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#23 Fundamentals--Fourier transform 常識集_傅立葉轉換

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In this post, we will demonstrate how to use Fourier transform to solve differential equations!

In this post, we will demonstrate how to use Fourier transform to solve differential equations!

其實在介紹Fourier transform之前本站就已經使用過了，在#13酵素的噴射引擎_中二，我們利用傅立葉轉換來解擴散方程式。關於傅立葉轉換的基本觀念，小編找到一個大陸人寫的文章，講解得非常平易近人(佩服至極)，所以在此只針對一些會用到的數學性質做簡單的補充說明。

(作者：韩昊)

即使是學過的人也很推薦，很多解說圖畫的很美。

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因為本站主要的關注點不是在訊號處理，所以我們比較常用到的是利用傅立葉轉換解微分方程。就如我們推薦的文章所述，傅立葉轉換可以把解微分方程的過程變成四則運算，我們現在要來把這一連串的數學描述清楚。

Today we will use Fourier transform to solve differential equations because Fourier transform could make the problem as simple as arithmetic. How does it work?

Today we will use Fourier transform to solve differential equations because Fourier transform could make the problem as simple as arithmetic. How does it work?

我們要從週期性的訊號開始。在上面的文章已經提到，對一週期為2pi的函數都可以寫成如下的形式：

We will start from a periodic function. Any periodic function with period 2pi could be written in the form as below:

因為正弦函數和餘弦函數具有下面的性質：We will start from a periodic function. Any periodic function with period 2pi could be written in the form as below:

Since sine and cosine functions possess the following properties:

所以將上式乘以sin或cos之後再積分從-pi到pi，我們可以知道

So if we multiply our functions with sine or cosine and integrate it from -pi to +pi, we will get:

那對於一週期為2L的函數，我們就可以寫成：

as for a periodic function with period 2L, it could be written as:

也就是說，假如我們讓wn = n*pi/L，我們可以把f(x)寫成像這樣子：

That is to say, if we let wn = n*pi/L, f(x) could then be written as:

我們假設That is to say, if we let wn = n*pi/L, f(x) could then be written as:

Assume that:

所以我們可以把f(x)寫成：

And f(x) could now become something like this:

And f(x) could now become something like this:

If L approaches infinity, the first term of the above equation becomes ignorable and we could change the summation into integral:

也就等於：

That is to say:

利用三角函數的和角公式，並且利用sin(wx-wv)是w奇函數，cos(wx-wv)是w的偶函數的事實，我們可以再化簡成：

Using the addition formula of trigonometric functions, and by knowing the fact that sin(wx-wv) & cos(wx-wv) are odd function and even function of w, respectively, we could further simplify the above equation:

到這裡我們已經很明顯可以看出來，f(v)可以先經過傅立葉轉換變成F(w)，再經過反傅立葉轉換變回f(x)，所以我們可以定義傅立葉轉換和反傅立葉轉換如下：

From the above we could notice that if we Fourier transform f(v) into another function, F(w), it could be inversely transformed back to f(x). That makes the definition of Fourier transform and inverse Fourier transform:

傅立葉轉換具有一些很好的性質，其中最棒的就是他讓微分變得很簡單：

There are lots of properties associated with Fourier transform. One of the best is that it makes differentiation relatively easy:

我們以後會頻繁地使用這個性質。需要例子的人，請參考#13酵素的噴射引擎_中二。

One of the example would be how we solve diffusion equation in #13 How enzymes propel themselves-III.

### Comments

小小勘誤一下，網誌中第三和第四個方程式的右手邊應該寫成π*δmn (Kronecker delta), 若m=n, δmn=1；否則δmn=0。

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