1807, the French mathematician Fourier wrote a basic paper on heat conduction, Heat Propagation, which was submitted to the Paris Academy of Sciences. However, after being reviewed by Lagrange, Laplace and Legendre, it was rejected by the Academy of Sciences. 18 1 1 year, he submitted a revised paper and won an Oscar, but it was not officially published. Fourier deduced the famous heat conduction equation in this paper, and found that the solution function can be expressed in the form of series composed of trigonometric functions when solving this equation, thus proposing that any function can be expanded into infinite trigonometric function series. Fourier series (that is, trigonometric series), Fourier analysis and other theories were founded.

1822, Fourier published the monograph "Analytical Theory of Heat" (Didot, Paris, 1822). This classic book developed the trigonometric series method applied by Euler and Bernoulli in some special cases into a rich general theory, and trigonometric series was later named after Fourier. Fourier uses trigonometric series to solve the heat conduction equation. In order to deal with the heat conduction problem in infinite region, the so-called "Fourier integral" is derived, which greatly promotes the study of boundary value problems of partial differential equations. But the significance of Fourier work goes far beyond this, which forces people to revise and popularize the concept of function, especially the discussion of discontinuous function; The convergence of trigonometric series stimulated the birth of set theory. Therefore, "thermal analysis theory" has influenced the process of analytical rigor in the whole19th century. Fourier 1822 became the lifelong secretary of the Academy of Sciences.

According to the principle of Fourier series, a periodic function can be expanded into the sum of a constant and a group of sine and cosine functions with the same period.

The periodic function f(t) with the period t satisfying Dirichlet condition can be expressed by the linear combination of trigonometric functions (Fourier series) at the following continuous points:

The above formula is called the Fourier series of f(t), where ω = 2π/t.

N is an integer, n & gt=0.

N is an integer, n>= 1.

At the discontinuity, the following equation holds:

A0/2 is the DC component of the signal f(t).

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C 1 is the amplitude of fundamental wave, and cn is the amplitude of n-th harmonic wave. C 1 is sometimes called the amplitude of the first harmonic. A0/2 is sometimes called the amplitude of the 0th harmonic.

The frequency of harmonics must be equal to an integer multiple of the fundamental frequency. Waves with 3 times the fundamental frequency are called third harmonics, waves with 5 times the fundamental frequency are called fifth harmonics, and so on. No matter how many harmonics, they are sine waves.