1
vote
1answer
30 views

$\sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}$

Hi I am trying to calculate the sum given by $$ \sum_{n=-\infty}^\infty e^{-\alpha n^2+\beta n}=\ = \sqrt{\frac{\pi}{\alpha}} e^{\beta^2/(4\alpha)} ...
3
votes
0answers
32 views

$\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5)$

Hi I am trying to calculate the infinite double sum $$ S:=\sum_{j,k=1}^\infty \frac{H_j(H_{k+1}-1)}{jk(k+1)(j+k)}=-\zeta(2)-2\zeta(3)+4\zeta(2)\zeta(3)+2\zeta(5),\quad H_n:=\sum_{k=1}^n\frac{1}{k}\ \ ...
2
votes
1answer
32 views

how ro prove f(x,y) is integrable in $[a,b]\times[c,d]$

If there exits a $f(x,y)$ in $\mathbb{R}^2$,and if we fix any $x$ in $[a,b]$, then $f(x,y)$ is increasing as $y$ increases. Also, if we fix any $y$ in $[c,d]$,the $f(x,y)$ is increasing as $x$ ...
3
votes
0answers
28 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
0
votes
1answer
15 views

Question about Riemann Integration and the indicator function

Let $S \subseteq \mathbb{R}^n$. Suppose $\chi_S$ is integrable and $\int_Q \chi_S = 1 $ for some rectangle $Q$ such that $S \subseteq Q $. Let $\epsilon > 0 $ be given, I want to ask how can I ...
0
votes
1answer
41 views

How to prove that $\max\{f,g\}$ is Riemann integrable? [duplicate]

If f(x) and g(x) are Riemann integrable in [a,b], why $h(x)=\max\{f(x),g(x)\}$ is still Riemann integrable in [a,b]? Or maybe it is wrong?
0
votes
1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
1
vote
1answer
46 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
2
votes
0answers
32 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
0
votes
0answers
20 views

Showing equivalence of weak convergence on closed and open intervals

Quick question. Let $I$ be an open bounded subset of $\mathbb{R}^{n}$. If I am given that $u_{m},u \in W^{1,\infty}(I)$ and I want to show that $u_{m} \rightharpoonup^{*} u$ in $L^{\infty}(I)$. Then I ...
1
vote
1answer
41 views

Compute $\int_{0}^x \vert \sin(t)\vert dt$ for $x\in \mathbb{R^+}$

Let $x\in \mathbb{R^+}$, compute $$\int_{0}^x \vert \sin(t)\vert dt$$ I tried like this : $$ \int_{0}^x \vert \sin(t)\vert dt=\int_0^{\lfloor \frac{x}{\pi}\rfloor \pi}\vert \sin(t)\vert ...
4
votes
0answers
96 views
+400

The long Integral with a nice result

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2Li_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) -\frac{\zeta(2)-2\log^2 ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
7
votes
3answers
102 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
2
votes
1answer
21 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
1
vote
1answer
32 views

Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.

I will post the assignment and then my attempt at solving it. Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
3
votes
3answers
72 views

Evaluate $\int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x$

My try, using $x = \sec(u)$ substitution: $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &=& \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &=& ...
-1
votes
0answers
38 views
+50

Question concerning the integrability of a function

Let $f: [0,1]^2 \to \mathbb{R}$ be a function such that $$ f(x,y) = \left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q} \\ 2y & : x \notin \mathbb{Q} \end{array} ...
2
votes
1answer
50 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
7
votes
1answer
121 views

Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
1
vote
2answers
132 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (v). (1) is the ...
1
vote
1answer
39 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
1
vote
1answer
47 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
2
votes
0answers
61 views
+400

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
0
votes
0answers
21 views

Integration over rectangles

"Prove that a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is integrable over every rectangle in $\mathbb{R}^n $ if and only if it is integrable over every ball in $\mathbb{R}^n$" So I'm stumped ...
0
votes
1answer
46 views

Double Integral Proof

Let function $f(x, y)$ be defined by $$f(x, y) =\begin{cases} 1,\text{ if }x = y,\\ 0,\text{ otherwise}.\end{cases}$$ Using the definition of the double integral show that the following integral exists ...
-3
votes
0answers
38 views

$\int_{a}^{b}{x^nf(x)dx}=0$ for all $n$ [closed]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Prove that if $\int_{a}^{b}{x^nf(x)dx}=0$ $\forall n\in \mathbb N$ then $f(x)=0$ for all $x\in [a,b]$
1
vote
1answer
22 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
0answers
23 views

