In different research fields, Fourier transform has many different variants, such as continuous Fourier transform and discrete Fourier transform. Firstly, Fourier analysis is proposed as a tool for thermal process analysis.
Laplace transform is an integral transform commonly used in engineering mathematics, also known as Laplace transform. Laplace transform is a linear transformation, which can transform a function with real number t(t≥ 0) into a function with complex number s.
Laplace transform is widely used in many engineering and scientific research fields, especially in mechanical systems, electrical systems, automatic control systems, reliability systems and random service systems.
Generally speaking, if the word "Fourier transform" is not preceded by any qualifier, it means "continuous Fourier transform". "continuous Fourier transform" represents the square integrable function as the integral form of complex exponential function;
The above formula actually represents the inverse transformation of continuous Fourier transform, that is, the function in time domain is expressed as the integral of the function in frequency domain. On the contrary, its forward transformation is just a function of frequency domain. Expressed as an integral form of a function in the time domain.
Generally, a function can be called an original function, while a function is called an image function of Fourier transform, and the original function and the image function form a Fourier transform pair.
When it is an odd-numbered function (or even-numbered function), the other chord (or sine) components are zero, and the transformation at this time can be called cosine transformation (or sine transformation).