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
/ 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+... 3 5 7. 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
{ 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... 3 4 f( t) si t+ si t+ si 3t+ si 4 t+... 3 4 3. 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 + + 58 4, Joh Bird
{ 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 4 4 4 + cos x cos3x cos5 x... (3) (5) sice cos( ) cos i.e. f(x) 4 + cos x+ cos3x+ cos5 x+... 3 5 4. 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 + + +... 3 5 4 i.e. + + + + +... 3 5 7 i.e. ad 4 + + + +... 3 5 7... 8 3 5 7 + + + + 5. Show that the Fourier series for the triagular waveform show is give by: y 8 siθ si 3θ + si 5θ si 7 θ +... 3 5 7 59 4, Joh Bird
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
( /)cos / si / + ( ) ( )cos si ( / )cos / si / + + cos cos / Thus, b b b 3 b 5 4 4 + + 4 4 4 4 8 + + 4 4 + 4 4 + 3 4 b 4 b 6 b 8, ad so o 4 4 + 3 3 3 3 4 4 4 4 8 + 3 3 3 3 3 4 4 + + 5 5 5 5 4 4 4 4 8 + + 5 5 5 5 5 It follows that b 7 8 7 ad so o Thus, y b si θ 8 8 8 8 siθ si 3θ + si 5θ si 7 θ +... 3 5 7 8 i.e. y siθ si 3θ + si 5θ si 7 θ +... 3 5 7 53 4, Joh Bird
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 + +, 4 4 3 3 cos si b3 + 3 3 3, (3) 53 4, Joh Bird
b 4 cos si + + 4 4, ad so o 8 8 Hece, i.e. f ( x) b si x si x+ si x si 3x si 4 x+... 4 3 8 f( x) si x+ si x si 3x si 4 x+... 4 9 8. 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 +... 3 5 533 4, Joh Bird
i.e. f( t) cost cos3t+ cos5 t... 3 5 (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 ( ), 3 3 3 b4 ( cos 4 cos ) ( ) 4 4, b 5 5 cos 5 cos ( ), 5 5 5 b6 ( cos 6 cos3) ( ), ad so o 6 6 3 f ( t) b si t si t si t+ si 3t+ + si 5t si 6 t+... 3 5 3 i.e. f( t) si t si t+ si 3t+ si 5t si 6 t+... 3 5 3 3. 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
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) () 4 6 8 ad so o 3 5 8 b3 + + + 3 (5) ( ) 3 (5) ( ) 3 5 (3)(5) (3)(5) b 5 4 5 35 8 + + + 5 (7) ( 3) 5 (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 8 8 8 8 si x si 3x si 5x si 7x 3 (3)(5) (3)(5)(7) (5)(7)(9) 535 4, Joh Bird
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 / + 8 8 + + 8 8 8 4 / x si x cos x si xsi x cos x + + / si cos si si cos cos + + + 536 4, Joh Bird
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) 36 36 (3) 8 a, ad so o ( cos 5 cos ) ( ) () (5) Thus, f ( x) a + a cos x cos x cos 6x cos x +... 4 (3) (5) i.e. cos 6x cosx f( x) cos x+ + +... 4 3 5 537 4, Joh Bird