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chapter5 - Chapter 5 Fourier Series 5.1 Periodic functions...

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Unformatted text preview: Chapter 5. Fourier Series 5.1 Periodic functions A function f ( x ) is called periodic if it is defined for all real x and if there is some positive number p such that f ( x + p ) = f ( x ) for all x. (1) The number p is called the period of f ( x ). 5.1.1 Graphs of periodic functions The graph of such a function can be obtained by pe- riodic repetition of its graph in any interval of length p . 2 MA1505 Chapter 5. Fourier Series For example, sine and cosine functions are periodic 2 π . f ( x ) = c , c constant, is a periodic function of period p for every positive number p . x, x 2 , x 3 , ··· ,e x , ln x are not periodic. 5.1.2 Some algebraic properties of periodic functions From (1), f ( x + 2 p ) = f (( x + p ) + p ) = f ( x + p ) = f ( x ) . 2 3 MA1505 Chapter 5. Fourier Series Thus (by induction) for any positive integer n , f ( x + np ) = f ( x ), for all x . Hence 2 p, 3 p, ··· are also periods of f . Further, if f and g have period p , then the function h ( x ) = af ( x ) + bg ( x ) with a, b constants also has period p . 5.1.3 Trigonometric series Our aim is to represent various periodic functions of period 2 π in terms of simple functions 1 , cos x, sin x, cos 2 x, sin2 x, ··· , cos nx, sin nx, ··· (2) which have period 2 π . The series that arises in this connection will be of the 3 4 MA1505 Chapter 5. Fourier Series form a + a 1 cos x + b 1 sin x + a 2 cos 2 x + b 2 sin2 x + ··· = a + ∑ ∞ n =1 ( a n cos nx + b n sin nx ) (3) where a ,a 1 ,a 2 , ··· ,b 1 ,b 2 , ··· are real constants. Series (3) is called a trigonometric series, and a n and b n are called coefficients of the series. The set of functions (2) is often called a trigonomet- ric system. We note that each term of the series (3) has period 2 π . Hence if the series converges, its sum will be a periodic function of period 2 π . 4 5 MA1505 Chapter 5. Fourier Series 5.2 Fourier Series Assume that f ( x ) is a periodic function of period 2 π and that it can be represented by a trigonometric series f ( x ) = a + ∞ X n =1 ( a n cos nx + b n sin nx ) . (4) That is, we assume that the series on the right con- verges and has f ( x ) as its sum....
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chapter5 - Chapter 5 Fourier Series 5.1 Periodic functions...

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