| | 分类于

### Euler’s formula:

$$\Large{e^{ix}=cosx+isinx \xrightarrow{x=\pi} e^{\pi i}+1=0}$$
This formula called created by god, because it contain $e\quad\pi\quad1$.

### How the formula come from?

$$\underset{x \to x_{0}}{\lim}f(x)=\sum_{i=0}^{n}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o(x^n)$$
when $x_0$ equals $0$, the formula transform to
$$\underset{x \to 0}{\lim}f(x)=\sum_{i=0}^{n}\frac{f^{(n)}(0)}{n!}x^n+o(x^n)$$
Here’s three items about $e^x\quad sinx\quad cosx$

$\small{e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+…+\frac{x^n}{n!}}$

$\small{sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}+O(x^5)}$

$\small{cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^4)}$

At the first equaltion we replace the partation of $x$ with $ix$
and the orgainal equaltion become to

$e^{ix}=1+ix-\frac{x^2}{2!}-i\frac{x^3}{3!}+\frac{x^4}{4!}+i\frac{x^5}{5!}+…+\frac{x^{in}}{in!}\Rightarrow e^{ix}=\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}+O(x^4)\right)+\left(ix-i\frac{x^3}{3!}+i\frac{x^5}{5!}+O(x^5)\right)$

that is $e^{ix}=cosx+isinx$

2019-01-21