Articles of calculus

Maximum area of a isosceles triangle in a circle with a radius r

As said in the title, I’m looking for the maximum area of a isosceles triangle in a circle with a radius $r$. I’ve split the isosceles triangle in two, and I solve for the area $A=\frac{bh}{2}$*. I have made my base $x$, and solve for the height by using the Pythagorean theorem of the smaller […]

Computing $\int (1 – \frac{3}{x^4})\exp(-\frac{x^{2}}{2}) dx$

How does one compute $$\int \left(1 – \frac{3}{x^4}\right)\exp\left(-\frac{x^{2}}{2}\right) dx?$$ Mathematica gives $(x^{-3} – x^{-1})\exp(-x^2/2)$ which indeed the correct answer, but how does one get there? Integration by parts gives $(y + y^{-3})e^{-y^2/2} + \int (y^2 + y^{-2})e^{-y^2/2} dy$, but I’m not sure what to do with the integral.

Non-centered Gaussian moments

I would like to find a (closed nice) expression for the non-centered Gaussian moments with mean $\mu$ and variance $\sigma$. In found something in wikipedia: $$E|X|^p = \sigma^p 2^{p/2} \frac{\Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi}} {}_1 F_1 \left(-\frac{1}{2}p, \frac{1}{2}, -\frac{1}{2}(\mu/\sigma)^2\right)$$ but I don’t understand this “confluent hypergeometric function” ${}_1F_1$ because it doesn’t seem to be well-defined for negative […]

Proving $x\leq \tan(x)$

How can i prove that $x\leq \tan(x)$ for any $x$ in $[0,\frac{\pi}{2})?$

Extending Cauchy's Condensation Test

The Cauchy test states $\sum a_n$ converges $\iff$ $\sum 2^k a_{2^k}$ converges. In Rudin 3ed, the (excerpt) proof is outlined as follows Can I modify (8) so we get For $n < 3^k$ $$s_n = (a_1 + a_2 + a_3) + (a_4 + \dots + a_9) \leq 3a_1 + 3^2a_3 + \dots + 3^{k+1} a_{3^k} […]

Hessian matrix for convexity of multidimensional function

To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the intuition behind the claim that, if the Hessian $H$ of a multidimensional differentiable function $f(x_1,…,x_n)$ is positive semi-definite, it must be […]

What is the relation between Fourier's Inversion theorem and the Dirac-Delta function?

This is a direct quote from page 472 of this book: From Fourier’s Inversion theorem $$f(t)= \int_{-\infty}^\infty f(u) \, \mathrm{d}{u} \left( \frac{1}{2\pi}\int_{-\infty}^\infty e^{-i\omega(t-{u})} \,\mathrm{d}\omega \right) \tag{1}$$ comparison of $(1)$ with the Dirac-Delta property: $$f(a)= \int f(x) \, \mathrm{d}x \, \delta(x-a)$$ shows we may write the $\delta$ function as $$\delta(t-u)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{i\omega(t-{u})} \, \mathrm{d}\omega$$ My question is […]

Equivalent Cauchy sequences.

Hi everyone I’m having a bad time with two questions in the Analysis book of Terry Tao. I finally finished one of the exercises and I’m wondering if the next reasoning is correct or maybe needs some changes: Definitions: Two sequence are equivalence $\iff$ $(\forall \varepsilon \in \mathbb{Q}^+\,) ( \, \exists N\in \mathbb{N}\,) \text{ s.t. […]

f is monotone and the integral is bounded. Prove that $\lim_{x→∞}xf(x)=0$

Question $f : [0,\infty] \to \Bbb R $ is monotone and $\displaystyle∫^∞_0f(x)\,dx$ converges. Note: we also proved before $\lim_{x→∞}f(x)=0$ Show that even $\lim_{x→∞}xf(x)=0$ Thanks!

Closed form for $\int \frac{1}{x^7 -1} dx$?

I want to calculate: $$\int \frac{1}{x^7 -1} dx$$ Since $\displaystyle \frac{1}{x^7 -1} = – \sum_{i=0}^\infty x^{7i} $, we have $\displaystyle(-x)\sum_{i=0}^\infty \frac{x^{7i}}{7i +1} $. Is there another solution? That is, can this integral be written in terms of elementary functions?