All Questions
Tagged with cv.complex-variables real-analysis
267 questions
7
votes
1
answer
204
views
Are $\log(\sigma(A(z))$ subharmonic functions?
Let $A$ be a matrix-valued entire function. It is then well-known that $\log \Vert A(z)\Vert$ is subharmonic. In particular, the operator norm is just the largest singular value of $A$.
Is it ...
- 124
1
vote
0
answers
112
views
A problem related to analytic function
Let, $z,w\in \mathbb{C}$. Let, $f(z)$ be an analytic function in $|z|<1$. Define, $f(z)= g(w)$ where $g(w)$ is analytic function in $\Re(w)>1/2$ and $w=\frac{1}{1+z^2}$ .
Question Prove that $$\...
user245973
2
votes
0
answers
65
views
On a question relating integral equation:
I don't know if the following question qualifies as research level. If it isn't, sorry.
Set the following terminology:
$ \alpha_1 =\alpha_1(t,x)=t(\tan^{-1}(x)+c)$
$\alpha_2=\alpha_2(s,x)=s(\tan^{-1}(...
- 41
1
vote
2
answers
106
views
'Partial boundedness' of continuously parametrised power series
Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space.
Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by
$...
- 463
6
votes
1
answer
181
views
Mittag-Leffler function
Let the Mittaq-Leffler function be defined by the expression
$$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$
Now let $n\in \mathbb ...
- 4,115
7
votes
1
answer
524
views
continued fraction for logarithmic integral
Does the logarithmic integral function $\operatorname{li}(x)$ have the continued fraction expansion
$$\operatorname{li}(x) = \cfrac{x}{\log x -1 -{}} \ \cfrac{1}{\log x -3 -{}} \ \cfrac{4}{\log x -...
- 4,985
4
votes
1
answer
389
views
Existence of periodic solution to ODE
We shall consider the matrix-valued differential operator
$$(L u)(x) :=u'(x) - \begin{pmatrix} 0 & \sin(2\pi x-\frac{\pi}{6})\\ - 2\sin(2\pi x+\frac{\pi}{6}) & 0 \end{pmatrix} u(x).$$
This is ...
- 192
12
votes
1
answer
858
views
Is this function concave?
Let
$$h(u):=u^3 \left|\int_u^\infty \frac{e^{-i t}}{t^3} \, dt\right|$$
for $u>0$. Is the function $h$ concave on $(0,\infty)$?
(For context, see Proposition 4.4.4 and formula (4.4.21) in this ...
- 128k
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 ...
- 14.6k
4
votes
0
answers
176
views
Finding roots of equation with gamma functions
Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
- 199
2
votes
1
answer
162
views
Max-root inequality for convex combination of real-stable monic polynomials (Kadison-Singer Problem)
In the paper "The Kadison-Singer Problem" by Marcin Bownik (https://arxiv.org/pdf/1702.04578.pdf), the following Lemma (3.8) is proven:
Lemma:
Let $p, q\in \mathbb{R}[x]$ be stable monic ...
- 203
3
votes
1
answer
179
views
Analytic or holomorphic extension of the ellipse perimeter function
Let ${\mathbb{R}^2}^+=\{(x,y)\in \mathbb{R}^2\mid x>0, y>0\}$.
Let $P:{\mathbb{R}^2}^+\to \mathbb{R}$ be the function with $P(a,b)=$ $\text{The perimeter of ellipse}\;\; \frac{x^2}{a^2}+\frac{y^...
- 356
5
votes
2
answers
338
views
Approximation of analytic function by a fixed number of monomials
This question seems simple but I can't manage to disprove it. Let $N\in \mathbb{N}$. We know that by its analyticity that this precise linear combination of monomials
$
\sum_{n=0}^K \frac1{n!} x^n
$
...
3
votes
0
answers
1k
views
On new (purely analytic) perspective towards theory of prime numbers
[I'm going to ask this question very carefully as a question similar to this received a critical response on this platform.
I myself am very skeptical about this but I want to know, from the experts' ...
- 375
7
votes
1
answer
396
views
Existence of complex function?
Motivated by a similar question Complex-doubly periodic function in two variables?, I would like to ask if there exists a non-zero function $(z_1,z_2) \mapsto f(z_1,z_2)$, where $z_1,z_2 \in \mathbb C$...
- 124
1
vote
0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
7
votes
6
answers
959
views
Intersection of two Jordan curves lying in the rectangle
Given a rectangle $ABCD$.
Let a Jordan curve $L_1$ joins the vertexes $A$ and $C$
and all points of $L_1$ belong the rectangle.
Let a Jordan curve $L_2$ joins the vertexes $B$ and $D$
and all points ...
