M257-316Notes_Lecture14

M257-316Notes_Lecture14 - Chapter 10 Lecture 14 - Even and...

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Chapter 10 Lecture 14 - Even and Odd Functions Even: f ( x )= f ( x ) Odd: f ( x f ( x ) 10.1 Integrals of Even and Odd Functions L Z L f ( x ) dx = 0 Z L f ( x ) dx + L Z 0 f ( x ) dx (10.1) = L Z 0 £ f ( x )+ f ( x ) ¤ dx (10.2) = 2 L R 0 f ( x ) dx f even 0 f odd . (10.3) Notes: Let E ( x ) represent an even function and O ( x ) an odd function. 1. If f ( x E ( x ) · O ( x ) then f ( x E ( x ) O ( x E ( x ) O ( x f ( x ) f is odd. 67

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Lecture 14 - Even and Odd Functions 2. E 1 ( x ) · E 2 ( x ) even. 3. O 1 ( x ) · O 2 ( x ) even. 4. Any function can be expressed as a sum of an even part and an odd part: f ( x )= 1 2 £ f ( x )+ f ( x ) ¤ | ±z } even part + 1 2 £ f ( x ) f ( x ) ¤ | ±z } odd part . (10.4) Check: Let E ( x 1 2 £ f ( x f ( x ) ¤ . Then E ( x 1 2 £ f ( x f ( x ) ¤ = E ( x ) even. Similarly let O ( x 1 2 £ f ( x ) f ( x ) ¤ (10.5) O ( x 1 2 £ f ( x ) f ( x ) ¤ = O ( x )odd . (10.6) 10.2 Consequences of Even/Odd Property for Fourier Series (I) Let f ( x ) be Even-Cosine Series: a n = 1 L L Z L f ( x ) cos | ±z } even ² nπx L ´ dx = 2 L L Z 0 f ( x ) cos ² nπx L ´ dx (10.7) b n = 1 L L Z L f ( x ) sin ² nπx L
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M257-316Notes_Lecture14 - Chapter 10 Lecture 14 - Even and...

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