$\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ where $M$ is the sup [duplicate]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Let $\displaystyle M=\sup_{x\in [a,b]}{f(x)}$. Prove that: $\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ I ...
2
votes
1answer
89 views

$ \lim\limits_{x \to +\infty}x\, e^{-x^2}\int_{0}^{x}e^{t^2}dt $

Hello every one Please I need your help for the 3rd question, I tried but i fail every time. for every real $ x $, we put $ f(x)=e^{-x^2}\int_{0}^{x}e^{t^2}dt $. Show that $ f $ is odd of class $ ...
3
votes
2answers
134 views

Real analysis question involving a linear ODE

Where do I start with this one? This question is really quite difficult..
2
votes
1answer
33 views

A bounded integral

I want to show that there exists $K\in\mathbb{R}^+$ such that $$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$ for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. ...
2
votes
2answers
66 views

Prove that integral of continuous function is continuously differentiable

Lots of things going on here. I immediately know that $F(x)$ does exist since $f$ is riemann integrable due to the fact that it is continuous. First I need to show that $F$ is continuous, then find ...
1
vote
1answer
27 views

Provide examples that satisfy the following cases

Provide examples that satisfy the following cases 1) $f_n: [0, ∞)$ → R that converges uniformly to the function $f (x) = 0$ on [0, ∞) but such that $\lim_{n→∞} \int_{0}^{∞} f_n(x) dx \neq ...
0
votes
0answers
57 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
1
vote
0answers
44 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
1
vote
0answers
43 views

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)d\theta$

I am trying to integrate the Log Sine Integral: $$ Ls_{n+1}=-\int_0^{\pi/3}\bigg[\ln\big(2\sin\frac{\theta}{2}\big)\bigg]^nd\theta $$ where n is a non-negative integer. This problem is strongly ...
0
votes
0answers
30 views

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ ...
1
vote
0answers
31 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
2
votes
1answer
121 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
0
votes
0answers
30 views

rising sun inequality [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
0
votes
1answer
70 views

Question about riemann integration

Let $Q \subseteq \mathbb{R}^n$ be a box and $f: Q \to \mathbb{R}$ be a integrable function with $f \geq 0$. Let $D$ be the set of discontinuities of $f$. Let $x \in Q \setminus D $, and choose a ball ...
0
votes
1answer
58 views

What are the values of the integrals?

Here is the problem: Let $R>0, \varepsilon>0$. For $t\geq 0$ define $$ \psi^+_\varepsilon(t) := \left\{ \begin{array}{ll} 1 & 0 \leq t \leq R\\ 1+(R-t)/\varepsilon ...
2
votes
1answer
75 views

Integrate $I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx$

Hi I am trying to evaluate the integral $$I=\int_0^{\pi/2}x^2\ln(\sinh x)\ln(\cosh x)dx.$$ Note we can write the integrand as $$ x^2 \ln\big(\frac{e^x-e^{-x}}{2}\big) ...
1
vote
0answers
23 views

Under which assumptions we have $f\in L^p$ for all $p\in\mathbb N$

So here is my question, I wanted to generalize, under what assumptions for some $f$ we have $f\in L^p(\mathbb R)\;\forall p\in\mathbb N.$ And I found the following, Let $f\in L^p(\mathbb R)$ for ...
1
vote
0answers
35 views

Higher-dimension integrability (over rectangles) well-defined

Here is the problem and my work toward a proof: Question: Prove that in the following definition, the value of $\int_E f dx$ is independent of the choice of rectangle $J$: Definition: ...
0
votes
0answers
18 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
2
votes
1answer
109 views

Integrate $\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta$

Hello I am trying to integrate $$ I=\int_0^\pi \theta^2 \ln^2\big(2\cosh\frac{\theta}{2}\big)d \theta $$ which is similar to Integral...$\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d ...
1
vote
1answer
27 views

What are the consequences of this simple property of $L^1$ functions?

I came across the following statement: Let $f\in L^1(\mathbb R,\mathbb R)$. Then $$\forall \varepsilon>0 \ \ \exists \delta>0 \ \ \text{such that for all open sets } U\subset\mathbb R \text{ ...
3
votes
1answer
21 views

Convergence in average on every set implies convergence?

Let's say we're working in a measure space $(X, \mathcal{B}, \mu)$, and let $f_n, f$ be measurable. Suppose I have that, for any measurable set $E$, $$ \int_E f_n d \mu \to \int_E f d \mu $$ Does that ...