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ch2 - Chapter 2 Fourier series In this chapter we deal with...

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Chapter 2: Fourier series In this chapter we deal with expansions of functions in terms of sines and cosines, as they occured in the last section of Chapter 1. In many respects it is simpler to work with the complex exponential function e ix instead of the trigonometric functions cos x and sin x . We recall that these functions are related by the formulas e ix = cos x + i sin x, (1) cos x = e ix + e - ix 2 , sin x = e ix - e - ix 2 . (2) 2.1. Fourier series of periodic functions Suppose that f : IR IC, (or f : IR IR), is (2 π )-periodic, that is, f ( x ) = f ( x +2 π ) for all x . We shall also assume that f is Riemann integrable on every bounded interval. This will be the case if f is bounded and is continuous except perhaps at finitely many points in each bounded interval. We wish to know if f can be extended in a series f ( x ) = 1 2 a 0 + n =1 ( a n cos nx + b n sin nx ) . (3) In view of the formulas (1) and (2), this can be rewritten as f ( x ) = + n = -∞ c n e ix , (4) where c 0 = 1 2 a 0 , c n = 1 2 ( a n - ib n ) , c - n = 1 2 ( a n + ib n ) , n = 1 , 2 , . . . . (5) Alternatively we have, taking into account cos( - nx ) = cos nx , sin( - nx ) = - sin nx , a 0 = 2 c 0 , a n = c n + c - n , b n = i ( c n - c - n ) , n = 1 , 2 , . . . . (6) Before analysing the formulas (3), (4) and in particular studying properties of con- vergence, let us first assume that f has a series expansion as above. In that case we can easily calculate the coefficients in terms of f : Let us multiply both sides of (4) by e - kx , and integrate from - π to π . Believing for the moment that it is allowed to integrate the series term by term, we obtain π - π f ( x ) e - ix dx = -∞ c n π - π e i ( n - k ) x dx. 7
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But π - π e i ( n - k ) x dx = 1 i ( n - k ) e i ( n - k ) x π - π = 0 if n = k, (7) π - π e i ( n - k ) x dx = π - π dx = 2 π if n = k. (8) Notice that via (5) one obtains from this easily the following formulas, which are useful otherwise, ( m, n IN), π - π sin nx sin mx dx = 0 for n = m, or n = m = 0 π for n = m = 0 , (9) π - π cos mx cos nx dx = 0 for n = m, π for n = m = 0 , (10) π - π sin mx cos nx dx = 0 . (11) In view of (7) and (8), the only term in the series that survives the integration is the term with n = k , hence π - π f ( x ) e - ikx dx = 2 πc k . In other words, relabeling the integer k as n , we find c n = 1 2 π π - π f ( x ) e - inx dx, n = 0 , ± 1 , ± 2 , . . . . (12) Using (6) and (2), this leads to a n = 1 π π - π f ( x ) cos nx dx, n = 0 , 1 , 2 , . . . , (13) b n = 1 π π - π f ( x ) sin nx dx, n = 1 , 2 , . . . . (14) To summarize: If f has a series expansion of the form (3) and (4), and the series converge decently so that term-by-term integration is permissable, then the coeffi- cients c n , a n and b n are given by (12), (13) and (14). On the other hand, if f is (2 π )-periodic and Riemann integrable over [ - π, + π ], then the integrals in (12),(13) and (14) make good sense, too. We are now in a position to make a formal definition. Definition 2.1. Suppose f is 2 π -periodic and integrable over [ - π, π ] . Then the numbers c n , a n and b n defined by (12),(13) and (14) are called the Fourier coefficients of f , and the corresponding series n = -∞ c n e inx or 1 2 a 0 + n =1 ( a n cos nx + b n sin nx ) 8
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is called the Fourier series of f .
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