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Fourier Sine and Cosine Series

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- 1. TARUN GEHLOT (B.E, CIVIL HONORS) 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
- 2. TARUN GEHLOT (B.E, CIVIL HONORS) 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 integrableon . 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
- 3. TARUN GEHLOT (B.E, CIVIL HONORS) (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 . Answer. We have and Therefore, we have
- 4. TARUN GEHLOT (B.E, CIVIL HONORS) Example. Find the Fourier Sine series of the function Answer. We have Hence TARUN GEHLOT (B.E, CIVIL HONORS) Find the Fourier Sine series of the function f(x) = 1 for .
- 5. TARUN GEHLOT (B.E, CIVIL HONORS) Example. Find the Fourier Sine series of the function Answer. We have 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 for functions defined on the interval [ TARUN GEHLOT (B.E, CIVIL HONORS) Find the Fourier Sine series of the function for > 1, we obtain periodic functions. 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( . the Fourier Sine and Cosine series (x)
- 6. TARUN GEHLOT (B.E, CIVIL HONORS) 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
- 7. TARUN GEHLOT (B.E, CIVIL HONORS)

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