Fourier integral is a transformation of integral in the operation process, which comes from the Fourier integral representation of function. The main content of Fourier analysis is to study many properties of functions by means of Fourier transform. Fourier transform has important applications in mathematics, physics and engineering technology. [1]
1、 Basic definitions and theorems
Basic definitions : If function f (x) meets the condition
① Continuous or only finite in any finite interval Type I breakpoint , and there are only limited extreme values; ② It is absolutely integrable on (- ∞,+∞), that is, finite; Then define [f (x) → C (ω)]
Is the (complex) Fourier transform of f (x); Note that C (ω)=F [f (x)]=f (ω), and call C (ω) the (complex) Fourier transform image function.
theorem : Based on the above definition, it can be proved that
(At the breakpoint, the integral on the right converges to the average of the left and right limits of f (x) at that point). This integral is called the Fourier complex integral of f (x); F (x) is the original function of C (ω) (inverse Fourier transform C (ω) → f (x)). Frequently recorded 。 II Fourier integral in real form
The form of real function corresponding to the theorem is:
It is called the (real form) Fourier integral of f (x). among The real Fourier transform is called f (x). 
