All Questions
Tagged with differential-calculus integration
16 questions
6
votes
1
answer
340
views
Law of unconsious statistician: application in characteristic function
Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...
- 5
5
votes
0
answers
266
views
Hadamard lemma without integration
Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero.
By the product ...
- 10.5k
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
2
votes
0
answers
130
views
Computing harmonic sum [closed]
I want to show the following equalities for harmonic sum
$$\sum_{k=1}^{\infty}\frac{(-1)^k}{k^3}=\lim_{n\to\infty}\int_0^{2n}\frac{-3}{x^4}([x]-2[\frac{x}{2}])dx$$
Any idea?
- 37
2
votes
0
answers
103
views
Combinatorial identity of Derivatives of super-Gaussian function
An asymptotic expansion I stumbled upon has real numbers $c^\alpha_{ij}$ as coefficients, where $i, j \in \mathbb{N}_0$ are non-negative integers and $\alpha \in \mathbb{N}_0^n$ is a multi-index. They ...
- 9,853
1
vote
1
answer
200
views
About $\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$
An integral has been pushed me over the edge for several weeks. It reads as:
$$\displaystyle\int_{\mathbb{R}_y^3}\int_{\mathbb{S}^2}e^{-\frac{1}{2}|x-[(x-y)\cdot\omega]\omega|^2}d\omega dy$$
I tried ...
- 21
1
vote
1
answer
278
views
Request for help with two integrals
It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n ...
- 1,893
1
vote
1
answer
1k
views
integrate exponential function of quadratic form over unit norm vectors
Let $A$ and $b$ be an $M\times M$ matrix and an $M\times 1$ vector, respectively.
I need to solve
$$\int_{\|x\|^2=1} \exp\left(-x^\ast Ax + 2\mathcal{R}\{x^\ast b\}\right)
\, dx$$
where $(\cdot )^\...
1
vote
1
answer
135
views
Prove the integral of multi-variable rational fraction is convergent
I have posted this problem in MSE long ago:
https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...
- 561
1
vote
1
answer
323
views
A Bessel-like integral
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
- 884
1
vote
0
answers
35
views
How to relate this integration with the integral expansion of special functions?
I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
- 11
1
vote
0
answers
726
views
The derivative of an integral function with indicator and max function as integrand
I encounter the following type of problem:
\begin{equation}
F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv
\end{equation}
where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...
- 35
0
votes
1
answer
167
views
Solution of this differential equation [closed]
I wonder if it is possible to solve analytically the following equation
$$
\dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2
$$
Where $\alpha_t$ is a complex function, $...
0
votes
0
answers
38
views
Symmetric expression of boundary term in integration by part
Suppose $\Omega\subset\mathbb{R}^2$ be a smooth domain. $f,g\in C^\infty(\Omega)$. We consider the integration by part here:
$$\begin{aligned}
\int_{\Omega}(\partial_1\partial_2f)g&=-\int_{\Omega}(...
- 421
0
votes
0
answers
67
views
Integration of matrix form of Vasicek variance (Python/Matlab)
$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
- 1