Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^{2}\left(x\right)\right)...
The following integral may seem easy to evaluate ...$$\int_{0}^{\pi/2}\arctan \left(2\tan^{2}\left(x\right)\right) \mathrm{d}x =\pi\arctan\left(\frac{1}{2}\right)$$Could you prove it ?.
View ArticleEquality in Young's inequality
Let's take a look at Young's inequality: If $u,v\geqslant 0$ and $p,q$ - positive real numbers such that $\frac{1}{p}+\frac{1}{q}=1$ then $$\dfrac{u^p}{p}+\dfrac{v^q}{q}\geqslant uv.$$It's easy to...
View ArticleProving Sum of Angles Identity analytically
Define functions $S$ and $C$ such that $S'(x)=C(x)$, $C'(x)=-S(x)$, $S(0)=0$, and $C(0)=1$. Is it possible to prove that for all $x,...
View ArticleProbability measures that are "mutually exclusive"
Let $\Omega=\{1,...,n\}$ be a finite set.A new definition: we say two measures $\mu_1$$\mu_2$ are mutually exclusive if:For all $\omega$, if $\mu_1(\omega)>0$, then $\mu_2(\omega)=0$.I wonder if...
View ArticleBuilding a sequence
Given sequences $\{a_n\}$ and $\{b_n\}$ that both approach zero as $n$ approaches infinity, is it possible to have: $\lim b_n/(b_n+a_n) = 2$ as $n$ approaches infinity?I thought of $a_n=1/n$ and $b_n=...
View ArticleLebesgue dominated convergence converse - is it true?
Suppose that $f_n:\Omega\to\mathbb{R}$ is a sequence of functions, where $\Omega\subset\mathbb{R}^N$ is a bounded measurable set such that for a.a. $x\in\Omega$ we have that:$$\lim\limits_{n\to\infty}...
View ArticleHow to deal with $\int \frac{\sin ^2nx}{\sin^2 x}\text d x$
Suppose function $f\in\mathcal{C}^1[0,\frac{\pi}{2}]$, and $f(0)=0$, calculate:$$\lim\limits_{n\to\infty}\frac{1}{\ln n}\int_{0}^{\frac{\pi}{2}}\frac{\sin ^2nx}{\sin^2 x}f(x)\text d x$$Because $f$ is...
View ArticleDefinition of class of continuously differentiable functions on a closed...
This is a technical simple question. The space $C^1([a,b])$ is defined as$$C^1([a,b])=\{f:[a,b]\to R: \exists\; f' \textrm{ and is continuous on } [a,b] \}.$$I think there is some explaining to do...
View ArticleWhy and how did someone come up with the concept of open sets? [closed]
I understand what open sets are and their significance in mathematics. However, I often wonder what motivated someone to define such a concept in the first place.
View ArticleProving Lipschitz continuity (Real analysis)
A function $h : A → \mathbb{R}$ is Lipschitz continuous if $\exists K$ s.t.$$|h(x) - h(y)| \leq K \cdot |x - y|, \forall x, y \in A$$Suppose that $I = [a, b]$ is a closed, bounded interval; and $g : I...
View ArticleBounding a summation of positive terms
Bound the summation of the form$ \sum_{k=\tau}^{t} (1+k)^{-v} $where $ \tau $ and t are positive integers and v $ \in $ (0.5, 1].This summation is bounded by the form $ (1 + t - \tau)^{-v} / (1 -v) $I...
View ArticleCondition for Product being Uniform Continuous
$f, g:R\rightarrow R$ satisfy the following conditions:$f(x)$ is uniform continuous over $R$$xf(x)$ is uniform continuous over $R$$g(x)$ is uniform continuous over $R$Prove or give a counterexample:...
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Rudin's proof of Schwarz inequality in $\mathcal{L}^2(\mu)$ space
I'm reading chapter 11 of Rudin's Principles of Mathematical Analysis, and here is Rudin's proof of Schwarz inequality in $\mathcal{L}^2(\mu)$ space.In the second equality, I believe Rudin used the...
View ArticleA simple(!) change of variables on an integral
Let $1\le p\le q <\infty$ and$$\|f\|_{m_{q}^{p}}:= \sup_{a\in\mathbb{R}^d,R\in(0,1)}|B(a,R)|^{1/p-1/q}\left(\int_{B(a,R)}|f(y)|^p dy\right)^{1/p}.$$If we take $f(x)=|x|^{-d/q}$,...
View ArticleHow can I solve this problem of Fourier series? [closed]
How can I solve this problem?In order to build a function whose Fourier series is convergent but not uniformly convergent, The following series needs to be accurately estimated.Using the formula...
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Real way$\int_0^\pi\frac{f(x):=\arctan(\ln\sin x/x)}{\ln^2(x^2+\ln^2\sin...
Relatively recently I came across an integral on some forum in this postComplex contour solutionThe target integral can be convolved into a more compact...
View ArticleQuestion about the limit of functions that agree almost everywhere...
I was looking at the following limit I saw online:$$ \lim_{(x, y) \to (1, 1)} \frac{36x^4 - 36y^4}{6x^2-6y^2}$$The idea is that we can find another function that agrees almost everywhere and find the...
View ArticleFinding maximum of function with two variables
I have question about finding maximum of function in two variables, the specifics are:$$V(a,b)=\frac{1}{r}\Big[a(e^{r(1-a-c_b b)})-r(1-a-c_b b)\Big]+\frac{1}{s}(2-c_b)b(e^{s(1-b-c_a a)}-1),$$where...
View ArticleBounding a sum of inverses of absolute values
I am studying the proof in Zygmund's Trigonometric Series of the existence of an integrable function whose Fourier series diverges everywhere. I am stuck in a part where it is said that if $n \in...
View ArticleBaby Rudin 3.9 -- should I use the root or ratio test? [closed]
I'm asked to find the radius of convergence of a bunch of different series.In the solutions, a mixture of the root and ratio tests are used. But, it is unclear to me which to use when.For example, one...
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