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0 votes
1 answer
124 views

Holomorphic functions of certain blow up at origin

Suppose that $D=\{z\in \mathbb C\,:\, |z|\leq 1\}$ and let $f$ be holomorphic on $D\setminus\{0\}$ such that $|f(z)|\leq e^{\frac{1}{|z|}}$ for all $0<|z|\leq 1$ and assume additionally that $\lim\...
1 vote
0 answers
144 views

integral over the unit sphere of $\Bbb C^n$

Please, is there a way to calculate this integral $$\int_{S_{2n-1}} \frac{e^{a \langle z, \zeta \rangle}}{|z - \zeta|^{\beta}} \, d\sigma(\zeta)$$ where $ z $ is a fixed point in the complex unit ball ...
-3 votes
1 answer
193 views

Bounding a number-theoretic integral

Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH. My attempt here is ...
0 votes
0 answers
57 views

Double-periodic functions with (possible) poles

Consider the set of double-periodic function $f:\mathbb C/(\mathbb Z+i \mathbb Z) \setminus \{z_0\} \to \mathbb C$, where $z_0$ is a fixed point inside $\mathbb C/(\mathbb Z+i \mathbb Z),$ that have a ...
0 votes
1 answer
127 views

Characterizing the integral as a function of $n$

Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
0 votes
0 answers
36 views

Derivate involving Bessel function of second type

Let. $$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$ Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
6 votes
0 answers
632 views

Generating functions in countable commutative monoids

Let $f: \mathbb{N}_0 \rightarrow \mathbb{C}$ be a function. The power series of $f$ can be viewed as the function $\mathscr{P}_f : q \mapsto \sum_{n \in \mathbb{N}_0}^{} f(n)q^n$ where $q \in \mathbb{...
4 votes
1 answer
144 views

Asymptotic decay rate of an oscillator integral

Question: I want to evaluate the decay estimate of the integral $I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $ for ...
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 ...
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 ...
2 votes
0 answers
259 views

Least number of circles required to cover a continuous function on $[a,b]$

I asked this question on MSE here. Given a continuous function $f :[a,b]\to\mathbb{R}$, what is the least number of closed circles with fixed radius $r$ required to cover the graph of $f$? It is ...
2 votes
4 answers
739 views

Is the hypergeometric function ${}_1F_2(1;a,a+\frac12;-x^2)$ an elementary function? How about its positivity, monotonicity, and convexity in $x$?

Is the generalized hypergeometric function ${}_1F_2\bigl(1;a,a+\frac12;-x^2\bigr)$ for $a>-1$ and $x>0$ an elementary function? How about the positivity, monotonicity, and convexity of the ...
1 vote
1 answer
175 views

Analyzable functions and accelero-summation

Is there a complete and rigorous, yet concise, definition of what an analyzable function is, along with the related notion of accelero-summation, both in the sense of Écalle? All of the definitions I ...
7 votes
1 answer
367 views

Duality of $H^1$ and BMO

While proving that the dual of $H^1$ is $BMO$ in Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, page 143, Stein says that we have $\left\Vert g \right\Vert_{H^1} \...
3 votes
1 answer
458 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
1 vote
1 answer
121 views

An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
0 votes
1 answer
102 views

On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
4 votes
2 answers
360 views

Functions with asymmetrically decreasing Fourier transform?

$\def\ii{{\rm i}}\def\bbR{\mathbb R}\def\bbC{\mathbb C}\def\bbNo{\mathbb N_0}\def\Fou{\mathscr F}$Specifically, I would like to have a compactly supported continuous function $f=u+\ii\,v:\bbR\to\bbC$ ...
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 ...
2 votes
1 answer
112 views

On compactly supported functions with prescribed sparse coordinates

Let $\{\phi_n\}_{n=1}^{\infty}$ be an orthonormal basis for $L^2((0,1))$ consisting of Dirichlet eigenfunctions for the operator $-\partial^2_x + q(x)$ where $q \in C^{\infty}_c((0,1))$ is fixed. ...
1 vote
1 answer
116 views

Examining the Hilbert transform of functions over the positive real line

$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
4 votes
2 answers
364 views

Nontrivial invariant transformations for heat equations

It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by $$ v(r,\theta) = u(\frac{1}{r},\theta)$$ is also harmonic for $r>0$. Note that the Kelvin ...
2 votes
1 answer
115 views

Bound for the $n$-th derivative of a proper rational function with no poles on the right half-plane

Suppose that $f$ and $g$ are polynomials with nonnegative coefficients, the degree of $g$ is greater than the degree of $f$, $g + f$ have no zeros on the right half plane $\mathbb{C}_+ = \{z \in \...
5 votes
1 answer
167 views

Upper bound for the $n$-th derivative of a rational function $\frac{f}{f+g}$

Let $f$ and $g$ be real polynomials with nonnegative coefficients. Let $$ h = \frac{f}{f+g}. $$ I want to prove that the $n$-th derivative of $h$ satisfies: There exists $C > 0$ such that $$ |h^{(...
0 votes
1 answer
414 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
-3 votes
2 answers
315 views

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
2 votes
2 answers
268 views

If $\inf\{b\in\mathbb{R}\mid\sum_{n=1}^{\infty}e^{-ax_n-by_n}<+\infty\}=1-a$ for all $a\in [0,1]$, does this equality hold for all $a\in\mathbb{R}$?

