拉普拉斯变换与傅立叶变换

拉普拉斯变换和傅立叶变换都是积分变换，是求解数学模型物理系统最常用的数学方法。这个过程很简单。一个复杂的数学模型被转换成一个简单的，可解的模型使用一个积分变换。对较简单的模型进行求解后，应用逆积分变换，从而为原模型提供解。

例如，由于大多数物理系统产生微分方程，因此可以通过积分变换将其转换为代数方程或低阶易解微分方程。那么解决问题就会变得更容易了。

什么是拉普拉斯变换？

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见函数的拉普拉斯变换。

什么是傅里叶变换？

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.

• 函数f（t）的傅里叶变换被定义为，而它的拉普拉斯变换被定义为。
• Fourier变换只定义为所有实数定义的函数，而Laplace变换不要求函数在设为负实数时定义。
• 傅里叶变换是拉普拉斯变换的一个特例。可以看出，对于非负实数，两者是一致的。（即，在拉普拉斯取s为iα+β，其中α和β为实数，eβ=1/√（2ᴫ））
• 每个有傅立叶变换的函数都有拉普拉斯变换，但反之亦然。