- 97
0
votes
1
answer
379
views
On some property of the zeros of $\zeta(s)$ in the complex plane
This property is rather elementary, and not at all specific to $\zeta$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well ...
- 3,098
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\...
- 43.1k
1
vote
0
answers
161
views
Justify $\int_0^\infty e^{-ax^2}\ \mathrm{d}x$ for complex $a$ and zero real part [closed]
(Reposted from math stack exchange)
I have searched and failed to find a rigorous proof showing that
$$\int_{0}^\infty e^{-ax^2}\ \mathrm{d}x = \frac{\sqrt{\pi}}{2\sqrt{a}}$$
is true for $\Re(a)=0$ ...
user161698
2
votes
0
answers
162
views
Hilbert transform on weighted Sobolev spaces
Let $\mathscr H\,f$ denote the Hilbert transform of a function $f \in L^2(\mathbb R)$. We know that $\mathscr H$ is an isometry on $L^2(\mathbb R)$, but I want to know to what is the mapping ...
- 4,115
-1
votes
2
answers
129
views
Is it possible for all of the smooth/continuous curves in $R^3$ to form a Hilbert space? [closed]
Under which condition can it form a Hilbert space? Or what space can it form?
You can write down certain condition to make it to be a Hilbert space, e.g., Let $$p(t)=[x(t),y(t),z(t)]^T\in \text{R}^3$$ ...
- 11
4
votes
1
answer
690
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
3
votes
1
answer
155
views
Inequality for generalized Laguerre polynomials
Please. Does anybody know a proof of this inequality
$$\Big|\frac{n!\Gamma(\alpha+1)}{\Gamma(n+\alpha+1)} L^{\alpha}_n(x)\Big|\leq e^{\frac{x}{2}}$$ where $\alpha\geq0$ and $x\geq0$ and
$L^{\alpha}_n$ ...
- 81
0
votes
1
answer
768
views
Square root of a continuous function
Is it square root of a real $\alpha-$Holder continuous function $f$ defined on $[0,1]$ a $\alpha/2$ Holder continuous, provided $\sqrt{f(x)}$ it exists and is continuous, i.e. whether $|f(x)-f(y)|\le ...
- 49
9
votes
1
answer
603
views
Interpolation theory and $C^k$-spaces
Consider the Banach spaces $C^k(M)$ ($k=0,1,2,\dots$), consisting of $k$times continuously differentiable functions $f:M\rightarrow \mathbb{C}$ on a closed manifold $M$ (or just the torus if that ...
- 779
11
votes
1
answer
411
views
A density question for the Hilbert transform
Let $\mathscr Hf$ denote the Hilbert transform of a function $f$ defined on the real-line $\mathbb R$. Are the set of functions
$$ \{(f+\mathscr Hf)_{|_{(0,1)}}\,:\, f \in C^{\infty}(\mathbb R)\quad \...
- 4,115
5
votes
1
answer
280
views
First order PDE in complex variables?
Consider the equation
$$f'(x)+ g(x)f(x)=0$$
This equation is an ODE and has a solution $$ f(x)=C e^{ \int_1^x g(x) \ dx}.$$
Similarly, we can look at complex variables and consider the equation and ...
- 536
1
vote
0
answers
84
views
A Riemann Hilbert problem on the unit square
Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on ...
- 4,115
29
votes
1
answer
1k
views
About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$
I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far.
I ...
- 483
4
votes
1
answer
1k
views
The cotangent sum $\sum_{k=0}^{n-1}(-1)^k\cot\Big(\frac{\pi}{4n}(2k+1)\Big)=n$
On the Wolfram Research Reference page for the cotangent function (https://functions.wolfram.com/ElementaryFunctions/Cot/23/01/), I saw the following partial sum formula
$$\sum_{k=0}^{n-1}(-1)^k\cot\...
- 195
1
vote
1
answer
287
views
Regular singular point of non-linear ODE: $\dot{x}(t) + t^{-1}Ax(t) = Q(x(t))$
Consider a system of ordinary differential equations of the form
$$
\dot{x}(t) + \frac{1}{t}Ax(t) = Q(x(t))
$$
where $x(t) \in \mathbb{C}^n$, $A \in \mathrm{Mat}_{n\times n}(\mathbb{C})$ is a constant ...
- 1,383
1
vote
0
answers
107
views
Comparison of two Fourier transforms
I am looking for $\delta>0$, such that
$$
\delta \int_{-\infty}^{\infty} \exp(its)
{ \Gamma\{2(it+1)/3\}\over \Gamma\{(it+1)/2\} }dt \le \\
\int_{-\infty}^{\infty} \exp(its)
{ \Gamma (it+1)\over \...
