All Questions
Tagged with cv.complex-variables real-analysis
267 questions
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 (...
- 356
7
votes
2
answers
588
views
$\sum_{k =1, k \neq j}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)\csc^2\left(\pi \frac{j-k}{N} \right)=?$
It is well-known that one can evaluate the sum
$$\sum_{k =1}^{N-1} \csc^2\left(\pi \frac{k}{N} \right)=\frac{N^2-1}{3}.$$
The answer to this problem can be found here
click here.
I am now ...
- 192
-1
votes
1
answer
119
views
Existence of a function with slow growth on derivatives
Does there exist a smooth compactly supported function $$f \in C^{\infty}_c((0,1))$$
such that
$$ \|D^k f\|_{L^{2}(0,1)} \leq \left\lfloor{\alpha\,k}\right \rfloor! \quad \forall\, k\in \mathbb N$$
...
- 4,115
6
votes
1
answer
525
views
Holomorphic extensions of a non-vanishing real-analytic function
Let f(z) be a holomorphic function defined on an open neighborhood $R$
of the interval $I=[0,1]\subset \mathbb{R}$. Assume $f$
does not vanish on $I$. Then $g(x) = |f(x)|$ is a real-analytic
function ...
- 20.1k
4
votes
3
answers
598
views
Meaning of divergent integrals
In quantum field theory, one usually encounters divergent integral when one calculates loop functions, such as the integral $\int_{0}^{\infty}dkk^{3}\frac{1}{(k^{2}-m^{2})^{2}}$ which is divergent. ...
1
vote
2
answers
243
views
Thinness and polarity
Let $D$ be a bounded open set in $\mathbb{R}^{n}$ with $n\geq2$ and $E$ a subset of the boundary $\partial D$ of $D$. $D$ is said to be thin at a point $y\in D$ if there is a superharmonic function $u$...
- 411
1
vote
0
answers
90
views
Is there any characterization of polynomials in terms of asymptotic properties of Taylor coefficients? [closed]
My formal question is
Let $f(z):=\sum_{n=0}^{\infty} c_n z^n$ be a formal power series.
Is there any characterization of polynomials in terms of the asymptotic properties the sequence $(c_n)$?
For ...
- 51
1
vote
1
answer
125
views
Asymptotics of the general second order affine recursion
What is the general method for finding the aymptotics of large $n$ of the sequence $(a_n)_{n=0}^\infty$ defined by the recursion
$$a_{n} = (\alpha_1n+\alpha_2) a_{n-1} + (\alpha_3n+\alpha_4) a_{n-2}+\...
- 2,239
0
votes
1
answer
362
views
A question about Riesz decomposition theorem
Let $D$ be a domain of $\mathbb{R}^{m}$ and let
$K(x)= \log|x|$ if $m=2$, and $K(x)=|x|^{2-m}$ if $m>2$. According to Riesz decomposition theorem (Hayman and Kennedy, "subharmonic functions", vol. ...
- 411
2
votes
0
answers
69
views
Extension of a $\delta$-subharmonic function that is subharmonic on a reduced domain
Suppose $B$ is a ball in $\mathbb{R}^{m}$ and $u$ and $s$ are subharmonic on $B$. Suppose there is a closed subset $F$ of the closure of $B$ with no interior such that $v=u-s$ is subharmonic on $B\...
- 411
1
vote
0
answers
39
views
The parameter regularity of power sum
Let $f(x,s)=\sum_{n=0}^\infty a_n(s)x^n$ where $|a_n(s)|\le1$ is a bounded function theory. Suppose for every $|x|<1$, $f(x,s)$ is Holder-$\alpha$ for $s$-variable, i.e. $|f(x,s_1)-f(x,s_2)|\le C|...
- 938
4
votes
0
answers
117
views
Korovkin subset of $C(\mathbb{T})$
Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\...
- 521
1
vote
0
answers
152
views
Is the normalized derivative of a holomorphic function Sobolev?
This question is a cross-post from MSE. it is also a special case of this question.
Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$, and let $f:B \to \mathbb{C}$ be holomorphic on the interior $B^o$, and ...
- 6,741
2
votes
1
answer
110
views
Proving that $\lim_{j \to i} Z_{ij} = [\ln(\frac{\Delta s_i}{2})-1]\Delta s_i$
If I have the following integral equation $$\phi(\vec{x})=\frac{1}{\pi}\int [\phi\frac{\partial (\ln r)}{\partial n} -\ln(r) \frac{\partial \phi}{\partial n}] ds$$
An approximate solution of $\phi$ ...
