All Questions
Tagged with cv.complex-variables sequences-and-series
83 questions
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) ...
- 25
8
votes
3
answers
616
views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 317
6
votes
1
answer
645
views
How many roots do $\tan(z)-z^n = 0$, $\sin(z)-z^n=0, \ \cos(z)-z^n=0, $ have?
I asked this question on MSE here.
I am investigating the number of roots of the equations:
$$\tan(z) - z^n = 0$$
$$\sin(z)-z^n=0$$
$$\cos(z)- z^n=0$$
within the vertical strip $|\text{Re}(z)| \leq \...
- 541
30
votes
1
answer
4k
views
Proof of "Possible new series for $\pi$" without use of physics
Related post: The post Possible new series for $\pi$ is about whether the identity is new, so to avoid confusion I was advised to ask this question separately.
I am looking for a proof of the ...
- 1,459
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 $\...
- 605
4
votes
2
answers
373
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 188k
7
votes
1
answer
268
views
Efficiently computing $\sum_k x^{k^2}$ modulo $p$
Let $p$ be prime. There is a whole host of "large" degree polynomials that can be computed efficiently modulo $p$. I was wondering if:
$$q(x) = \sum_{k=0}^{p-1} x^{k^2}$$
is a polynomial ...
- 591
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 ...
- 515
6
votes
3
answers
535
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}...
- 43.1k
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 ...
- 2,679
24
votes
1
answer
2k
views
Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
- 2,679
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}{...
- 7,300
2
votes
2
answers
260
views
Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,730
3
votes
1
answer
261
views
Where does the Weierstrass expansion of $\operatorname{sn}$ come from?
In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$')
$$\operatorname{sn}u=\frac{B}{A}$$
where $...
- 41
7
votes
1
answer
335
views
If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
- 1,730
0
votes
1
answer
138
views
proving inequality in Riemann zeta function
Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...
- 305
6
votes
0
answers
171
views
Computing residues at $\infty$
As an initial note, let me show by example what I mean by the terminology 'residue at $\infty$' I use in the title. I assume there is some standard terminology for this stuff, so I'd appreciate it if ...
- 1,730
4
votes
0
answers
159
views
Correct way to extend a sequence defined on the naturals into the complex plane
Preamble
Sequences $a_n$ defined on the natural numbers are clearly not uniquely interpolated by only one function. In particular, given an interpolation $f(n) = a_n$, then $f(n) + \sin(2\pi n)$ is ...
- 1,730
9
votes
1
answer
657
views
Is anything known about the power series $\sum x^p$ for $p$ prime?
I'm interested in information about the power series
$$\sum_{\text{$p$ prime}} x^p$$
and the related power series
$$\sum_{n=1}^\infty (-1)^n x^{p(n)}$$
where $p(n)$ is the nth prime.
Immediately, the ...
- 1,730
6
votes
1
answer
241
views
Fractional integrals and $\sum f(n) n^x$
Preamble
The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
- 1,730
2
votes
1
answer
474
views
Value of $\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}$
I wonder if any of you knows how to find the value
of the series $$\sum_{n=1}^{\infty}\frac{e^{-bn}}{n^2+z^{2}}.$$
This function shows up while solving a magnetostatic problem
with complex-valued ...
- 29
9
votes
0
answers
313
views
Switching the order of a summation and replacing a series by its analytical continuation
Background
A useful trick when trying to analyze a series $\sum_{n=0}^\infty f(n)$ is to expand $f(n)$ as some kind of series, swap the order of summation, and then evaluate the inner infinite sum. ...
- 1,730
10
votes
2
answers
597
views
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
- 2,679
7
votes
0
answers
306
views
Gottfried Helms' tetra-eta series
Here Gottfried Helms introduces the following fascinating divergent series
$$ T_2(x)=- \sum_{n=1}^\infty (-1)^n n^{n^x}$$
The terms don't go to zero, so technically the series does not converge ...
- 1,730
21
votes
6
answers
1k
views
What is the relationship between $\sum_{n=0}^\infty f(n) x^n$ and $-\sum_{n=1}^\infty f(-n) x^{-n}$?
Background
Taking a relatively arbitrary combination of exponential and polynomial terms, for instance
$$\sum_{n=0}^\infty \left(n^{2}\sin\left(n\right)+n\cos\left(3n-2\right)\right)\cos\left(5n+1\...
- 1,730
4
votes
2
answers
430
views
Is there a recurrence for the coefficients of the Laurent series expansion of $\frac{1}{1-e^{e^x - 1}}$?
So there's an elementary (but in my opinion quite interesting!) result which is that the Laurent series expansion of
$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \...
- 2,679
6
votes
1
answer
324
views
Has anyone characterized the zeroes of the Bell numbers?
I was reading this post about the Bell Numbers where users Lucian and Vladimir Reshetnikov give us Dobiński's formula for the Bell numbers
$$ B(x) = \frac{1}{e} \sum_{k=1}^{\infty} \frac{k^x}{k!}. $$
...
- 2,679
5
votes
3
answers
342
views
Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
- 2,679
13
votes
3
answers
1k
views
Is anything known about the series $\sum_{n=0}^{\infty} x^{\sqrt{n}} $?
It's well known that there are a shocking number of identities for the usual Jacobi theta function $$ \theta_3(x) = \sum_{n=-\infty}^{\infty} x^{n^2}. $$
So I wanted to turn my attention to slowly ...
