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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.

Definition 1 [

and are called the Fourier coefficients of

Definition 2 [

Lemma 1 [

series with period of

Theorem 1 Let

where

Proof It was clear that the period of

Let

So we get

Let

Therefore, we obtain

In the same way, we have

In a word, while

Thus in this case, the expansion reduces to

Theorem 2 Let

where

Proof The period of

Let

So we get

Let

We obtain

In the same way, we have

Thus in this case, the expansion reduces to

where

Theorem 3 Let

(1) While

where

(2) While

where

Proof (1) We use the method of periodic extension to

Because

Let

Let

We obtain

Thus in this case, the expansion reduces to

where

(2) In the same way, we can prove Theorem 3 (2).

Similarly, we can prove the following Theorem 4.

Theorem 4 Let

(1) While

where

(2) While

where

Suppose the function is defined on

use odd and periodic extension, we can get two forms of Fourier series as Theorem 3. If we use symmetry ex-

tension about the line

I would like to thank the referees and the editor for their valuable suggestions.