CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES
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1 CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period. The square wave show is a eve fuctio sice it is symmetrical about the f(x) axis. Hece, the Fourier series is give by: f(x) a + a cos x (i.e. the series cotais o sie terms) a a f ( x )dx / { d x d x + / { / / [ x] + [ x] [ () + [( ) ( )] ] f ( x )cos x d x / { cos x d x cos x d x + / 56 4, Joh Bird
2 / si x si x + / si( / ) si( / ) + si( / ) 4 si Whe is eve, a 4 Whe is odd, a 4 ad a for, 5, 9,... for 3, 7,,... Hece, a 4, a 4 4 3, a5, ad so o 3 5 Hece the Fourier series for the above waveform is give by: f(x) 4 cos x cos 3x+ cos 5x cos 7 x Obtai the Fourier series of the fuctio defied by: f(t) t+, t t, t which is periodic of period. Sketch the give fuctio. The periodic fuctio is show i the diagram below. Sice it is symmetrical about the origi, the fuctio is odd, ad f ( t) b si t 57 4, Joh Bird
3 { b f ( t)si t d t ( t )si t d t ( t )si t dt + + t cos t si t cos t t cos t si t cos t by itegratio by parts cos cos + + cos( ) cos( ) cos( ) cos( ) cos cos + + Hece, b, b, b 3, b 4, ad so o 3 4 i.e. f(t) i.e. si t si t si 3t si 4 t f( t) si t+ si t+ si 3t+ si 4 t Determie the Fourier series defied by f(x). x, x + x, x which is periodic of period The periodic fuctio is show i the diagram below. Sice it is symmetrical about the f(x) axis, the fuctio is eve, ad f ( x) a a cos x + a f( x)d x f( x)dx due to symmetry x ( ) d x x x ( ) { a f ( x)cos x d x ( x)cos x d x ( x)cos x d x , Joh Bird
4 { cos x x cos x d x cos x x cos x d x ( ) + ( + ) si x xsi x cos x si x xsi x cos x by itegratio by parts cos( ) cos cos( ) cos + + ( cos ) Whe is eve, a 4 () Whe, a ( ) 4 (3) (3) Whe 3, a3 ( ) Whe 5, a5 ( ) f ( x) a a cos x + 4 (5) (5) ad so o cos x cos3x cos5 x... (3) (5) sice cos( ) cos i.e. f(x) 4 + cos x+ cos3x+ cos5 x I the Fourier series of Problem 3, let x ad deduce a series for /8 Whe x i the series of Problem 3, f(x), 4 cos cos hece, + cos i.e i.e. ad Show that the Fourier series for the triagular waveform show is give by: y 8 siθ si 3θ + si 5θ si 7 θ , Joh Bird
5 The fuctio is periodic of period The equatio of the fuctio betwee ad / is of the straight lie form y mθ + c where gradiet, m ad itercept, c / Hece, equatio of the lie betwee ad / is y θ The equatio of the fuctio betwee / ad 3/ is of the straight lie form y mθ + c where gradiet, m Whe θ, ad y ad sice y mθ + c the + c from which, c. Hece, equatio of lie betwee / ad 3/ is y θ + The equatio of the fuctio betwee 3/ ad is of the straight lie form y mθ + c where gradiet, m Whe θ, ad y ad sice y mθ + c the / () + c from which, c 4. Hece, equatio of lie betwee 3/ ad is y θ 4 The triagular wave is a odd fuctio sice it is symmetrical about the origi Hece, the Fourier series is give by: f(θ) ( b si θ) i.e. a a b f ( x )si x d x θ si θdθ + + si θd θ / θ / θ cos θ si θ θ cos θ si θ cos θ + + / / / 53 4, Joh Bird
