# Fourier Sine and Cosine Series Recall that the Fourier series of f(x) is defined by where We have the following result:

Theorem. Let f(x) be a function defined and integrable on interval .

(1)
If f(x) is even, then we have and (2)
If f(x) is odd, then we have and This Theorem helps define the Fourier series for functions defined only on the interval . The main idea is to extend these functions to the interval and then use the Fourier series definition.

Let f(x) be a function defined and integrable on . Set and Then f1 is odd and f2 is even. It is easy to check that these two functions are defined and integrable on and are equal to f(x) on . The function f1 is called the odd extension of f(x),
while f2 is called its even extension.

Definition. Let f(x), f1(x), and f2(x) be as defined above.

(1)
The Fourier series of f1(x) is called the Fourier Sine series of the function f(x), and is given by where (2)
The Fourier series of f2(x) is called the Fourier Cosine series of the function f(x), and is given by where Example. Find the Fourier Cosine series of f(x) = x for . and Therefore, we have  Example. Find the Fourier Sine series of the function f(x) = 1 for . Hence  Example. Find the Fourier Sine series of the function for . which gives b1 = 0 and for n > 1, we obtain Hence  Special Case of 2L-periodic functions.
As we did for -periodic functions, we can define the Fourier Sine and Cosine series for functions defined on the interval [-L,L]. First, recall the Fourier series of f(x) where for .
1.
If f(x) is even, then bn = 0, for . Moreover, we have and Finally, we have 2.
If f(x) is odd, then an = 0, for all , and Finally, we have The definitions of Fourier Sine and Cosine may be extended in a similar way. [Geometry] [Algebra] [Trigonometry ]
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