- 111
1
vote
0
answers
87
views
An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,115
1
vote
3
answers
207
views
Existence of solution to linear fractional equation
We consider the equation
$$ \sum_{j=1}^n \frac{\lambda_j}{x-x_j} =i$$
where $\lambda_j>0$ and $x_j$ are real distinct numbers.
I want to show that if $\lambda_k$ is small compared to the ...
user121558
2
votes
1
answer
140
views
Must $q$ be analytic?
I have a continuous function $q:\mathbb{R}^+ \to \mathbb{R}^+$. An interesting property of this function is that
$$F(s) = \frac{e^{-q(s)}q(s+1)}{1-e^{s-q(s)}}$$
which also takes $\mathbb{R}^+ \to \...
- 388
3
votes
0
answers
235
views
Chern number of projection-Topological magic in physics
I enclosed a computation from a well-known paper in the field of mathematical physics where the Chern number of the first Landau level is computed (the result claimed is $-1$) and the full paper can ...
- 31
0
votes
1
answer
375
views
Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213
2
votes
0
answers
304
views
Prove conjugation law for Exp(z) using only Exp(x + y) = Exp(x)Exp(y) [closed]
Starting with just the property $E(x + y) = E(x)E(y)$, one can prove quite a lot of the main properties of the exponential function on real numbers. For example, $E(0) = 1$, and $E'(X) = E(X)$, and $E(...
- 21
1
vote
1
answer
150
views
Riesz measure of a smooth subharmonic function on a ball
Let $B(x_{0},r)$ be the ball of center $ x_{0} $ and radius $r>0$ in $ \mathbb{R}^{k} $ ($ k\geq2 $), and $u$ a subharmonic function on an open neighborhood of the closure of $B(x_{0},r)$. Let $\mu$...
- 411
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
0
votes
0
answers
124
views
Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]
Recall a very famous theorem due to Hartogs for complex analytic functions of several variables.
Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
- 671
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
1
vote
1
answer
136
views
Conditions to obtain a real logarithm of a unitary unimodular complex matrix?
The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...
- 113
3
votes
1
answer
169
views
On the values of an entire function
Let $0<q<1$ and consider the entire function $f(z)=\displaystyle \sum_{k=0}^\infty q^{k^2}z^k$. For $a>1,$ denote $m_j=f(a^j),\; j=0,1,2,\dots.$
Question: Does there exist an entire function ...
- 783
10
votes
0
answers
844
views
Witt's proof of Gelfand-Mazur / Ostrowski's Theorem
Previously asked on Math Stackexchange without answers.
Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
- 2,444
3
votes
0
answers
171
views
Nekrasov Partition function and the leading term of Prepotential
I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...
- 425
4
votes
0
answers
210
views
Inclusion of Hardy spaces
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality.
It is also known that for $p>1$ it holds that $L^p(\mathbb R)...
7
votes
3
answers
602
views
Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$
During research involving the Born–Jordan quantization I came across the expression
$$
\frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1
$$
for $k\in\mathbb N_0$. It is not too ...
1
vote
0
answers
100
views
Higher order derivative of negative power of cosine function
This is a question I encountered in my own research on Generalized
Hyperbolic Secant (GHS) distributions. It is known that the Laplace transform of the
basis measure for this family is
$$L\left( \...
- 984
0
votes
1
answer
243
views
a counter-example of a holomorphic extension
Suppose $f$ a function holomorphic on the unit bidisk $\mathbb{D}\times \mathbb{D}$, such that $f$ is $\mathcal{C}^{\infty}$ on $]-1,1[\times\partial\mathbb{D}$, and has holomorphic extension on $\...
2
votes
2
answers
207
views
A problem on the maximal modulus
Let $f$ be a transcendental entire function, we know that
$\log M(r, f)$, with $M(r,f)=\max_{|z|=r}|f(z)|$, is a convex function with respect to $\log r$ and
$\lim\limits_{r\rightarrow\infty}\frac{\...
- 1,706
2
votes
0
answers
119
views
question about sequences and series (complex analysis may be only elementary real analysis)
I would like to have your help about the proof of the following statement:
If the sequences of complex numbers $\{F_N\}_{N \in \mathbb{N}},\{G_N\}_{N \in \mathbb{N}}$ have the following properties:
(...
- 51
7
votes
1
answer
304
views
Argument principle for matrices
Let $f,g$ be entire functions, then the argument principle teaches us that
$$\frac{1}{2\pi i}\int_{\mathbb{C}} g(z) \frac{f'(z)}{f(z)} dz$$
is equal to $g$ evaluated at the zeros of $f.$
Now, let ...
