All Questions
1,033 questions with no upvoted or accepted answers
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130
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summability and analytic continuation
Let $d_n=LCM(1,\cdots,n)$. It is well-known that $d_n=e^{\Psi(n)}$ where $\Psi$ est the second Chebyshev function. One knows that $\Psi(x)=\sum_{k\le x}\Lambda(k)$ where $\Lambda$ is the Von Mangold ...
- 3,998
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260
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Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
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116
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Dimension of the set of the polynomial growth harmonic function on the hyperbolic plane
We consider the hyperbolic plane and the harmonic function there. Pick any point $p$. Let $H_n, n \in\mathbb N$ be the set of the harmonic functions $f$ such that $|f(x)|\leq c(1+ d(x,p))^n$.
What is ...
- 5,063
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109
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solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
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49
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Non interacting complex unit
How to work with two non interacting complex units say i and j. These two imaginary complex unit represent different quantities. For example i is for periodicity in theta and j is for frequency or ...
- 1
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90
views
Derivative of a conjugation of matrices
Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
- 758
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256
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Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
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65
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what is the first non-constant term in the Kronecker Limit formula?
The Kronecker Limit formula gives the constant term in the Laurent expansion about s=1 of the Eisenstein series E(s,\tau). What is the next term? I.e., the coefficient of the first power of (s-1)? I ...
- 153
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161
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Asymptotic analysis of a sum of complex summands using integral
I'm trying to find the exact asymptotics of a sum:
$$A = \sum^n_{i=0} \begin{pmatrix} 2n \\ i \end{pmatrix} x^{i} y^{2n-i} $$
as $n\rightarrow\infty$. Here $x,y$ are complex numbers, $|x|\leq1, |y|\...
- 93
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309
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Holder continuous analytic function
Assume that $0<\alpha<1/2$ and $f$ is analytic in the unit disk $D$ with $|f(z)-f(w)|\le M|z-w|^\alpha$. Can we state that in general $$\int_D |f'(z)|^2 dx dy <\infty?$$
- 33
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332
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Examples of functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition
there are examples of lacunary functions with natural boundary that do not satisfy Fabry or Hadamard gap theorem condition.I want to know more examples of those functions,the more the better,...
- 3,104
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134
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Mittag-Leffler function and Laplace Integral
Let $E_{\alpha}(z)\triangleq \sum_{n=0}^{\infty} \frac{z^n}{\Gamma(\alpha n + 1)}$ be the Mittag-Leffler function.
I am looking for a full proof of the following fact (a reference to a proof in the ...
- 424
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142
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Arakelian's approximation theorem
I have a difficulty in understanding how one gets relation (2) in the proof of the Theorem, in the nice paper
[Jean-Pierre Rosay and Walter Rudin, Arakelian's Approximation Theorem, The American ...
- 1
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173
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Derivative of a function related to Dedekind zeta function
Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$.
From the ...
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141
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Boundary behavior of Harmonic functions
Assume that $f$ is harmonic in the unit disk $|z|<1$, with boundary function of bounded variation, such that $$\lim_{r\to 1}f(re^{it})= 0$$ for $t\in[0,\pi]\setminus \mathbf{Q}$, where $\mathbf{Q}$ ...
- 303
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381
views
Help with an irregular integral
I am looking for help with doing the following integral :
$$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in \...
- 181
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191
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Differentiability of minimax objective function with respect to a decision variable
I have the following optimization problem:
$$\text{find } x= \min_{a} \max_{\lambda\in\Lambda} |R(h\lambda)|$$
where $\Lambda$ is some finite, fixed set of complex numbers and $R(z)$ is a ...
- 1,253
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100
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Is this set of curves discrete?
Let $\alpha_1, \dots, \alpha_n$ be complex numbers whose sum is zero and $u_1, \dots, u_{2g-2+n}$ be pariwise distinct nonzero complex numbers. Consider the the set of smooth genus g curves with n+1 ...
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191
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Asymtotic Complexity Analysis using logarithms and binomial coefficients
On page 11 of "Smaller decoding exponents: ball-collision decoding" by Berstein et.al. they have the formula \begin{equation}\lim_{n \rightarrow \infty} \frac{1}{n}\log_{2}\left(\dbinom{k_{1}}{p_{1}}\...
- 101
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173
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Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.
This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...
- 975
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169
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Is degree a "strict -transform" birational invariant for surfaces in the complex projective 3-space?
(Edit) My question is as follows: My previous question was about a kind of "strict-transform birational" invariant not birational invariants as usual. So I just delete the question since it ...
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213
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unbounded plurisubharmonic function
Reading Demailly's Algebraic and Complex geometry in particular chapter 3 about positive currents, tha author defines the unbounded locus $L(u)$ of a plurisubharmonic function $u$ to be the set of ...
- 75
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624
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zeros of a holomorphic function in several variable
Let $f$ be a holomorphic function in $n$ complex variables on a domain $D\subset \mathbb C^n$. Let $S$ be a subset of $D$ such that for a polynomial $P$ in $n$ variable, $P(S)=(a,b)$ for an interval $...
- 415
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268
views
Is the absolute value of the j-invariant bounded from below on an annulus
Let $j:\mathbf{H}\to \mathbf{C}$ be the $j$-invariant. It's a modular function for $\Gamma(1) = \textrm{PSL}_2(\mathbf{Z})$.
For $\epsilon>0$ small, let $B(\epsilon)$ be the image of the strip $$\{...
- 225
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819
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Possible application of Rouche's theorem to aproblem of complex roots of polynomials
The following holds:
Let $P(x)$ be a polynomial in one variable $x$ of degree $3$ with complex coefficients
such that
a)
$$
P(-1)=P(1)=0
$$
Then
b)
the formal derivative $P^{'}(x)$ has a root in ...
- 1,641
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841
views
covariant derivative complex manifold
Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...
- 23
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520
views
Motivation of proof of Riemann-Roch for elliptic curve and generalizations
Given a lattice $L \subseteq \mathbb{C}$, Alain Robert defines a theta function as a meromorphic function such that $\theta(z+\omega)=a(\omega) e^{\pi h(\omega)(z+\frac{\omega}{2})} \theta(z)$ for all ...
- 15.4k
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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:
$$ \overline{\...
- 4,115
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votes
1
answer
226
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Subspaces of $H^{\infty}(\mathbb{D})$ which contains a nontrivial weak* closed subalgebra
Let $H^{\infty}(\mathbb{D})$ denotes the Banach space of bounded holomorphic functions in the unit disc. Consider the weak* topology on $L^{\infty}(\mathbb{T})$
that it inherits as the dual of $L^{1}(\...
- 175
-1
votes
1
answer
215
views
Best approximation of the modulus function
While there is extensive study regarding the best approximation of function with polynomial functions in the real domain, the study of approximation of complex variables becomes much sparse. See this ...
- 149
-1
votes
2
answers
87
views
Limits of integral series
Suppose we have the series of functions:
\begin{equation}
F(x)=\sum_{n=1}^{\infty} f_n(x)
\end{equation}
where convergence is uniform.
Additionally, consider the partial functions of the series:
\...
-1
votes
1
answer
124
views
Borel summation
If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by
$$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$
Suppose that $f$ and ...
- 139
-1
votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
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