Abstract
Fourier series is an important mathematical concept. It is well known that we need too much computation to expand the function into Fourier series. The existing literature only pointed that its Fourier series is sine series when the function is an odd function and its Fourier series is cosine series when the function is an even function. And on this basis, in this paper, according to the function which satisfies different conditions, we give the different forms of Fourier series and the specific calculation formula of Fourier coefficients, so as to avoid unnecessary calculation. In addition, if a function is defined on [0,a], we can make it have some kind of nature by using the extension method as needed. So we can get the corresponding form of Fourier series.
Highlights
too much computation to expand the function into Fourier series
The existing literature only pointed that its Fourier series is sine series
we give the different forms of Fourier series
Summary
Definition 1 [1]-[3] Let f ( x) be an integrable function on [−l,l]. Are called the Fourier coefficients of f ( x). Definition 2 [1]-[5] Let f ( x) with the period 2l be an integrable function on [−l,l] , trigonometric series. Zhang with the Fourier coefficient are called Fourier series of f ( x) , denoted by f (x) a0 2. Lemma 1 [6] Let f ( x) be an integrable function on [−l,l] with period of 2l , the Fourier coefficients are ( ) calculated according to period of 2kl k ∈ N +. The calculation indicates there are same results between Fourier ( ) series with period of 2l and 2kl k ∈ N +
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