欧拉公式の推导
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欧拉公式の推导

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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?

This answer is Taloyr
$$
\underset{x \to x_{0}}{\lim}f(x)=\sum_{i=0}^{n}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n+o(x^n)
$$
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 $e^{ix}=cosx+isinx$

看了这么久,不请我喝杯茶吗?

2019-01-21