Supplementary mathematics/Fourier series

The Fourier series is a periodic expansion for functions such as f(x) in terms of the sum of the infinity of the sine and cosine functions, and the expansion is exponential. is also used in the Fourier series. The study of the Fourier series is one of the calculus of calculus and is known as the analysis of harmonics. Of course, this topic can be an arbitrary tonic function in a trigonometric set. and integral to match.

In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved for a single sinusoid, the solution of an arbitrary function is immediately available by expressing the original function in Fourier form. Is. Connect the series and then solve for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytical solutions.

Any set of functions that form a complete orthogonal system has a generalized Fourier series similar to the Fourier series. For example, using the orthogonality of the roots of a Bessel function of the first kind yields a so-called Fourier-Bessel series.

Fourier series for periodic functions
For example, expressions such as sine, cosine and exponential eikx can be used to define the Fourier series. It provided square waves (1 or 0 or -1) which are good examples, along with delta functions in the derivative. In order to present a spike and a step function, and a sloping surface, etc., Bayez looked at how their expansion should be found theoretically and mathematically. You should start with sin x first. This Fourier series theory has a period of 2π of sin(x + 2π) = sin x. It is an odd function because sin(-x) = - sin x, and vanishes at x = 0 and x = π. Every sin nx function has these three properties, and Fourier looked at infinite combinations of sines

$$S(x)=b_1 \sin x+b_2 \sin 2x+b_3 \sin 3x+...=\sum_{k=1}^{N} b_n \sin nx $$

If we assume that the numbers in the Fourier series decrease quickly enough, the set S(x) will have three characteristics. In this hypothesis, the importance of the decay rate and code is predicted.

Periodic S(x + 2π) = S(x) Odd S(−x) = −S(x) S(0) = S(π) = 0

200 years ago, Joseph Fourier, a French mathematician, expanded the Fourier series with an interesting suggestion. Joseph Fourier realized that the series of the function S(x) with those properties can be written as an infinite series. He expressed sine and cosine. This idea started the huge and important development and development of the Fourier series. The first step for us and you in the Fourier series is to calculate the number bk that multiplies sin kx or cos kx from S(x).

If we assume that the expansion of $$S(x)=\sum b_n \sin(nx)$$, both sides can be multiplied by sin kx. If it is integrated from 0 to π, it is integral:

$$\int_{0}^{\pi}S(x)=\sin(kx)dx=\int_{0}^{\pi} b_1\sin(x) \sin(kx)dx+\int_{0}^{\pi} b_2\sin(2x) \sin(kx)dx+...+\int_{0}^{\pi} b_k \sin(kx) \sin(kx)dx$$

On the right, all integrals are zero except for the highlighted integral with n = k.This "perpendicular" feature will dominate the entire chapter. They make sinuses 90◦ angles in function space, when their inner products are integral from 0 to π:

$$\int_{0}^{\pi} \sin(nx) \sin(kx)dx $$