Evaluation of $\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$
How to evaluate the following integral$$\int_0^{\pi/4} \sqrt{\tan x} \sqrt{1-\tan x}\,\,dx$$It looks like beta function but Wolfram Alpha cannot evaluate it. So, I computed the numerical value of...
View ArticleClik here to view.
Mapping the north pole to the equator
Given the standard equirectangular map of planet earth, say $M((\theta, \phi))$, for $0\leq \theta \leq 2\pi, -\pi/2 \leq \phi \leq \pi/2$.Now rotate the planet by $90°$ such that a new map...
View ArticleEquivalence of the least upper bound property and the boundedness of...
In a calculus course we prove a lot of theorems as fundamental consequences of the least upper bound property of R. In some cases, such as the Intermediate Value Theorem and the Bolzao-Weierstrass...
View ArticleHow many non-measurable sets can we construct?
I understand there are $|\mathbb{P} (\mathbb{R})|$ many non-measurable sets. However the sets that occur through the Vitali construction are the only sets (that I am aware of), which we can construct...
View ArticleBounded integrals are invariant under composition by homeomorphisms
Suppose that $f$ is a measureable function and that$$\int_1 ^\infty |f(x)| dx \leq C<\infty$$then is it true that$$\int_1^\infty |f(\sqrt x)| = \int_1^\infty |f(x)|$$I suspected this to be true...
View ArticleProve multiplication of Dedekind cuts is associative.
I'm trying to prove that multiplication of Dedekind cuts is associative. I've read some proof online, but they all have a step I feel uncomfortable with. They say that a rational $q$ is in their...
View ArticleFinite number of limit points in NULL SET
$\mathbf{Question:}$ Let $ E \subseteq [a,b] $ be any subset which has only finite number of limit points. can we say $E$ is a null set ?$\mathbf{My \hspace{1mm} attempt:}$If $A \subset \mathbb{R}$ is...
View ArticleClosed form for $\sum_{n=1}^{\infty}\frac{n^k}{2n!}$
I'm looking if there's a general closed form for the following function:$$f(k) = \sum_{n=1}^{\infty}\frac{n^k}{2n!},$$for $k\in \Bbb{N}$.For specific $k$ values I can utilize the fact that$$\cosh(x) =...
View ArticleUncountable set has uncountably many limit points. (Proof Checking Request.)
Show that any uncountable subset of the reals has uncountably many limit points.Let $S\subseteq \mathbb R$ be uncountable and let $L$ be the set of all the limit points of $S$.Assume on the contrary...
View ArticleIs a function uniformly continuous?
Let $F(t, p) = a(t)|p|$ for $(t, p) \in (0, T) \times \mathbb R$, where $T > 0$ and $0 < a(t) < 1$ is a Lipschitz continuous function.In this case, is $F$ uniformly continuous in $t$,...
View ArticleFind functions $u(x), v(x) \in L^1(\Bbb{R})$ such that their convolution...
This exercise (with the same wording as above) is found in my professor's lecture notes. I am having trouble finding two such functions.From Young's inequality, we know that, for any $p,q,r \in\ [1,...
View ArticleIs a composition $\phi(t,x(t))$ of a $C^{1}$ function $\phi(t,x)$ with a...
I'm reading a book about optimal control theory, but I don't understand an argument.Take any Lipschitz continuous arc $x \colon [S,T] \to \mathbb{R}^{n}$. It's true that $t\mapsto \phi(t,x(t))$ is a...
View ArticleEvaluate $\int\limits_0^\infty \frac{\ln^2(1+x)}{1+x^2}\ dx$
This problem was already solved here (in different closed form).But how can you prove $\ \displaystyle\int\limits_0^\infty\frac{\ln^2(1+x)}{1+x^2}\ dx=2\Im\left(\operatorname{Li}_3(1+i)\right)\ $Where...
View ArticleAnother Gambler's Ruin Problem
Let $(X_n)_n$ be a sequence of i.i.d. random variables with $$\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = -1) = 1/2.$$ Define $S_n = X_1 + \cdots + X_n$ for $n \geq 1$ and $S_0 = 0$.We can easily show that...
View ArticleImage of a line and a plane through affine map
Let $f:\mathbb{R}^3 \to \mathbb{R}^3, f(x,y,z) = (x+y+z,2x-y+3,3x+z+1)$, $$\pi : x+y-z = 1$$and $$d:\frac{x-1}{2} = \frac{y-1}{0} = \frac{z}{3}.$$I want to find $f(d)$ and $f(\pi)$.I found that...
View ArticleContinuity of optimal value of a functional on a Hilbert space
It is now cross-posted on overflow: https://mathoverflow.net/questions/477254/continuity-of-optimal-value-of-a-functional-on-a-hilbert-spaceLet $f:X\times Y\to \mathbb{R}$ be a bounded continuous...
View ArticleMoving the derivative inside the integral when the integral has a singularity
If $M(x,y)$ is differentiable on a bounded subset $\Omega\times \Omega\subset \mathbb{R}^2\times \mathbb{R}^2$, can we move the derivative under the integral sign in...
View ArticleIf $\nu \ll m$ and $f\in L^1(\mathbb{R}; m)$, does it then follow necessarily...
Let $m$ be a measure on the space $(X, \mathcal{F})$, $f\in L^1(\mathbb{R}; m)$ and $\nu$ be a measure such that $\nu\ll m$. Does it then follow that $f\in L^1(\mathbb{R};\nu)$? I have a feeling that...
View ArticleHow to formally prove that a $C^2$ boundary can't "oscillate" too much?
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^2$-boundary S meaning that for every $x\in S$ there exists an open neighborhood $V\subset \mathbb{R}^n$ containing $x$ and a real-valued...
View ArticleIf a homeomorphism preserves analyticity of curves, does it preserve...
Let $f: \mathbb R^n \rightarrow \mathbb R^n$ be a homeomorphism that sends every analytic curve to an analytic curve. Does it send every smooth curve to a smooth curve?I came to this problem when I was...
View Article