> For the complete documentation index, see [llms.txt](https://docs.rosin-tech.com/latest/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.rosin-tech.com/latest/orchestra-vst/orchestravst-shi-yong-shou-ce/shu-wei-xun-hao-chu-li-x-shu-wei-xiao-guo-qi/ou-la-gong-shi.md).

# 歐拉公式

在高中數學有提到複數， $$z = a +bi$$ ，極式為 $$z = r(cos\theta + isin\theta)$$，如下圖所示： 

<figure><img src="https://i.imgur.com/zltVoSw.png " alt=""><figcaption></figcaption></figure>

虛數 $$i$$ 這個概念在中學時期就接觸，一開始我們只知道它是 `-1` 的平方根，但 $$i$$ 真正的意義是什麼呢?&#x20;

<figure><img src="https://i.imgur.com/xrUAaSA.jpg" alt=""><figcaption></figcaption></figure>

這裡有一條數軸，在數軸上有一個紅色的線段，它的長度是 `1`。當它乘以 `3` 時，它的長度發生變化，變成藍色的線段，而當它乘以 `-1` 時，就變成綠色的線段，或者說線段在數軸上圍繞原點旋轉 180 度。

我們知道乘 `-1` 其實是乘兩次 $$i$$ 使線段旋轉 180 度，那麼乘一次 $$i$$ 是什麼呢？很簡單，旋轉 90 度。&#x20;

<figure><img src="https://i.imgur.com/rArsvx9.jpg" alt=""><figcaption></figcaption></figure>

同時，我們得到一個垂直的虛數軸。實數軸與虛數軸共同搆成一個複數的平面，也稱複平面。於是我們可發現，乘上虛數 $$i$$ 的作用就是在複平面之上進行「旋轉」。

那歐拉公式到底是什麼呢？

$$
\begin{gather\*} e^{ix} = \cos x + i\sin x \end{gather\*}
$$

這個公式關鍵的作用，是**將弦波統一成了簡單的指數形式**。我們來看看圖像上的涵義：&#x20;

<figure><img src="https://i.imgur.com/bMBjHXO.jpg" alt=""><figcaption></figcaption></figure>

歐拉公式所描繪的，是一個隨著時間變化，在複平面上做圓周運動的點，隨著時間的改變，在時間軸上就成了一條螺旋線。如果只看它的實數部分，也就是螺旋線在左側的投影，就是一個最基礎的餘弦函數。而右側的投影則是一個正弦函數。


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