Tagged Questions
3
votes
1answer
239 views
Discontinuity in function; order of integration matters
I'm struggling a bit with Chapter 10 of Rudin's "Principles of Mathematical Analysis," and I was hoping to get some help here.
I'll post the problem and my current progress.
Exercise 2: For $i = 1, ...
1
vote
0answers
44 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
4
votes
1answer
66 views
$\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not ...
2
votes
0answers
33 views
Closed curves question
Can you give me some help on the following problem?
Given two closed curves $\alpha, \beta : \mapsto \mathbb{R}^3$ we define $\phi_{\alpha \beta}: I^2 \mapsto \mathbb{R}^3$ as $\phi_{\alpha \beta} ...
1
vote
3answers
90 views
$\|f*g\|_q\leq \|g\|_q \|f\|_1$ and $\|f*g\|_\infty\leq \|g\|_q \|f\|_{q^{'}}$, $(1/q+1/q^{'}=1)$?
Now I'm reading the Young inequality. It says that if $f \in L^p(R)$, $g \in L^q(R)$, $1\leq p,q\leq \infty$, $1/p+1/q\geq 1$. Then how could we have the following inequalities:
$$\|f*g\|_q\leq ...
1
vote
1answer
160 views
analytically calculate value of convolution at certain point
i'm a computer science student and i'm trying to analytically find the value of the convolution between an ideal step-edge and either a gaussian function or a first order derivative of a gaussian ...
0
votes
0answers
44 views
Simpson's rule characteristics
I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:
...
0
votes
1answer
30 views
Integral over a Hypercube
Let $C_R$ be a hypercube in $\mathbb{R}^n$ of side length $2R$ and $B_R$ a ball in $\mathbb{R}^n$ of radius $R$.
I know I can say that
$$\int_{B_R}f(x)\,dx \sim \int_{C_R} f(x)\,dx $$
Could anyone ...
2
votes
1answer
84 views
Interchanging supremum and integral (another question)
Suppose that $\varphi$ is a smooth $\mathcal{K}_{\infty}$-function and $\bar{\textbf{B}}$ is the unit ball set in $\mathbb{R}^{n}$. Let $x : \mathbb{R}_{\geq 0} \times \mathbb{R}^{n} \to ...
1
vote
0answers
36 views
Compactness in $L^p$
I am studying this article:
http://arxiv.org/pdf/0906.4883.pdf
There is a little part that I do not understand, in the proof of theorem 5, page 4.
Let P be the projection map of $L^p(\mathbb{R}^n)$ ...
2
votes
1answer
200 views
Integration by Parts and Leibniz Rule for Differentiation under the Integral Sign
Basically a friend of mine and I have had this hot debate for a little too long, I contend that these two tools are not only logically unconnected but they require different assumptions (I believe one ...
2
votes
1answer
50 views
Interchange differential operator with Lebesgue integral.
Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and ...
5
votes
0answers
65 views
What is the relation of $\int f dx^1\wedge dx^2\wedge …\wedge dx^n=\int f dx^1…dx^n$
In a book "calculus on manifolds" it is defined that $\int f dx^1\wedge dx^2\wedge ...\wedge dx^n=\int f dx^1...dx^n$ but how it is possible the relate the integrand of a multilinear function ...
1
vote
1answer
65 views
Concept of integration to differential form
How to integrate differential form actually. As far as I know, a differential form is a multilinear function mapping from a vector space to a real number. Let's take $\int_c fdx+gdy$ as an example. It ...
3
votes
1answer
89 views
Riemann integrable proof
Let $f:[-1,1]$ be Riemann integrable and $\psi(x)=x\ sin(\frac{1}{x})$ for $0<x\leq1$ and $\psi(0)=0$. Show that $x\mapsto f(\psi(x))$ is Riemann integrable over $[0,1]$.
Also, just curious ... ...
0
votes
1answer
61 views
Whether convergence in L2 norm implies convergence a.e.? [duplicate]
How to prove or disprove$$\lim_{n\to\infty}\|f_n-f\|=0\;\Rightarrow \;\lim_{n\to\infty}f_n(x)=f(x)\; a.e.?$$ Any hint is appreciated.
1
vote
1answer
39 views
Proof of Lemma 4.2 in [G-T] pg 55
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n (n\geq 3)$ and let $\Omega_0$ be any domain containing $\Omega$ for which the divergence theorem is true. Let $f$ be bounded and locally Holder ...
1
vote
1answer
41 views
Lebesgue integrable function and limit
Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$.
My solution which is ...
