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by r9m
Fri Jul 28, 2017 9:01 pm
Forum: Calculus
Topic: A series involving Harmonic numbers
Replies: 2
Views: 1475

Re: A series involving Harmonic numbers

This is closely related to problem 11993 from American Mathematical Monthly Journal . Now the problem presented in that integral form can be dealt with rather easily and one avoids having to calculate the last Euler Sum I left off .. :) There's an old blog post of mine with spoilers for this problem...
by r9m
Mon May 29, 2017 7:28 am
Forum: Calculus
Topic: Root integral
Replies: 1
Views: 1016

Re: Root integral

Making the change of variable $x \mapsto x^2$, the integral changes to: $\displaystyle \mathcal{J}(k) = \frac{1}{2}\int_1^4 \frac{x^k}{\sqrt{(x-1)(4-x)}}\,dx$. Let, $\displaystyle f(z) = \frac{z^k}{\sqrt{(z-1)(4-z)}}$ and consider the integral of $f(z)$ about a positively oriented dumbell contour ($...
by r9m
Sat May 27, 2017 5:48 pm
Forum: Functional Analysis
Topic: Lemma
Replies: 2
Views: 1420

Re: Lemma

Since, $U : H \to H$ satisfies $\lVert U \rVert \le 1$, then $$\left<(I-U)h,h\right> = \lVert h \rVert^2 - \left< Uh,h \right> \ge \lVert h \rVert^2 (1 - \lVert U \rVert) \ge 0 \text{ for all } h \in H$$ We claim that, $N(I-U) = R(I-U)^{\perp}$ If, $h \in N(I-U)$ then, $\left<(I-U)(h - th'), h - th'...
by r9m
Thu May 25, 2017 8:10 am
Forum: Calculus
Topic: An infinite product
Replies: 1
Views: 940

Re: An infinite product

Using the infinite product for cosines: $\displaystyle \cos (\pi x) = \prod\limits_{n=1}^{\infty} \left(1-\frac{x^2}{\left(n-\frac{1}{2}\right)^2}\right)$, \begin{align*}\prod\limits_{n=1}^{\infty} \left(1 + \frac{x^2}{n^2+n-1}\right) &= \prod\limits_{n=2}^{\infty} \frac{\left(n-\frac{1}{2}\right)^2...
by r9m
Wed May 24, 2017 8:11 pm
Forum: Measure and Integration Theory
Topic: A problem from Rudin's Real and Complex Analysis
Replies: 0
Views: 985

A problem from Rudin's Real and Complex Analysis

Problem : Let, $f$ be a real-valued Lebesgue measurable function on $\mathbb{R}^k$, prove that there exists Borel functions $g$ and $h$ such that $\mu_k(\{x \in \mathbb{R}^k: g(x) \neq h(x)\}) = 0$ and $g(x) \le f(x) \le h(x)$ a.e. $[\mu_k]$. (where, $\mu_k$ is the Lebesgue measure on $\mathbb{R}^k...
by r9m
Sun May 21, 2017 8:48 am
Forum: Linear Algebra
Topic: Linear isometry
Replies: 3
Views: 1768

Re: Linear isometry

An alternative approach: It suffices to show that $f$ is surjective and preserves mid-points, i.e., if $x,y \in \mathbb{R}^2$ then, $$f\left(\frac{x+y}{2}\right) = \frac{f(x) + f(y)}{2}$$ then, by simple induction one can extend the above identity to $f(rx+(1-r)y) = rf(x) + (1-r)f(y)$ where, $r$ is ...
by r9m
Sat May 20, 2017 11:24 am
Forum: Meta
Topic: New sub-forum for Measure and Integration theory
Replies: 1
Views: 1172

New sub-forum for Measure and Integration theory

Would it be a possible to have a sub-forum for 'Measure and Integration Theory' that is covered in most graduate classes?
by r9m
Sat May 20, 2017 11:07 am
Forum: PDE
Topic: An integral identity for Harmonic Functions
Replies: 0
Views: 1084

An integral identity for Harmonic Functions

If $u$ be a Harmonic Function in a open connected set $\Omega \subset \mathbb{R}^n$ and $\overline{B(x_0,R)} \subset \Omega$ (the closed ball of radius $R$ centered at $x_0 \in \Omega$). (i) Show that: $$\int_{\partial B(0,1)} u(x_0 + ry)u(x_0 + Ry)\,d\sigma(y) = \int_{\partial B(0,1)} u^2(x_0 + cy)...
by r9m
Thu May 18, 2017 11:24 pm
Forum: Functional Analysis
Topic: Closed linear subspace
Replies: 1
Views: 1054

Re: Closed linear subspace

If, $p = 1$, then $E_1$ is clearly a closed subspace of $L^1([0,\infty))$ as: If $f_n \overset{L^1}{\longrightarrow} f$ for $f_n \in E_1$, then $\displaystyle \left|\int_{0}^{\infty} f\,dx\right| = \left|\int_{0}^{\infty} (f-f_n)\,dx\right| \le \int_{0}^{\infty} \left|f - f_n\right|\,dx \underset{n ...
by r9m
Thu May 18, 2017 7:58 pm
Forum: Functional Analysis
Topic: An application of Banach-Steinhaus theorem
Replies: 0
Views: 849

An application of Banach-Steinhaus theorem

Suppose $1 < p < \infty$ and $p,q$ are conjugate indices, i.e., $\displaystyle \frac{1}{p} + \frac{1}{q} = 1$. If $(\mathbb{R},\mu)$ be the Lebesgue measure. If the following properties hold: (i) $\displaystyle g \in L^{q}_{\text{loc}}(\mu)$. (ii) $\displaystyle \int_{\mathbb{R}} |fg|\,d\mu < \infty...