CHAPTER : FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION POT PERIODIC GRAPH
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:
Eample Eve uctio a b cos Odd uctio a b si Neither odd or eve uctio a b e
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
Product Fuctios: i. Eve Eve = EVEN ii. Eve Odd = ODD iii. Odd Odd = EVEN iv. Neither Odd = NEITHER v. Neither Eve = NEITHER
SKETCH A PERIODIC FUNCTION Deiitio Fuctio deied i the iterval periodic with i ad oly i. is said to be a
Eample Sketch graph or the uctio betwee rage
Solutio: =, Betwee rage, - -,- -
Solutio: =, Betwee rage, - - - -,- -
Eample 4 Sketch graph or the uctio betwee rage 4 4 4
Solutio: =, Betwee rage -, - -,- sice the graph is symmetry about the origi, we ca say that is a odd uctio
Solutio: =, Betwee rage 4 4 -, -4-4 -,-
Eample Sketch graph or the uctio betwee rage
Solutio: =, Betwee rage -,, - sice the graph is symmetry about the y-ais, we ca say that is a EVEN uctio
Solutio: =, Betwee rage -,, -
Eample 6 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio. 4 4, 4
Eample 7 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio. 4 4, 4
Eample 8 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,
Eample 9 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,
Eample 9 Sketch graph or the uctio betwee rage hece determie whether it is a odd, eve or either uctio.,
CHAPTER : FOURIER SERIES FOURIER SERIES COEFFICIENTS FOURIER SERIES EXPANSION
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
FOURIER SERIES COEFFICIENTS d b d a d a si cos
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
Solutio: a = - -
Solutio: b Fourier series coeiciets: = d d d a
Solutio: b Fourier series coeiciets: = si si si si si cos cos cos d d d a
Solutio: b Fourier series coeiciets: = d d d b cos cos cos cos cos si si si
Solutio: c Fourier series epasio: = si si si cos si cos b a b a a
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
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
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.
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
Where the Fourier coeiciets are: a d a b cos d
Solutio iii 4 4 4 4 d d d a
Solutio iii si 8 si 4 si 4 si 4 si 4 cos 4cos cos d d d a
Solutio iv cos si 8 cos si 8 4 si cos b a a
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
Solutio i = -9-6 - 6 9 - ii is a odd uctio sice the graph is symmetry about origi.
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
Where the Fourier coeiciets are: a a b si d
Solutio iii 4 4 6 6 cos 6 cos 6 cos 6 si si d d b
Solutio iv si 4 si b
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
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