CHAPTER 5: FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION PLOT PERIODIC GRAPH

Transcription

1 CHAPTER : FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION POT PERIODIC GRAPH

2 PROPERTIES OF EVEN AND ODD FUNCTION Fuctio is said to be a eve uctio i: Fuctio is said to be a odd uctio i: Fuctio is said to be either eve or odd uctio i:

3 Eample Eve uctio a b cos Odd uctio a b si Neither odd or eve uctio a b e

4 Eample Determie whether the ollowig uctio is eve, odd or either uctio e e d c b e a si cos si si cos cos

5 Product Fuctios: i. Eve Eve = EVEN ii. Eve Odd = ODD iii. Odd Odd = EVEN iv. Neither Odd = NEITHER v. Neither Eve = NEITHER

6 SKETCH A PERIODIC FUNCTION Deiitio Fuctio deied i the iterval periodic with i ad oly i. is said to be a

7 Eample Sketch graph or the uctio betwee rage

8 Solutio: =, Betwee rage, - -,- -

9 Solutio: =, Betwee rage, ,- -

10 Eample 4 Sketch graph or the uctio betwee rage 4 4 4

11 Solutio: =, Betwee rage -, - -,- sice the graph is symmetry about the origi, we ca say that is a odd uctio

12 Solutio: =, Betwee rage 4 4 -, ,-

13 Eample Sketch graph or the uctio betwee rage

14 Solutio: =, Betwee rage -,, - sice the graph is symmetry about the y-ais, we ca say that is a EVEN uctio

15 Solutio: =, Betwee rage -,, -

16 Eample 6 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio. 4 4, 4

17 Eample 7 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio. 4 4, 4

18 Eample 8 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,

19 Eample 9 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,

20 Eample 9 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,

21 CHAPTER : FOURIER SERIES FOURIER SERIES COEFFICIENTS FOURIER SERIES EXPANSION

22 FOURIER SERIES A epasio o a periodic uctio i terms o a iiite sum o cosie ad sie. Theorem: et be a uctio deied i the iterval < < ad be periodic with period, hece Fourier series or is give by: a a cos b si where a, a ad b are called the Fourier Series coeiciets o

23 FOURIER SERIES COEFFICIENTS d b d a d a si cos

24 Eample A periodic uctio is deied by, i Sketch the graph o the uctio over ii Fid the Fourier coeiciets correspodig to the uctio iii Write the correspodig Fourier series

25 Solutio: a = - -

26 Solutio: b Fourier series coeiciets: = d d d a

27 Solutio: b Fourier series coeiciets: = si si si si si cos cos cos d d d a

28 Solutio: b Fourier series coeiciets: = d d d b cos cos cos cos cos si si si

29 Solutio: c Fourier series epasio: = si si si cos si cos b a b a a

30 Eample A periodic uctio is deied by, i Sketch the graph o the uctio over ii Fid the Fourier coeiciets correspodig to the uctio iii Write the correspodig Fourier series

31 Eample A periodic uctio is deied by, i Sketch the graph o the uctio over ii Based oi, determie whether is a odd or eve uctio. State the reaso or your aswer iii Fid the Fourier coeiciets correspodig to the uctio iv Write the correspodig Fourier series 4

32 Solutio i = π 4 -π -π/ -π -π/ π/ π π/ π ii is a eve uctio sice the graph is symmetry about y-ais.

33 FOURIER SERIES FOR EVEN FUNCTION I is a eve uctio, the: Theorem Fourier Series o Eve Fuctios I is a eve uctio the b =. Thereore, Fourier series is give by, d d cos a a

34 Where the Fourier coeiciets are: a d a b cos d

35 Solutio iii d d d a

36 Solutio iii si 8 si 4 si 4 si 4 si 4 cos 4cos cos d d d a

37 Solutio iv cos si 8 cos si 8 4 si cos b a a

38 Eample A periodic uctio is deied by, 6 i Sketch the graph o the uctio over ii Based oi, determie whether is a odd or eve uctio. State the reaso or your aswer iii Fid the Fourier coeiciets correspodig to the uctio iv Write the correspodig Fourier series 9 9

39 Solutio i = ii is a odd uctio sice the graph is symmetry about origi.

40 FOURIER SERIES FOR ODD FUNCTION I is a odd uctio, the: d Theorem Fourier Series o Odd Fuctios I is a odd uctio the a = ad a =. Thereore, Fourier series is give by, b si

41 Where the Fourier coeiciets are: a a b si d

42 Solutio iii cos 6 cos 6 cos 6 si si d d b

43 Solutio iv si 4 si b

44 Eample 4 A periodic uctio is deied by, i Sketch the graph o the uctio over ii Based oi, determie whether is a odd or eve uctio. State the reaso or your aswer iii Fid the Fourier coeiciets correspodig to the uctio iv Write the correspodig Fourier series

45 Eample A periodic uctio is deied by, 6 6 i Sketch the graph o the uctio over ii Based oi, determie whether is a odd or eve uctio. State the reaso or your aswer iii Fid the Fourier coeiciets correspodig to the uctio iv Write the correspodig Fourier series