3
votes
2answers
64 views
Suppose $f(x)$ is such that $\int_{-\infty}^\infty e^{tx} f(x)dx = \arcsin (t - \sqrt{1/2})$
Suppose $f(x)$ is such that
$$\int_{-\infty}^\infty e^{tx} f(x)dx = \arcsin(t - \sqrt{\frac{1}{2}})$$ for all $t$ where the right-side expression is defined.
Compute $$\int_{-\infty}^\infty ...
1
vote
0answers
157 views
Double Fourier Series $\cos(nx)\cos(my)$
Let $f(x,y) = xy$ on the square $[0, \pi]^2$. Find the Fourier cosine-cosine series of $f$.
I am working on this question with a group and one of us gets all the coefficients as zero. Is this correct ...
0
votes
0answers
38 views
Stieltjes integration with step function
Assume $F:[a,b]\rightarrow R$is bounded and right continuous at $a$ and $\alpha$ is the step function given by $\alpha(a)=A, \alpha(x)=B, a<x\leq b.$ Show that $f\in R(\alpha)$ on $[a,b]$ and
...
1
vote
1answer
51 views
Taylor expansion with integral?
I have looked at a version of a Taylor expansion that has an integral- for the first time. Is this the same as the usual version of a Taylor expansion without integrals? Also, do the $\alpha's$ have ...
5
votes
1answer
69 views
Prove that $\lim_{t \rightarrow 0} t \int_{0}^{\infty} e^{-tx} f(x) dx =1$
I am trying to solve Rudin 8.11:
Suppose $f$ is Riemann-integrable on $[0,A]$ for all $A<\infty$, and $f(x) \rightarrow 1$ as $x \rightarrow \infty$.
Prove that
$$\lim_{t \rightarrow 0} ...
1
vote
2answers
61 views
Evaluate the Riemann-Stieltjes integral
Given that $f$ is continuous and of bounded variation on $[a,b]$, evaluate $\int^b_a f(x)df(x).$
1
vote
2answers
37 views
Riemann-Stieltjes integration with floor function
Evaluate $$\int_{\frac{2}{3}}^8 f(x)d\alpha(x)$$ where $\alpha$ is continuous and $f$ is the floor function, that is $f(x)$ is the greatest integer less than or equal $x$.
0
votes
0answers
23 views
Rieman Stieltjes linear properties
Prove that if $f\in R(\alpha)\cap R(\beta)$ on $[a,b]$ then $f\in R(c\alpha+d\beta)$ on $[a,b]$ and
$$\int_a^bfd(c\alpha+d\beta)=c\int^b_afd\alpha+d\int^b_afd\beta$$
1
vote
2answers
193 views
Deriving that piecewise continuous functions are integrable
Suppose $g$ is integrable on $[a,c]$ and $h$ is integrable on $[c,b]$, then show
$$f(x) = \begin{cases} g & x\in[a,c], \\ h & x\in[c,b].\end{cases}$$
is integrable on $[a,b]$.
...
3
votes
1answer
174 views
Showing $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$ using complex integration
Recently I had to use the fact that the Dirichlet integral evaluates as
$$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$ a couple of times.
There already is a question that specifically ...
1
vote
1answer
63 views
Show that this equation is true.
Consider the following function in $\mathbb{R}^n (n\geq 3)$:
\begin{equation}
H(y)=2b_n\int_{0}^{\infty}e^{\ as}D_n\Phi(y-\tilde{x}+bs)\text{ d} s,\quad (x, y\in\mathbb{R}_{+}^{n}, x\neq y),
...
2
votes
1answer
108 views
Riemann-Stieltjes integrability criterion
I am currently reading through chapter 11 of Rudin's Principles of Mathematical Analysis, and I'm trying to solve problem 7:
Find a necessary and sufficient condition that $f \in \mathfrak R(\alpha)$ ...
2
votes
2answers
160 views
Prove $|\int_a^b$$f(x)dx| \leq \int_a^b$$|f(x)|dx$
Prove $$\left|\int_a^b f(x)dx\right| \leq \int_a^b |f(x)|dx.$$
My thoughts: first I think we must show that if $f \geq 0$ is Riemann integrable on $[a,b]$, then $\int_a^b f(x)dx \geq 0$. Then we ...
0
votes
0answers
28 views
Integrating Dirac of scalar function of (x,y,z) over volume dxdydz
I want to solve this integral:
\begin{equation}
\int_{\mathbb{R}^3} \delta \left (A - b(\frac{x^2}{m_x}+\frac{y^2}{m_y}+\frac{z^2}{m_z})\right) dx dy dz
\end{equation}
($\delta$ is the Dirac delta ...
6
votes
4answers
172 views
why $\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$ means surface area?
