<p>I am a second-year CS undergrad at NIT Jalandhar. I have completed a course on Complex Analysis. I have developed interest for Mathematical Modelling of Physics and Geography.</p>
<p>My collaborative distance with every mathematician or researcher in general is <span class="math-container">$\int_{1}^{\infty} {1}/{x^p}\mathrm dx$</span> such that <span class="math-container">$p \le 1$</span>, but I'm trying hard to push <span class="math-container">$p$</span> out of <span class="math-container">$(-\infty, 1]$</span>.</p>
<hr />
<p><span class="math-container">$$\begin{aligned}&amp;\int \arctan\left(\frac{2\cos^2\theta}{2-\sin(2\theta)}\right)\sec^2\theta \mathrm d\theta =\int (\arctan u-\arctan(u-1) )\mathrm du , u=\tan\theta \end{aligned}$$</span></p>
<hr />
<p>Few results compiled:</p>
<p><span class="math-container">$1$</span>. (<strong>Gauss' circle problem</strong>) Number of ordered pairs <span class="math-container">$(x,y), x,y\in\mathbb{Z}:x^2+y^2\le r^2$</span> is given by <span class="math-container">$N(r)$</span>. <span class="math-container">$[.]$</span> denotes <span class="math-container">$\lfloor x\rfloor $</span>. <span class="math-container">$$\boxed{N(r)=1+4\sum_{i=0}^{\infty}\left(\left[\frac{r^2}{4i+1}\right]-\left[\frac{r^2}{4i+3}\right]\right)}$$</span></p>
<p><span class="math-container">$2$</span>. Number of points with integer coordinates strictly inside the triangle bounded by <span class="math-container">$x+y=n$</span> and the coordinate axes is given by <span class="math-container">$N(r)$</span>.</p>
<p><span class="math-container">$$\boxed{N(r)=\sum_{k=1}^{n-2}(n-k)=\sum_{k=1}^{n-2}k=\frac{(n-1)(n-2)}{2}}$$</span></p>
<p><span class="math-container">$3.$</span> (<strong>Glasser's master theorem</strong>)(<strong>Cauchy–Schlömilch transformation</strong>) (<a href="https://math.stackexchange.com/q/457231/478779">link to proof</a>) If <span class="math-container">$f(x)$</span> is a continuous function on <span class="math-container">$\mathbb{R}$</span> and the line integral of <span class="math-container">$f(x)$</span> on <span class="math-container">$\mathbb{R}$</span> exists and <span class="math-container">$a \gt 0$</span>.
<span class="math-container">$$\boxed{\int_{-\infty}^{+\infty}f(x)\mathrm dx=\int_{-\infty}^{+\infty}f\left(x-\frac{a}{x}\right)\mathrm dx}$$</span></p>
<p><span class="math-container">$4.$</span> (<strong>Frullani integral</strong>) <span class="math-container">$f(x)$</span> is a function over <span class="math-container">$x\ge 0$</span> and <span class="math-container">$\lim f(x)$</span> as <span class="math-container">$x\to \infty$</span> exists, then the Frullani definite integral can be evaluated as follows.</p>
<p><span class="math-container">$$\boxed{\int_{0}^{\infty}\frac{f(ax)-f(bx)}{x}\mathrm dx=\left(f(0)-\lim_{x\to\infty}f(x)\right)\ln \frac{b}{a}}$$</span></p>
<p><span class="math-container">$5$</span>. (<strong>Ramanujan's nested radical</strong>) Ramanujan gave the general result for the general nested radical which holds for all <span class="math-container">$a$</span>, <span class="math-container">$n$</span> and <span class="math-container">$x$</span> in <span class="math-container">$\mathbb{R}$</span>.</p>
<p><span class="math-container">$$\boxed{\sqrt{ax+(n+a)^2+x\sqrt{a(x+n)+(n+a)^2+(x+n)\sqrt{.}}}=x+n+a}$$</span></p>
<p><span class="math-container">$6$</span>. (<strong>Herschfeld's convergence theorem</strong>) For real, nonnegative terms <span class="math-container">$x_n$</span> and real <span class="math-container">$p$</span> with <span class="math-container">$0&lt;p&lt;1$</span>, the following expressions converges iff <span class="math-container">$\exists M\in\mathbb{R}| M\ge(x_{n})^{p^{n}}$</span>.</p>
<p><span class="math-container">$$\boxed{\lim_{k\to \infty}x_{0}+(x_{1}+(x_{2}+(\ldots +(x_{k})^{p})^{p})^{p})^{p}}$$</span></p>
<p><a href="https://stackexchange.com/users/11723754/paras-khosla"><img src="https://stackexchange.com/users/flair/11723754.png" width="208" height="58" alt="profile for Paras Khosla on Stack Exchange, a network of free, community-driven Q&amp;A sites" title="profile for Paras Khosla on Stack Exchange, a network of free, community-driven Q&amp;A sites" /></a></p>

Paras Khosla