User:Dionyziz/Sandbox

Linking

 * Foo
 * /Foo
 * Buda/../Foo
 * Foo
 * ~

Number theory
$$ \phi(mn) $$

Math stuff
$$ \{ 1_5, 2_n, 10_{79} \} $$

$$ \frac{\displaystyle\sum_{i=0}^{32} 210 000 \lfloor \frac{50. 10^8}{2^i} \rfloor }{10^8} $$

$$ \epsilon' = \epsilon \frac{ts(block_n) - ts(block_{n-2016})}{\text{2 weeks}} $$


 * User:Dionyziz/Countable rationals
 * User:Dionyziz/Physics

Fifeth Delirium
Achiath ladre, fifeth delirium, ostiras de mire ochiath. Midre mode it es okun ebalath, osimas de lipis est. Ese tes vi mosiris de tos, fosium ost en ete nes. Katilas ves, me tathicas vis est vive sunt. Is es, ethisis, cas mones licis thesiris mis. Et kyra thes de vive ques mosiris ebal es. Fies sunt quere de it es kanibi ost sunte ese tes. Cent marvelis osun sunt et rival ete est. Opus fertis sanite de ikus verte lores ices vis.

Amore tis, okuris verte omniris. Amore tis, tilere fertis fies olte ires sunt, larite dei sunt. De kaliris fithes lithiris mes tilte feres nes, mi vive karti oputis fithes me ques.

Oniris - osiris - ostiris.

Lepita.

u(x)
Let $$ u(x) = a^i, a \in \mathbb{R}, a > 0 $$.

Then, let a number $$ k \in \mathbb{C} $$ so that $$ a^i = e^k $$

We have:


 * $$ a^i = e^k $$


 * $$ ln{a^i} = ln{e^k} $$


 * $$ iln{a} = k $$


 * $$ a^i = e^k = e^{ilna} $$

From Taylor series (the expansion of $$ e^k $$ also applies for $$ k \in \mathbb{C} $$) we have:


 * $$e^k = \sum_{n=0}^{\infty}{{1 \over n!}k^n} = e^{iln(a)} = \sum_{n=0}^{\infty}{{1 \over n!}{ilna}^n} = \sum_{n=0}^{\infty}{{1 \over n!}i^nln^n(a)}$$

But $$ a^i = e^k $$ and therefore:


 * $$ a^i = \sum_{n=0}^{\infty}{{i^n \over n!}ln^n(a) } $$

So that:


 * $$ u(x) = \sum_{n=0}^{\infty}{{i^n \over n!}ln^n(x) } $$


 * $$ x^i = u(x) = \lim_{p \to \infty} [\sum_{n=0}^p{{i^n \over n!}ln^n(x)}] $$

L'Hosptial's Rule
L'Hôpital's rule

Assymptotic Notation
A term of notable significance in algorithmics, mathematics-wise, is that of assymptotic comparison between functions.

Take two functions, a(x) and b(x) which we want to compare assymptotically*


 * If we had wanted to assymptically compare two functions of two or more variables, say, for instance f(x, y) and g(x, y), we would have to choose which argument to use for the comparison and set a constant value for the rest. Hence, if we chose the argument x for the comparison we would define j(x) = f(x, y1) and k(x) = g(x, y2) with some constants y1 and y2.