0
votes
0answers
55 views
New differintegral formula: how is it related to other differintegral formulas?
Lets define new differintegral formula as
$$\mathbb{D}^s_xf(x)= \sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
or, equivalently,
$$\mathbb{D}^s_xf(x …
1
vote
1answer
199 views
Integral inequality for convex function
Let $u(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Suppose that for some real numbers $a,b$ with $a < b$ the following equality is true:
\begin{equation}
\frac …
0
votes
0answers
53 views
Some questions regarding Ramanujan summation — Part I
The Ramanujan Summation method, is a method through which divergent series can be summed to convergent values.
I have several questions regarding this summation method. For more …
0
votes
0answers
88 views
Green’s fuction
Hi;
Please I have to find the Green function of a third order boundary value problem, and i don't know if this is the same method that in the case of second order BVP ?
Thank yo …
0
votes
0answers
36 views
Young function bahave like a $t^k$ near the origin?
Suppose $\Phi(t)=\int_0^t \phi(s)ds$ is an Young function with $\phi \in C^1 (0,\infty)$. Assume also that ($\Delta_2$ condition)\begin{equation}\tag{1}(1+\Gamma_1)\Phi(t)\leq t \p …
0
votes
0answers
93 views
Harmonic analysis on the Heisenberg group
It is well known that,
\noindent
Theorem 1. For $f\in L^{2}(\mathbb H_{n}=\text{The Heisenberg group of dimesion } 2n+1)$ we have the expansion
$$f(z, s)= (2\pi)^{-n} \sum_{k=0}^{ …
0
votes
1answer
116 views
Conditions to partially take a limit of composition of function
Here is my question.
I have two continuous functions $f_n(x)$ and $a_n(x)$. I know for a fact that $\lim_{n\rightarrow\infty}a_n(x) = a(x)$, and I would like to perform a partial …
11
votes
1answer
574 views
The unreasonable effectiveness of Pade approximation
I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. …
2
votes
2answers
253 views
Monotonic bijections of rational numbers
How can one characterize monotonic bijections from $\mathbb{Q}$ to
$\mathbb{Q}$? It is easy to see that piecewise linear functions which are
strictly monotonic and surjective will …
2
votes
1answer
66 views
approximation methods in integral equations
Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence l …
-2
votes
0answers
59 views
Behaviour of the gradient w.r.t. boundary conditions for elliptic PDEs
Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$.
Suppose that
\begin{align}
& …
0
votes
0answers
97 views
fourier transform [closed]
The fourier series is decieded by the derivatives of a point, but many theorems says all continuous
function defined in finite span has fourier tansform equal to the original funct …
0
votes
0answers
165 views
An interesting summation [closed]
I wonder if there is a simple closed form solution to the following sum: $\sum_{k = 1}^n \frac{(1/2)^k}{k}$? Wolfram Alpha expresses it in terms of the Lerch transcendent, but I wo …
3
votes
4answers
668 views
Another Chicken or Egg: Sequence or Series
This is a side question which is more motivated by teaching than research.
First, I am trying to convince myself that sequences appear before series (as numerical approximations t …
0
votes
0answers
104 views
How to show whether this double sum is divergent/convergent?
I want to show whether the following series diverge
$\sum_{V=0}^{\infty }\sum_{P=0}^{\infty }\frac{1}{P!V!}\left [ \frac{ig}{6} \int d^{4}x\left ( \frac{1}{i} \frac{\delta }{\delt …

