拉普拉斯變換與傅立葉變換
拉普拉斯變換和傅立葉變換都是積分變換，是求解數學模型物理系統最常用的數學方法。這個過程很簡單。一個複雜的數學模型被轉換成一個簡單的，可解的模型使用一個積分變換。對較簡單的模型進行求解後，應用逆積分變換，從而為原模型提供解。
例如，由於大多數物理系統產生微分方程，因此可以通過積分變換將其轉換為代數方程或低階易解微分方程。那麼解決問題就會變得更容易了。
什麼是拉普拉斯變換？
Given a function f (t) of a real variable t, its Laplace transform is defined by the integral (whenever it exists), which is a function of a complex variable s. It is usually denoted by L {f (t)}. The inverse Laplace transform of a function F(s) is taken to be the function f (t) in such a way that L {f (t)} = F(s), and in the usual mathematical notation we write, L -1{F(s)} = f (t). The inverse transform can be made unique if null functi*** are not allowed. One can identify these two as linear operators defined in the function space, and it is also easy to see that, L -1{ L {f (t)}} = f (t), if null functi*** are not allowed.
下表列出了一些最常見函數的拉普拉斯變換。
什麼是傅里葉變換？
Given a function f (t) of a real variable t, its Laplace transform is defined by the integral (whenever it exists), and is usually denoted by F { f (t)}. The inverse transform F -1{F(α)} is given by the integral . Fourier transform is also linear, and can be thought of as an operator defined in the function space.
Using the Fourier transform, the original function can be written as follows provided that the function has only finite number of discontinuities and is absolutely integrable. 
- 函數f（t）的傅里葉變換被定義為，而它的拉普拉斯變換被定義為。
- Fourier變換隻定義為所有實數定義的函數，而Laplace變換不要求函數在設為負實數時定義。
- 傅里葉變換是拉普拉斯變換的一個特例。可以看出，對於非負實數，兩者是一致的。（即，在拉普拉斯取s為iα+β，其中α和β為實數，eβ=1/√（2ᴫ））
- 每個有傅立葉變換的函數都有拉普拉斯變換，但反之亦然。
