tutorial 4 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 4 (Solution Notes) Preliminaries Introductory to Fourier Series 80
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One More Application of Fourier Series Fourier Series has so many amazing applications, now let us look at one much simpler but useful application. Parseval’s Identity 1 L ± L L [ f ( x )] 2 dx =2 a 2 0 + ² n =1 ( a 2 n + b 2 n ) , where a 0 , a n , b n are Fourier coefficients of f ( x ), and 2 L is the period of f ( x ). Proof. Suppose f ( x )= a 0 + ² n =1 ( a n cos nx + b n sin nx ), then we substitute into the integral ± L L [ f ( x )] 2 dx ,andget : ± L L [ f ( x )] 2 dx = ± L L f ( x ) · f ( x ) dx = ± L L f ( x ) · a 0 + ² n =1 ( a n cos nx + b n sin nx ) dx = ± L L a 0 · f ( x )+ ² n =1 ( a n · f ( x )cos nx + b n · f ( x )sin nx ) dx = ± L L a 0 · f ( x ) dx + ² n =1 ³± L L a n · f ( x nx dx + ± L L b n · f ( x nx dx ´ = a 0 · ± L L f ( x ) dx + ² n =1 ³ a n · ± L L f ( x nx dx + b n · ± L L f ( x nx dx ´ = a 0 · (2 L · a 0 ² n =1 a n · ( L · a n b n · ( L · b n ) = L · 2 a 2 0 + ² n =1 ( a 2 n + b 2 n ) Thus, 1 L ± L L [ f ( x )] 2 dx a 2 0 + ² n =1 ( a 2 n + b 2 n ). ± Remark: This identity can be used to solve Practise exercise 1.2 (ii). 83
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Some Words Before Tutorial Someone told me he was lost in reading the lecture notes, the text book, and felt confused about Fourier Series. Here I summarize your lecture notes ±rst, and actually, all the contents that you are required to know have been listed in the following: 1. Only periodic functions have their Fourier series. 2. If a function is de±ned on an interval, but not de±ned on the other parts 17 ,thenwe can write out the Fourier series of the function by using half-range expansions . Check details in question 6. 3. Remember (in mind or in help sheet) the formula of Fourier series. Do not su²er yourself to remember how to get the formula, just make sure you can apply the formula when you are given a periodic function. 4. There are two kinds of formulas for Fourier series: one for period 2 π , and the other for period 2 L . You need to know both, but the latter formula is more general, when you substitute L by π , you can get the formula for period 2 π . In this sense, you just need to know the formula for period 2 L : f ( x )= a 0 + ± n =1 ( a n cos L x + b n sin L x ) , and the coefficients are given by the Euler’s formulas : a 0 = 1 2 L ² L L f ( x ) dx a n = 1 L ² L L f ( x )cos L xdx n =1 , 2 , ··· b n = 1 L ² L L f ( x )sin L n , 2 , 5. Still some small skills and tricks, check the tutorial questions and past year’s papers. For example, for even functions and odd functions, the formulas are much simpler (Fourier cosine series and Fourier sine series). You will ±nd this chapter is not so difficult, but just substitute your function into the given formula, and then do some integrations.
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This note was uploaded on 05/05/2009 for the course MA MA1505 taught by Professor Ma1505 during the Spring '09 term at National University of Juridical Sciences.

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tutorial 4 solution - NATIONAL UNIVERSITY OF SINGAPORE...

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