6 ( /)cos / si / + ( ) ( )cos si ( / )cos / si / + + cos cos / Thus, b b b 3 b b 4 b 6 b 8, ad so o It follows that b ad so o Thus, y b si θ siθ si 3θ + si 5θ si 7 θ i.e. y siθ si 3θ + si 5θ si 7 θ , Joh Bird
7 EXERCISE 365 Page 79. Determie the half-rage sie series for the fuctio defied by: f(x) x, x, x The periodic fuctio is show i the diagram below. Sice a half-rage sie series is required, the fuctio is symmetrical about the origi ad f ( x) b si x / { / ( )si d x x x si d by itegratio by parts cos si b f x x x x x x + cos si Hece, b + +, b cos si + ( ) cos si + +, cos si b , (3) 53 4, Joh Bird
8 b 4 cos si , ad so o 8 8 Hece, i.e. f ( x) b si x si x+ si x si 3x si 4 x f( x) si x+ si x si 3x si 4 x Obtai (a) the half-rage cosie series ad (b) the half-rage sie series for the fuctio f(t), t, t (a) The periodic fuctio is show i the diagram below. Sice a half-rage cosie series is required, the fuctio is symmetrical about the f(t) axis ad f ( t) a a cos t + a dt [ t] / / si si si t a cos d t t / / Whe is eve, a 3 5 si si si ad a, a 3 3 3, a5, ad so o 5 5 Thus, f ( t) a + a cos t cost + cos3t cos5 t , Joh Bird
9 i.e. f( t) cost cos3t+ cos5 t (b) The periodic fuctio is show i the diagram below. Sice a half-rage sie series is required, the fuctio is symmetrical about the origi ad f ( x) b si x cos t b si t d t cos cos / /, Hece, b cos cos ( ) Thus, b ( cos cos) ( ), 3 b3 cos 3 cos ( ), b4 ( cos 4 cos ) ( ) 4 4, b 5 5 cos 5 cos ( ), b6 ( cos 6 cos3) ( ), ad so o f ( t) b si t si t si t+ si 3t+ + si 5t si 6 t i.e. f( t) si t si t+ si 3t+ si 5t si 6 t Fid the half-rage Fourier sie series for the fuctio f(x) si x i the rage x. Sketch the fuctio withi ad outside of the give rage. The periodic fuctio is show i the diagram below. Sice a half-rage sie series is required, 534 4, Joh Bird
10 the fuctio is symmetrical about the origi ad f ( x) b si x b si xsi x d x ( cos x)si x d x (si x cos xsi x)d x si x si( x ) si( x ) d x ( + ) cos x cos( + x ) cos( x ) + ( + ) ( ) cos cos( + ) cos( ) + + ( + ) ( ) ( + ) ( ) Hece, 8 8 b (3) () (3) () 3 3 b b b b (4) () (4) () ad so o b (5) ( ) 3 (5) ( ) 3 5 (3)(5) (3)(5) b (7) ( 3) 5 (7) ( 3) (3)(5)(7) (3)(5)(7) 8 It follows that b7 ad so o (5)(7)(9) Thus, f ( x) b si x si x si 3x si 5x si 7x 3 (3)(5) (3)(5)(7) (5)(7)(9) 535 4, Joh Bird
11 8 si x si 3x si 5x si 7x i.e. f( x) si x... ()(3) ()(3)(5) (3)(5)(7) (5)(7)(9) 4. Determie the half-rage Fourier cosie series i the rage x to x for the fuctio defied by: f(x) x, x ( x), x The periodic fuctio is show i the diagram below. Sice a half-rage cosie series is required, the fuctio is symmetrical about the f(x) axis ad f ( x) a a cos x + / { d ( )d / / x x a x x+ x x x + / { cos d ( )cos d / a x x x + x x x / / x si x cos x si xsi x cos x + + / si cos si si cos cos , Joh Bird
12 cos si si cos + cos cos Whe is odd, a 8, () 4 4 ad a ( cos cos ) ( ) a4 ( cos cos 4) ( ) (4) 6, 8 a, 6 a 8, ( cos3 cos 6 ) ( ) (6) (3) 8 a, ad so o ( cos 5 cos ) ( ) () (5) Thus, f ( x) a + a cos x cos x cos 6x cos x (3) (5) i.e. cos 6x cosx f( x) cos x , Joh Bird
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