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Paras Khosla

I'm a student currently in 12th standard. I have completed a course on Complex Analysis. I have developed interest for Mathematical Modelling of Physics and Geography.

My collaborative distance with every mathematician or researcher in general is $\int_{1}^{\infty} {1}/{x^p}\mathrm dx$ such that $p \le 1$, but I'm trying hard to push $p$ out of $(-\infty, 1]$.


Few results compiled:

$1$. (Gauss' circle problem) Number of ordered pairs $(x,y), x,y\in\mathbb{Z}:x^2+y^2\le r^2$ is given by $N(r)$. $[.]$ denotes $\lfloor x\rfloor $. $$\boxed{N(r)=1+4\sum_{i=0}^{\infty}\left(\left[\frac{r^2}{4i+1}\right]-\left[\frac{r^2}{4i+3}\right]\right)}$$

$2$. Number of points with integer coordinates strictly inside the triangle bounded by $x+y=n$ and the coordinate axes is given by $N(r)$.

$$\boxed{N(r)=\sum_{k=1}^{n-2}(n-k)=\sum_{k=1}^{n-2}k=\frac{(n-1)(n-2)}{2}}$$

$3.$ (Glasser's master theorem)(Cauchy–Schlömilch transformation) (link to proof) If $f(x)$ is a continuous function on $\mathbb{R}$ and the line integral of $f(x)$ on $\mathbb{R}$ exists and $a \gt 0$. $$\boxed{\int_{-\infty}^{+\infty}f(x)\mathrm dx=\int_{-\infty}^{+\infty}f\left(x-\frac{a}{x}\right)\mathrm dx}$$

$4.$ (Frullani integral) $f(x)$ is a function over $x\ge 0$ and $\lim f(x)$ as $x\to \infty$ exists, then the Frullani definite integral can be evaluated as follows.

$$\boxed{\int_{0}^{\infty}\frac{f(ax)-f(bx)}{x}\mathrm dx=\left(f(0)-\lim_{x\to\infty}f(x)\right)\ln \frac{b}{a}}$$

$5$. (Ramanujan's nested radical) Ramanujan gave the general result for the general nested radical which holds for all $a$, $n$ and $x$ in $\mathbb{R}$.

$$\boxed{\sqrt{ax+(n+a)^2+x\sqrt{a(x+n)+(n+a)^2+(x+n)\sqrt{.}}}=x+n+a}$$

$6$. (Herschfeld's convergence theorem) For real, nonnegative terms $x_n$ and real $p$ with $0<p<1$, the following expressions converges iff $\exists M\in\mathbb{R}| M\ge(x_{n})^{p^{n}}$.

$$\boxed{\lim_{k\to \infty}x_{0}+(x_{1}+(x_{2}+(\ldots +(x_{k})^{p})^{p})^{p})^{p}}$$

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