Why the following integral means the area of surface $f(x,y)=z$?
$$\iint \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}\:dx\:dy$$
0
votes
1answer
76 views
approximating a riemann integrable function by sequences of step functions and sequences of continuously differentiable functions
Suppose that $f$ is Riemann integrable on $[0,M]$.
How can I show that a) $f$ can be approximated uniformly by a sequence of finite step functions? and b) by a sequence of continuously differentiable ...
8
votes
4answers
131 views
limit of an integral of a sequence of functions
Suppose that $f$ is continuous on $[0,1]$. ($f'(x)$ may or may not exist).
How can I show that
$$\lim_{n\rightarrow\infty} \int\limits_0^1 \frac{nf(x)}{1+n^2x^2} dx = \frac{\pi}{2}f(0)\;?$$
My ...
1
vote
0answers
60 views
(Real Analysis) Integration of two functions
Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...
2
votes
0answers
103 views
A proof of Stirling's Formula
I need to gain understanding of a proof of Stirling's formula. I have read through Tim Gowers' and Terence Tao's but I'm struggling to follow them. How rigorous is this proof, if at all? Thank you.
...
8
votes
5answers
280 views
the sum: $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$ using Riemann Integral and other methods
I need to prove the following:
$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+(-1)^{n+1}\frac{1}{n}+\cdots=\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$$
Method 1:)
The series $\sum_{n=1}^\infty ...
2
votes
3answers
56 views
Convergence of improper integral involving exponential
How to show that $\int_0^{\infty} e^{\lambda x}x^{r} dx$ converges when $\lambda$ is negative and $r$ a positive integer?
2
votes
1answer
154 views
baby rudin, chapter 10, (differential forms) theorem 10.27
I'm having difficulties with the reasoning in the proof of theorem 10.27 (regarding integration over oriented simplexes).
say ...
1
vote
0answers
77 views
1
vote
1answer
109 views
Integration of even (and odd) function
Suppose that $a>0$ and that $f$ is integrable on $[-a,a]$. Show that if $f$ is even then
$$
\int_{-a}^0 fdx = \int_0^a fdx
$$
using the Riemann sum definition of Riemann integrability.
This is ...
6
votes
2answers
183 views
$f$ is integrable, but $f$ has no indefinite integral
Let
$$f(x)=\cases{0,& $x\ne0$\cr 1, &$x=0.$}$$
Then $f$ is clearly integrable, but has no antiderivative (primitive), at least on the entire domain of $f$, since any antiderivative function ...
0
votes
1answer
75 views
Proof of integration-by-substitution (two questions)
Here's a version of the theorem:
$$\int_a^b f(g(x))g'(x)dx=\int_{g(a)}^{g(b)} f(u)du$$ provided that:
$f$ is continuous on an interval $I$,
$g'$ is continuous on $[a,b]$,
$g[a,b]=I$ ...
0
votes
0answers
42 views
variational problem
I have: $\Omega \subset R$, be open and bounded, assume that $q \in L^{\infty}(\Omega)$ satisfies $q\geq 0$ a.e in $\Omega$, and let $f :\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function such ...
5
votes
3answers
81 views
Stuck on this integral involving exp and the floor function
Here is the integral
$$\int_0^\infty \lfloor x \rfloor e^{-x}dx$$
Here is what I have so far:
$$I = \sum_{n=0}^\infty \int_n^{n+1} n e^{-x}dx$$
$$ = \sum_{n=0}^\infty -ne^{-n-1} + ne^{-n}$$
$$ = ...
1
vote
1answer
78 views
How to find the range and inverse of this linear operator?
Given $T \colon C[0,1] \to C[0,1]$ defined by $$Tx(t):= \int_0^t x(r) dr$$ for each $t\in [0,1]$, where $C[0,1]$ is the normed space of continuous real-valued (or complex-valued) functions defined on ...
1
vote
0answers
116 views
Step functions dense in Integrable functions with respect to $L_2$
Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
0
votes
1answer
66 views
Is the integral of a smooth function continuous?
Suppose I have a function $f(a,x):\mathbb{R}^2\rightarrow\mathbb{R}$ that is smooth (i.e. infinitely differentiable) over its entire domain $\mathbb{R}^2$. Let $I(a)=\int_{-\infty}^{\infty}f(a,x)dx$. ...
0
votes
1answer
37 views
On the convergence of a specific sequence of integrable functions
Let $\{f_n\}$ a sequence of measurable non-negative functions on $\mathbb{R}$ converging point-wise on $\mathbb{R}$ to $f$, and let $f$ integrable over $\mathbb{R}$. If $\displaystyle ...