- 2,679
1
vote
2
answers
113
views
$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$
Assume $A$ is an operator on a Hilbert space with discrete spectrum. Assume $B$ is a positive operator on the same Hilbert space also with a discrete spectrum. Also assume $A$ and $B$ commute.
I'm ...
- 1,150
20
votes
2
answers
729
views
Multiple roots of polynomials with coefficients $\pm 1$
Question P. Can a polynomial $P(x)=\sum_{n=0}^ma_nx^n$ with coefficients $a_n\in\{-1,1\}$ (and $P(1)=0$) have a multiple root in the interval $(\tfrac12,1)$?
Also I am interested in a similar question ...
- 41.8k
7
votes
1
answer
268
views
A differential equation governing compositional inversion
Looking for references for the following theorem.
Given the formal Taylor series/exponential generating function
$$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$
for which the indeterminates $a_n$ and ...
- 10.5k
6
votes
0
answers
394
views
Is there a residue sum formula in quaternionic analysis?
In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.
If the function $f: \mathbb{C} \to \mathbb{C} $ ...
- 4,829
4
votes
0
answers
241
views
Karamata's Abelian/Tauberian Theorem in the complex plane
The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels):
Fix $c, \rho>0$. If ...
- 406
1
vote
1
answer
163
views
Upper bound on double series
We consider the sum
$$ \sum_{m \in \mathbb Z^2} \frac{1}{(3 m_1^2+3m_2^2+3(m_1+m_1m_2+m_2)+1)^2}. $$
Numerically, it is not particularly hard to see that the value of this series is well below $4$, ...
- 73
2
votes
1
answer
180
views
As-closed-as-possible formula for an integral and/or sum
I need to find the solution of this integral:
$$\int^\pi_{-\pi}{d\varphi\,{}\frac{\Gamma(n,2\pi\varphi)}{(1+ia\varphi)^n}},\tag{1}\label{1}$$
where $a\in(0,1)$ and $n$ is a positive integer (not zero)....
2
votes
1
answer
177
views
Another combinatorial identity
Is it true that
$$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$
for all natural $n$ and all natural $p\ge2n$, where
$$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)!
(p-r+i)! (n-r+i)! ...
- 128k
2
votes
1
answer
423
views
Cauchy's integral formula and essential singularities
Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral
$$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$
where $\gamma_{z_0}$ is a small circle around $z_0$.
By ...
- 99
5
votes
1
answer
673
views
Is this infinite product entire?
Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
- 73
4
votes
1
answer
317
views
Taylor coefficients of Hadamard product
I imagine this to be a very classical question in complex analysis:
Consider the Hadamard product
$$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$
where $E_1(z):=(1-z)e^z$ is the first elementary ...
- 73
1
vote
2
answers
164
views
Natural boundary with non-zero "thickness"
Many series (in fact, "most" Taylor series) have a natural boundary at the unit circle. This boundary is only 1 dimensional, in the sense that only along the unit circle is it true that ...
- 1,730
4
votes
0
answers
268
views
Four infinite series involving Riemann zeta function
Can you provide a proof for at least one of the claims given below?
It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
- 2,661
4
votes
1
answer
396
views
Limit at infinity of infinite series
Let
$$
f(x) = \sum_{n=0}^{\infty} a_n x^n
$$
and suppose that the radius of convergence of this series is infinite. Is there a general method to know whether $\lim_{x \rightarrow \infty} f(x)$ exists ...
- 1,608
5
votes
0
answers
225
views
Validity of $\ln z=\frac{\pi}{\operatorname{AGP}(\theta_2^2(1/z),\theta_3^2(1/z))}$
Definitions
For the definitions of $\operatorname{AGP}$ and $\operatorname{AGO}$, see here or here. $\theta_2(z)$ and $\theta_3(z)$ are defined as follows:
$$\theta_2(z)=\sum_{n=-\infty}^\infty z^{(n+...
- 83
9
votes
0
answers
321
views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
- 83
1
vote
0
answers
76
views
Second question on a real sequence
I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
- 903
3
votes
0
answers
233
views
Sequence unifomly bounded
Let $f(\lambda,z)$ be a continuous function on $\Bbb R^2$ such that
I) For $n\in\Bbb N$ and $x\in\Bbb R^*_+$ we have : $f(n,x)=\cos(nx)+ x O\big(\frac{1}{n}\big)$ as $n\to\infty$ and $x\in[n^{-1}\...
- 81
4
votes
1
answer
690
views
An asymptotic expansion of a infinite sum
I am interested in the asymptotic expansion in $t$($t>0$) when $t\to 0^+$ of the following series
$$
\sum_{k\ge 0}e^{-k^{2/n}t}
$$
for integer $n>2$ (n=1 follows from Poisson summation formula ...
- 193
-1
votes
1
answer
87
views
Inferring polynomial rate of convergence from polynomial bound
Let $x_n$ be a non-negative valued sequence and suppose that the following hold:
$\lim\limits_{n\to\infty} x_n =0$
There exists some polynomial function $p$ of degree at-least $1$ such that:
$$
\|x_n\...
- 5,407
3
votes
1
answer
144
views
Coefficient problem in the class $\Sigma$
Let $\Sigma$ be the class of univalent (injective) holomorphic functions on $\mathbb{C}\backslash \mathbb{D}$ where $\mathbb{D}$ is the closed unit disk. Analogous to the famous Bieberbach conjecture ...
- 133