# Fourier Series

• I. N. Bronshtein
• K. A. Semendyayev

## Abstract

Fundamental concepts. In many problems (differential equations, vibration theory) it is sometimes necessary to replace, accurately or approximately, a given periodic function f(x) with period T by a trigonometric sum
$$s_{n}(x)=\frac{1}{2}\;a_{0}+a_{1}\;\textup{cos}\;\omega x+a_{2}\;\textup{cos}\;2\omega x+\cdots +a_{n}\;\textup{cos}\;n\omega x+b_{1}\;\textup{sin}\;\omega x+b_{2}\;\textup{sin}\;2\omega x+\cdots +b_{n}\;\textup{sin}\;n\omega x$$
, where $$\omega =\frac{2\pi} {T}$$ (if T = 2π, then ω = 1). The sum s n (x) is the best approximation of the function f(x) (in the sense of p. 728), if we take for a k and b k the so-called Fourier coefficients defined by the following formulas due to Euler (Euler formulas):
$$a_{k}=\frac{2}{T}\int\limits_{0}^{T}f(x)\;\textup{cos}\;k\omega x\;dx=\frac{2}{T}\int\limits_{x_{0}}^{x_{0}+T}f(x)\;\textup{cos}\;k\omega x\;dx=\frac{2}{T}\int\limits_{0}^{T/2}[f(x)+f(-x)]\;\textup{cos}\;k\omega x\;dx,\;\;\;k=0,1,2,\cdots ,n$$
,
$$b_{k}=\frac{2}{T}\int\limits_{0}^{T}f(x)\;\textup{sin}\;k\omega x\;dx=\frac{2}{T}\int\limits_{x_{0}}^{x_{0}+T}f(x)\;\textup{sin}\;k\omega x\;dx=\frac{2}{T}\int\limits_{0}^{T/2}[f(x)+f(-x)]\;\textup{sin}\;k\omega x\;dx,\;\;\;k=1,2,\cdots ,n$$
.

## Keywords

Fourier Series Periodic Function Trigonometric Series Harmonic Vibration Vibration Theory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

1. (1).
At points of discontinuity we assume $$f(x)=\frac{f(x-0)+f(x+0)}{2}$$.Google Scholar
2. (1).
The coefficient b6 can be discarded, for, as can easily be observed, the corresponding term of the series is immaterial for the values of the function at the considered points.Google Scholar