All Questions
140
questions
2
votes
1
answer
41
views
A question about switching the order of $\int$ and $\lim$ for a series of complex functions
When I took undergraduate complex analysis, the instructor was trying to prove an inequality, and he used a technique as below:
$$\int \lim_{n\rightarrow \infty} f_n dz= \lim_{n\rightarrow \infty} \...
- 1,131
0
votes
1
answer
34
views
Measure Theory: Proof regarding a measurability of a function [closed]
Is my proof for the following problem correct?
Let $f:X\rightarrow[0,\infty)$ and $X=\bigcup_{i\in\mathbb{N}}A_i$ be measurable.
Prove: $f$ is measurable $\iff$ $f\cdot\chi_{A_i}$ is measurable.
Proof:...
user1197316
5
votes
1
answer
279
views
First fundamental theorem of Calculus continuity not necessary?
I know if $f:[a,b] \to \mathbb{R}$ is continuous, then $g(x) = \int _{a}^{x}f(t) \, dt$ is differentiable on $[a,b]$. Furthermore, $g'(x) = f(x)$.
This is known as the first fundamental theorem of ...
- 1,706
0
votes
0
answers
60
views
An integrand of indefinite integral
I need to find some examples of a function $\xi \left( x\right) > 0$, where $x \in [x_{0},\infty ),x_{0}>e
$, such that the integral $$I =\int \frac{1}{x-\frac{x}{\ln \left( x\right) }+\xi \...
- 315
0
votes
1
answer
28
views
Estimate on integral of cosine on a set of positive measure
I would like to find an estimate of the form
$$
\left|\int_S \cos(\theta)d\theta\right|\leq A<|S|
$$
for a positive measure set $S\subset [0,2\pi)$ and $|S|$ the Lebesgue measure of the set $S$. I....
1
vote
0
answers
95
views
L1 distance between functions and integral of symmetric difference between sublevel sets
I have come across this question : Can I write the integral of a function in terms of its level sets? and it makes me wonder if the L1 distance between $f$ and $g$, two bounded function over a compact ...
- 55
5
votes
1
answer
47
views
Complicated change of variables formula with CDF of normal distribution instead of diffeomorphism
I have a parameter $\theta\in[0, 10]^4$ and a variable $z\in\mathbb{R}^m$. Consider the following integral
$$
I(\theta, z) := \int F(\theta, z) \mathbb{I}(\|f(\theta, z)\| \leq 1) p(\theta, z) dz d\...
- 5,247
0
votes
0
answers
23
views
Fundamental Theorem of Calculus over an infinite interval such as $F(x)=\int_{-\infty}^x f(t)dt$ then $F'(x)=f(x)$ [duplicate]
The Fundamental Theorem of Calculus states that if $f$ is continuous on $[a,b]$, then if we take $F(x)=\int_a^x f(t)dt$ for $x\in [a,b]$, we have $F'(x)=f(x)$ for all $x\in (a,b)$.
My question is how ...
- 5,601
0
votes
1
answer
33
views
Is the integral $\int_{[0, 1]} a^{-(b-x)} \ d\nu(x)$ small independently of the measure $\nu$?
Let $\nu$ be a positive measure, $b>1$ and let $M\gg 0$ be fixed. Assume $a\ge M$. As part of a more challenging exercise, I am trying to evaluate the integral
$$\int_{[0, 1]} a^{-(b-x)} \ d\nu(x).$...
- 425
0
votes
0
answers
76
views
Variable change in functional integral
Suppose I have following functional integral
$$ \int \mathcal{D}x\ f(x) \tag{1}$$
where $$\mathcal{D}x=\prod_t dx(t) \tag{2}$$
is the functional measure.
Now, I want to expand $x(t)$ as
$$x(t)=\sum_n ...
- 225
0
votes
1
answer
99
views
bounded function in $\mathbb{R}^d$ [duplicate]
There is a classic problem in Analysis stating that for integrable $f$ and $g$ bounded and measurable then their product is integrable. Now I am trying to show if $g$ is measurable on $\mathbb{R}^d$ ...
- 225
2
votes
2
answers
83
views
Calculating the flux of the vector field + Gauss
Calculate the flux $\int_BF · ν dS$ of the vector field $F : \mathbb{R}^3 \to \mathbb{R}^3, F(x, y, z) := (yz, xz, x^2)^T$ through the northern hemisphere B of radius 1 with 0 as its center, i.e. the ...
- 205
1
vote
0
answers
92
views
Find the area with integrals
I need to calculate the area of the following surface $A\subset \Bbb R^2$ using Fubini’s Theorem:
$$A := \left\{ (x,y) \in \Bbb R^2 ~:~ 1\le xy\le 2 \text{ and } 1 \le \frac yx \le 2\right\}$$
Can I ...
- 205
1
vote
1
answer
212
views
Integral with triangle vertices
$\int_Bxy +\frac1{(1 + x + y)^2}dλ^2$
$B$ denotes the interior of the triangle with vertices $(0,0)$, $(0, 1)$ and $(2,2)$.
I followed the same method got a double integral $$\int_0^2\int_0^yxy +\frac{...
- 205
0
votes
1
answer
30
views
Find the values of $p\ge 1$ such that $|x|^{-\frac{4}{5}p}(1+|x|^3)$ is integrable on unit ball and complement of unit ball, respectively
Consider the function $w:\mathbb{R}^4\setminus\{0\}\to \mathbb{R}, w(x)=|x|^{-\frac{4}{5}p}(1+|x|^3)$. Firstly, I am asked to find the values of $p\ge 1$ such that $w\in L^1(B_1(0))$ and then I am ...
- 885