5,225
reputation
3
6

## Paras Khosla $(function() {$(".js-rank-badge").addSpinner().load("/users/rank?userId=478779"); });

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 $$0, the following expressions converges iff $$\exists M\in\mathbb{R}| M\ge(x_{n})^{p^{n}}$$.

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

0
2
questions
~223
people reached
• Jalandhar, Punjab, India
• Member for 3 years, 1 month
• 16 profile views
• Last seen 8 hours ago

Score 0
Posts 2
Score 0
Posts 1
Score 0
Posts 1
Score 0
Posts 1
Score 0
Posts 1

3

6