Let $\left\{x_n\right\}_{n=1}^{+\infty},\left\{y_n\right\}_{n=1}^{+\infty}\subset [0,+\infty)$ be two sequences of non-negative real numbers. Suppose there exist $\lambda\ge 1, c\ge 0$ such that $\...
-1 votes
2 answers
87 views

Limits of integral series

Suppose we have the series of functions: \begin{equation} F(x)=\sum_{n=1}^{\infty} f_n(x) \end{equation} where convergence is uniform. Additionally, consider the partial functions of the series: \...
11 votes
3 answers
1k views

"Simple" integral equation

Let $H(z)$ be a continuous solution of the problem $$ H(z)=\frac1{1-z}\int_z^1 \frac{2\zeta}{1+\zeta} H(\zeta^2)\,d\zeta,\ \ \ z\in[0,1);\ \ \ H(1)=1. $$ Is it true that $H(0)=1-\ln2$? The question ...
74 votes
15 answers
18k views

$f(f(x))=\exp(x)-1$ and other functions "just in the middle" between linear and exponential

The question is about the function $f(x)$ so that $f(f(x))=\exp (x)-1$. The question is open ended and it was discussed quite recently in the comment thread in Aaronson's blog here http://...
0 votes
1 answer
116 views

Sufficient conditions for ensuring that a monic polynomial in $\mathbf{Z}[x]$ possesses exclusively simple roots

I am seeking sufficient conditions to ensure that a monic polynomial, denoted as $f$ in $\mathbf{Z}[x]$, possesses exclusively simple roots. Based on an old paper (this reference), it has been ...
1 vote
1 answer
90 views

The number of roots of pseudo-exponential polynomials

Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
3 votes
0 answers
244 views

Norm on the space of real analytic functions

The space $C^{\omega}(\Omega)$ of real-valued real analytic functions on the open bounded set $\Omega\subset \mathbb R^n$ does not have any obvious or natural metric which would make it a Fréchet ...
1 vote
0 answers
113 views

Computing a limit for the Weierstrass function

Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
1 vote
0 answers
143 views

Analyticity of a function in two complex variables

Let $f$ be a function defined on $\mathbb{C}^2$ given by $$ f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
4 votes
1 answer
305 views

Holomorphic extension of the Fourier transform of a measure

If an entire holomorphic function $f(z)$ is given by the analytic continuation of $f(x)=\int_\mathbb{R}e^{-ix\xi}\,d\mu(\xi)$ with a finite Borel measure $\mu$ on $\mathbb{R}$, then $g(x):=\int_\...
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 ...
6 votes
3 answers
536 views

A need for analytic continuation of a finite sum function

Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$. I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum) \begin{align*} {\color{red}...
3 votes
1 answer
166 views

A limit arising from Mellin Inversion: How to compute a specific term of an asymptotic series?

So I am wondering if there exists a general procedure for the following problem: given a monotonically increasing function $f(n)$ which is nonegative on the interval $[0,\infty)$ and grows faster than ...
3 votes
0 answers
75 views

Separate holomorphicity implies holomorphicity on analytic varieties

Suppose that $M$ and $N$ are two complex analytic varities and suppose that $f\colon M\times N \to \mathbb{C}$ is a map. Further assume that $f$ is such that for every $p\in M$ the map $f(p,\cdot)\...
2 votes
1 answer
185 views

Local equality of functions implies global equality?

The following question arised in my research, and I was unable to settle it after playing with it for sometime. Let $\{a^k_i\}_{i\geq 1}$ (for $k\in \{1,2,3,4\}$) be four sequences of real numbers. ...
0 votes
0 answers
102 views

Asking a reference about the $p$-Laplacian of $|\nabla u|^p$

It is well-known that for a harmonic function $u$, i.e. $$ \Delta u=0, $$ the quantity $|\nabla u|^2$ is subharmonic, i.e. $$\Delta (|\nabla u|^2) \geq 0. $$ Reason: $$\Delta (|\nabla u|^2)= 2 \nabla (...
2 votes
0 answers
70 views

Differentiable functions on analytic varieties

Let $\iota\colon X\to \Omega\subseteq \mathbb{C}^n$ be a complex analytic variety $X$ in an open subset $\Omega$ of $\mathbb{C}^n$. If $N$ is a smooth manifold and $h\colon M\to X$ is a continuous map,...
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 ...
6 votes
1 answer
300 views

Log-convexity of determinant

Let $f(z):=\langle g(z),g(z)\rangle,$ where $z \mapsto g(z)$ is holomorphic and $\langle \bullet,\bullet\rangle$ is an inner-product on some function space, such as $L^2$, such that $\langle g(z),g(z)\...
0 votes
0 answers
111 views

Sum power series not continuous unit circle

This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there. Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
1 vote
2 answers
588 views

Inequality between coefficients of a polynomial and its supremum

For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all ...
4 votes
1 answer
201 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 $\psi: U_2 \to W_2$ ...
6 votes
1 answer
408 views

On an asymptotic integral decay

Let $f\in C^{\infty}([-1,0])$ be real-valued and suppose that $$ \left| \int_{-1}^{0} f(t)\,e^{\lambda t} \,{\rm d} t \right| \leq e^{-\sqrt{\lambda}},$$ for all $\lambda > 0$. Does it follow that $...

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