- 93
6
votes
2
answers
1k
views
Analyzing the decay rate of Taylor series coefficients when high-order derivatives are intractable
This could be a soft question. I am trying to show that the $n$-th Taylor series coefficient of a function is $O(n^{-5/2})$. However, because the function is a function composition of another function ...
- 212
6
votes
1
answer
239
views
Positivity of $ \int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it} { \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt$
I have the following function
$$
\int_{-\infty}^{\infty} \left\{{2^{1/\beta-1/2} \over v}\right\}^{it}
{ \Gamma\{(it+1)/\beta\}\over \Gamma\{(it+1)/2\} }dt
$$
where $1<\beta<2$, $v>0$. Need ...
- 93
0
votes
1
answer
322
views
Injectivity of analytic functions
Suppose $f : \mathbb{R} \rightarrow \mathbb{R}^n$ is a real analytic function on $(a, \infty)$. I have two questions:
Suppose $||f(x)|| \rightarrow \infty$ as $x \rightarrow \infty$. I know without ...
- 431
7
votes
1
answer
298
views
Functions $f \geq 0$ on $\mathbb{R}$ which are of the form $f = |g|^2$ for some entire function $g$
I think the answer to this question must be well known. Is it possible to characterize those functions $f \colon \mathbb{R} \to \mathbb{R}_+$ which are of the form $f(x) = |g(x)|^2, x \in \mathbb{R},$ ...
- 2,101
1
vote
1
answer
182
views
Analyze a function defined in terms of an integral
Here is a question that really has puzzled me for quite a while. I happened to see this function defined in terms of an integral
$$f(x):=\int_0^{\pi/2}\frac{2e^{x+e^x\cos y}}{1+\left(e^{e^x\cos y}\...
- 1,350
0
votes
0
answers
211
views
Understanding Krantz's proof of Hefer's lemma in $\mathbb{C}^2$
Note: I initially phrased the question in a different way, and it did not receive much attention. In the hope to make it more interesting, I have included a (long) introduction to contextualize and ...
- 11
4
votes
1
answer
119
views
minimizing weighted length of closed curves
Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...
- 1,350
1
vote
0
answers
203
views
Construction of weight function to satisfy condition on given 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 are going to consider
The following two conditions should meet for $\...
- 375
3
votes
2
answers
497
views
if $f\circ f=g$ has no solution does this imply $f\circ f=g+g^{-1}$ also has no solution with $g^{-1}$ being a compositional inverse of $g$?
This question is related to solving $f(f(x))=g(x)$.
Assume that $g$ is a bijective function $g:\mathbb{R}\to \mathbb{R}$. If there is no continuous function $f : \mathbb R \to \mathbb R\,$ for which $...
user147204
2
votes
1
answer
103
views
A density question
Suppose $\Omega= (0,1)\times(0,1)\subset \mathbb R^2$. Assume that $f, g \in C^{\infty}(\Omega)$ and that
$$ \int_\Omega \left(f(x_1,x_2)- \frac{m}{(n+1)}g(x_1,x_2)\right) x_1^n \,x_2^m \,dx_1\,dx_2 = ...
- 4,115
3
votes
1
answer
237
views
Isoperimetric inequality for analytic functions on an annulus
Let $f$ be an anylytic function on the unid disk $|z|<1$. It is well known that
$$\left (\int_0^{2\pi}f(e^{i\theta})d \theta \right)^2 \geq 4\pi \iint_{|z|<1} |f(r e^{i\theta})|^2r dr d \theta.$...
- 434
1
vote
0
answers
196
views
Asymptotic of a functional as $x\rightarrow \infty$
Consider the following functional :
$$
I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1},
$$
where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
- 375
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
0
votes
0
answers
77
views
Construction of (general class of) function(s), which sieves out primes, w.r.t. given conditions:
Consider the function $F(x)$ defined in following manner:
$F(n)$ is finite (likely $F(x)\in[0,1]$) if $n$ is prime and zero otherwise:
It has to satisfy following conditions:
(1) $F(x)$ is ...
- 375
0
votes
1
answer
335
views
On partial sum estimate on the series $S(p,q;s)=\sum_1^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$ and other Generalizations
Consider the following sum :
$$S(p,q;s)=\sum_{n=1}^q\frac{\sin^2(\frac{p\Gamma(n)}{n})}{n^s}$$
Here , $p$ is a variable w.r.t which we are going to analyse the sum.
$s$ is another parameter with ...
- 375
4
votes
1
answer
247
views
Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?
Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
- 6,976
21
votes
2
answers
2k
views
Real rootedness of a polynomial
Let's consider $m$ and $n$ arbitrary positive integers, with $m\leq n$, and the polynomial given by:
$$ P_{m,n}(t) := \sum_{j=0}^m \binom{m}{j}\binom{n}{j} t^j$$
I've found with Sage that for every $...
- 1,889