x}+\textrm{i} \,\partial_y)$ and let $D$ be the unit disc in the complex plane. For each $\lambda \in \mathbb C$, consider the problem:
$$ \overline{\...
- 4,115
12
votes
1
answer
988
views
Zeros of an infinite series
Let $\sum_{j=1}^{\infty}a_{j}$ be a convergent series of positive numbers and $\{z_{j}\}_{j=1}^\infty$ a closed discrete subset of the open unit disc $\mathbb{D}$. Then $h(z):=\sum_{j=1}^{\infty}\frac{...
- 401
4
votes
1
answer
538
views
Rouché's Theorem in complex analysis on the relation of the number of zeros and poles of meromorphic functions in a region [closed]
This question is from my son referenced in my earlier question, Need advice or assistance for son who is in prison. His interest is scattering theory . He asked me to post this question:
Hello and ...
- 545
2
votes
0
answers
141
views
Are Poisson integrals uniquely determined by their radial limits?
Let $\mu$ be a complex Borel measure on the unit circle, and suppose its Poisson integral $u$ satisfies $\lim_{r\to 1-}u(re^{i\theta})=0$ for every $\theta$. Does it follow that $\mu=0$?
This is of ...
- 199
1
vote
0
answers
49
views
Uniform boundedness and boundedness of Maclaurin coefficients [closed]
Suppose $f(z)$ is a bounded in some region $|z|\leq R>1 $ and analytic function and consider the Maclaurin series for $f$:
$$ f(z)= \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}z^n.$$ My question is:
...
- 826