This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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. We say the right hand side of (4) is the Fourier series of f ( x ). Given f ( x ), our task now is to determine the coeffi cients a n and b n . 5 6 MA1505 Chapter 5. Fourier Series 5.2.1 Determine a We integrate both sides of (4) from π to π : Z π π f ( x ) dx = Z π π ( a + ∞ X n =1 ( a n cos nx + b n sin nx )) dx. Assuming that term by term integration is allowed, we obtain Z π π f ( x ) dx = a Z π π dx + ∞ X n =1 ( a n Z π π cos nxdx + b n Z π π sin nxdx ) = 2 πa + ∞ X n =1 • a n sin nx n ‚ π π + • b n cos nx n ‚ π π ¶ = 2 πa So a = 1 2 π Z π π f ( x ) dx....
View
Full Document
 Winter '11
 yap
 Math, Fourier Series

Click to edit the document details