All Questions
Tagged with integration sequences-and-series
10 questions with no upvoted or accepted answers
7
votes
0
answers
299
views
An integral for the tribonacci constant and the general case
When I asked for integrals involving the tribonacci constant $T$, user @nospoon gave the nice answer,
$$\int_0^{\infty} \eta( i \, x)\,\eta(i \,11 x) \,dx = \frac{ \ln T}{\sqrt{11}} $$
However, the ...
- 12.6k
5
votes
0
answers
254
views
Is there a practical application of natural integral or differintegral?
The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$f^...
- 10.1k
2
votes
0
answers
379
views
Is this double integral of Fourier series always real?
Consider $f(x)$ a function from $\mathbb{R^+}$ to $\mathbb{C}$ such that $f(x) \sim_0 x$ and $\int_{0}^{\infty} f(x) dx=\int_{0}^{\infty} x^2 f(x) dx=0$
Can we demonstrate that following integral is ...
- 1,199
1
vote
0
answers
102
views
Conjectured closed form of $\int_0^1\frac{\ln^3(1+x)\,\ln^3x}x\mathrm{d}x$
I posted this question on Math Stack Exchange, but there were no helpful comments or answers
https://math.stackexchange.com/q/4874446/1298448
How to integrate $${\displaystyle \int_0^1\frac{\ln^3(1+x)\...
- 224
1
vote
0
answers
210
views
Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
1
vote
0
answers
233
views
Fubini: can we interchange integration order on this double integral (with Fourier series product)
Can we interchange the order of integration of following double integral ?
$$I = \int_{0}^{1} \int_{0}^{\infty} F(x,y) \overline{R(x,y)} - R(x,y) \overline{F(x,y)} \; dx \; dy$$
Where $F(x,y)= \...
- 1,199
0
votes
0
answers
117
views
integral of exponential of Fourier series
I have encountered the following integral:
\begin{equation}
\int_0^{1} e^{-i F(x)} dx, \quad F(x) = \sum_{k=1}^L a_k \sin(2\pi k x) + b_k \cos(2\pi k x).
\end{equation} I have found several great ...
- 9
0
votes
0
answers
75
views
Dense subspace of square integrable functions on the complex disc
Denote by $L^{2}(D,(1-|z|^{2})^{a-1}|z|^{2b-2}dx dy)$ the set of square integrable functions on the complex disc $D= \lbrace z \in C, \; |z| <1 \rbrace$ with respect to the measure $(1-|z|^{2})^{a-...
0
votes
0
answers
146
views
Does the following sequence $\{g_n\}$ converge?
Consider a function sequence $\{f_n(t)\}$ ($n\in\mathbb{N}^+$) defined on the interval $(\frac{1}{2},1)$, where
\begin{eqnarray}\label{eqn:constraint1}
f_n(t)=\frac{\exp\left(n\left(\log R(h_t) - th_t\...
- 550
0
votes
0
answers
185
views
Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral
The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
