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# Fourier series basic results

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Fourier series basic results

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### Fourier series basic results

1. 1. TARUN GEHLOT (B.E, CIVIL HONORS) Fourier Series: Basic Results Recall that the mathematical expression is called a Fourier series. Since this expression deals with convergence, we start by defining a similar expression when the sum is finite. Definition. A Fourier polynomial is an expression of the form which may rewritten as The constants a0, ai and bi, , are called the coefficients of Fn(x). The Fourier polynomials are -periodic functions. Using the trigonometric identities we can easily prove the integral formulas (1) for , we have
2. 2. TARUN GEHLOT (B.E, CIVIL HONORS) for n>0 we have (2) for m and n, we have (3) for , we have (4) for , we have Using the above formulas, we can easily deduce the following result: Theorem. Let We have
3. 3. TARUN GEHLOT (B.E, CIVIL HONORS) This theorem helps associate a Fourier series to any -periodic function. Definition. Let f(x) be a -periodic function which is integrableon . Set The trigonometric series is called the Fourier series associated to the function f(x). We will use the notation Example. Find the Fourier series of the function Answer. Since f(x) is odd, then an = 0, for . We turn our attention to the coefficients bn. For any , we have We deduce
4. 4. TARUN GEHLOT (B.E, CIVIL HONORS) Hence Example. Find the Fourier series of the Answer. We have and We obtain b2n = 0 and TARUN GEHLOT (B.E, CIVIL HONORS) Find the Fourier series of the function
5. 5. TARUN GEHLOT (B.E, CIVIL HONORS) Therefore, the Fourier series of Example. Find the Fourier series of the function function Answer. Since this function is the function of the example above minus the So Therefore, the Fourier series of TARUN GEHLOT (B.E, CIVIL HONORS) Therefore, the Fourier series of f(x) is Find the Fourier series of the function function Since this function is the function of the example above minus the constant So Therefore, the Fourier series of f(x) is constant .
6. 6. TARUN GEHLOT (B.E, CIVIL HONORS) Remark. We defined the Fourier series for functions which are wonder how to define a similar notion for functions which are Assume that f(x) is defined and The function F(x) is defined and integrableon of F(x) Using the substitution Definition. Let f(x) be a function defined and integrable of f(x) is where TARUN GEHLOT (B.E, CIVIL HONORS) We defined the Fourier series for functions which are -periodic, one would wonder how to define a similar notion for functions which are L-periodic. ) is defined and integrable on the interval [-L,L]. Set ) is defined and integrableon . Consider the Fourier series , we obtain the following definition: ) be a function defined and integrable on [-L,L]. The Fourier series periodic, one would . Consider the Fourier series ]. The Fourier series
7. 7. TARUN GEHLOT (B.E, CIVIL HONORS) for . Example. Find the Fourier series of Answer. Since L = 2, we obtain for . Therefore, we have
8. 8. TARUN GEHLOT (B.E, CIVIL HONORS)TARUN GEHLOT (B.E, CIVIL HONORS)