- 167
18
votes
2
answers
1k
views
Characterisation of bell-shaped functions
This is an open problem that I learned from Thomas Simon. I will completely understand if the question is judged as non-research level (and it is indeed not related to my research), but I believe a ...
- 17.2k
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
0
votes
1
answer
185
views
Meromorphic solutions to Legendre's equation
I just saw the following question that was asked yesterday on math overflow on meromorphic solutions to ODEs
Although, I understand the answers and comments to the questions, I did not understand how ...
- 501
2
votes
2
answers
258
views
Meromorphic extension of solutions to ODEs
I encountered the following question in my studies:
Let us assume we have a real anlaytic solution to an ODE on $\mathbb{R}$ of Schr\"odinger type
$-\psi''(x)+V(x)\psi(x)=\lambda \psi(x)$
but we ...
- 167
2
votes
0
answers
60
views
Finding a function in contour integration involving Riemann mapping
Let $T$ be a rectifiable Jordan curve in $\mathbb{C},$ $G$ be the interior of $T,$ and $\Phi$ be a conformal map of the unit disk $\mathbb{D}$ onto $G.$ Let $\mathcal{P}_{n}$ be the space of algebraic ...
4
votes
0
answers
261
views
Is the following integral positive or not?
Let $n$ be a given even positive integer. We have the following integral
\begin{eqnarray}
&&\int_0^1\cdots\int_0^1\prod\limits_{i=1}^n\prod\limits_{j=1}^n(x_i-y_j)dx_1\cdots dx_ndy_1\cdots ...
- 1,997
4
votes
3
answers
667
views
Regularity for the roots of (characteristic) polynomials with given multiplicity
A classical result states that roots of a polynomial are continuous functions of its coefficients.
This is, for exemple, a direct consequence of Rouché's theorem.
Using the implicit function ...
- 2,135
1
vote
1
answer
179
views
One trig "survives" a binomial summation: why?
I've seen many trigonometric identities but here is one that I encountered for which I did not find a reference.
In case you wonder where this came from, I was investigating certain $q$-series in ...
- 43.1k
0
votes
1
answer
662
views
A polynomial and its reciprocal expansion [closed]
Suppose $f(x)=\prod_{k=1}^n(x-a_k)$ where all $a_k>0$.
Expand the function $\frac1f$ at $\infty$ so that
$$\frac1{f(x)}=\frac{b_n}{x^n}+\frac{b_{n+1}}{x^{n+1}}+\cdots.$$
Does it follow that each $...
- 309
5
votes
1
answer
694
views
Number of zeros of a real analytic function
Let $f(x,y,t):[-1,1]^3\to \mathbb{R}$ be a real-analytic function. Assume that for any fixed $x,y$, $f(x,y;t)$ is not a constant function $[-1,1]\to \mathbb{R}$. Since the zeros of a non-constant real-...
- 187
7
votes
1
answer
1k
views
The sinc function strikes again [duplicate]
Recall $\text{sinc}(x)=\frac{\sin x}x$. It's a familiar exercise that $\int_0^{\infty}\text{sinc}(x)\,dx=\frac{\pi}2$.
But, at present, I wish to ask about the following claim on a "sinc-ing" ...
- 43.1k
2
votes
0
answers
136
views
To find a positive function with compact spectrum
Let
$e_1=(0,1)^T$,
$$
S=\left\{x\in \mathbb{R}^2\Big| \frac{|\langle x, e_1\rangle|}{|x|}>\delta>0\right\},
$$
is a cone in $\mathbb{R}^2$.
I want to find a non-trivial smooth function ...
- 29
7
votes
1
answer
450
views
Convergence of Lagrange interpolation polynomials to entire functions
Consider an entire function $\ f:\mathbb C\rightarrow\mathbb C.\ $ Let $\ (a_n\in\mathbb C:n=0\ 1\ \ldots)\ $ be an infinite sequence, where $\ a_k\ne a_n\ $ whenever $\ k\ne n.\ $ Let $\ L_n\ $ be ...
- 7,500
3
votes
1
answer
205
views
Understand the properties of this function
We define a function
$f(t):=\sum_{n=0}^{\infty}e^{-nt}= \frac{1}{1-e^{-t}}= \frac{e^{\frac{t}{2}}}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}=\frac{2e^{\frac{t}{2}}}{\sinh\left(\frac{t}{2} \right)}$
observe ...
- 31
1
vote
0
answers
252
views
Contour integration and application of residue theorem [closed]
I found the following contour integration done in an article, but I do not fully comprehend what has actually been calculated here? Contour integration
The argument of the function $s$ is supposed to ...
- 11