Chapter5

Chapter5 - Chapter 5 Fourier Series and Fourier Integrals c...

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Chapter 5 Fourier Series and Fourier Integrals c s Sadhal 2009 5.1 Introduction Fourier series and Fourier integrals form a powerful mathematical tool for solving linear partial di f erential equations. While there are numerous other applications, the focus here is mainly towards developing the groundwork for solving partial di f erential equations. Simply put, Fourier series consists of a representation of an arbitrary piecewise continuous function as a series of sine and cosine functions. The theoretical basis of Fourier series is easier to understand from the perspective of periodic functions and we shall initially concentrate on such functions. 5.2 Periodic Functions A periodic function is one that repeats itself. While the term periodic implies repetition in time, it is generically applied to spatially repetitive functions. An example is given below. ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½½ f ( x ) 5.3 Even and Odd Functions The knowledge about symmetry or antisymmetry of a function can be useful in de f ning what type of expansion is applicable. For this purpose we de f ne even and odd functions. An even function is one that is symmetric, i.e., f ( x )= f ( x ). An odd one is antisymmetric with the property that g ( x g ( x ). Some examples are given here. 83
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84 CHAPTER 5. FOURIER SERIES AND FOURIER INTEGRALS c s SADHAL 2009 Even Functions [ f ( x )= f ( x ) ] ½ ½ ½ ½ ½ ½ Z Z Z Z Z Z .............................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ... ............................................................................................................................................... ..... . Odd Functions [ g ( x g ( x ) ] ................................................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ............................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ ½ Z Z Z Z Z Z Z Z Z Z Z Z
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5.3. EVEN AND ODD FUNCTIONS 85 Every function can be represented as a sum of even and odd functions. Consider F ( x )= F ( x )+ F ( x ) 2 + F ( x ) F ( x ) 2 = f ( x g ( x ) , where we de f ne f ( x F ( x F ( x ) 2 (5.1) and g ( x F ( x ) F ( x ) 2 . (5.2) By replacing x with x in equation (5.1), we see that f ( x F ( x F ( ( x )) 2 = F ( x F ( x ) 2 = f ( x ) Since f ( x f ( x ), we conclude that f ( x )isanevenfunct ion . Similarly, by replacing x with x in equation (5.2), we have g ( x F ( x ) F ( ( x )) 2 = F ( x ) F ( x ) 2 = X F ( x ) F ( x ) 2 ~ = g ( x ) , which proves that g ( x ) is an odd function. Thus, every function, regardless of (anti)symmetry can be expressed as a sum of an even and an odd function. EXAMPLE 5.1 Consider F ( x e x + x 2 +2 x = f ( x g ( x ) . The decomposition would give the even part as f ( x 1 2 [ F ( x F ( x )] = 1 2 ± ( e x + x 2 x )+( e x + x 2 2 x ) = =c o s h x + x 2 , and the odd part as g ( x 1 2 [ F ( x ) F ( x )] = 1 2 ± ( e x + x 2 x ) ( e x + x 2 2 x ) = =s i n h x x.
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86 CHAPTER 5. FOURIER SERIES AND FOURIER INTEGRALS c s SADHAL 2009 5.4 Periodic Functions as Series of Periodic Functions 5.4.1 Odd Functions Consider an odd function, g ( x ), which has a period of 2 π . 2 ππ 0 π 2 π 3 π g ( x ) ....................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The function sin x is odd and has a period of 2 π . π 0 π 2 π sin x .
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Chapter5 - Chapter 5 Fourier Series and Fourier Integrals c...

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