# Quotations & equations of the month

Here you can find citations and equations already displayed in the widget in the right column.

##### October 2018 :

“A mathematician is a machine that turns coffee into theorems.” – P. Erdös

Ramanujan’s Pi Serie : $$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty}{\frac{(4k)!(1103+26390k)}{(k!)^{4} 396^{4k}}}$$

##### November 2018 :

“A problem can be solved by thinking about how it was created.” – A. Einstein

Eulerian product : $$\zeta (s) = \prod_{p \in \mathcal{P}}^{+\infty}{\frac{1}{1-p^{-s}}}$$

##### December 2018 :

“It is better to aim the perfection and miss it rather than aim the imperfection and reach it.” – B. Russell

Newton’s binomial formula: $$(a+b)^{n} = \sum_{k=0}^{n}{{n\choose{k}}a^{k}b^{n-k}}$$

##### January 2019 :

“Mathematics consists to prove obvious things with complex means.” – G. Polya

Hilbert inequality: $$a_i,b_i (0 \leq i\leq n)\in \mathbb{C}, \Bigg | \sum_{k,j=0}^n \frac{a_k \overline{b_j}}{1+j+k} \Bigg | \leq \pi \sqrt{\sum_{p=0}^n |a_p|^2} \sqrt{\sum_{p=0}^n |b_p|^2}$$

##### February 2019 :

“Doing mathematics, it’s often getting lost into a jungle and try to gather all information to find new ways. With luck, we get out of this.” – M. Mirzakhani

Arc length in Riemannian metric:

$$\ell=\int_a^b\|\gamma'(t)\| {\mathrm d} t=\int_a^b\sqrt{g(\gamma'(t),\gamma'(t))} {\mathrm d} t$$

##### March 2019 :

“Mathematicians do not study objects, but relations between these objects.” – H. Poincaré

Cauchy’s repeated integral formula ($$f$$ is a real continuous function):

$$f^{[n]}(x) = \frac{1}{(n-1)!}\int_{a}^{x}{(x-y)^{n-1}f(t)\mathrm{d}t}$$

##### April 2019 :

“The art to ask right questions, in mathematics, is more important than the art to solve them.” – G. Cantor

Gauss’ integral:

$$\int_{\mathbb{R}}{\mathrm{e}^{-\alpha x^2}}\mathrm{d}x = \sqrt{\frac{\pi}{\alpha}}$$

##### May 2019 :

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” – J. Von Neumann

Poisson’s probability law ($$k \in \mathbb{N}, \lambda > 0$$) :

$$p(k) = P(X = k) = \frac{\lambda^{k}}{k!}\mathrm{e}^{-\lambda}$$

##### June 2019 :

“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” – W.P.Thurston

Volume of a $$n$$-dimension euclidian ball ($$k = \frac{n}{2}$$):

$$V_n = \frac{\pi^{k}R^n}{\Gamma(k + 1)}$$

##### January 2020 :

“No great discovery was ever made without a bold guess.” – I. Newton

Stirling equivalent :

$$n! \underset{+\infty}{\sim} \sqrt{2\pi n}(\frac{n}{e})^n$$

##### April 2020 :

“In mathematics, we don’t understand things, we’re settling into.” – J.von Neumann

Rank-Nullity Theorem ($$u \in \mathcal{L}(E,F)$$ ) :

$$\mathrm{rg}(u) = \mathrm{dim}\ E \ – \ \mathrm{dim}\ \mathrm{Ker}(u)$$

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