radians A function f ( x ) is called periodic if it is defined for all real x and if there is some positive number P such that:

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Gertrude Baker
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1 Fourier Series. Graph of y Asix ad y Acos x Amplitude A ; period 36 radias. Harmoics y y six is the first harmoic y y six is the th harmoics 3. Periodic fuctio A fuctio f ( x ) is called periodic if it is defied for all real x ad if there is some positive umber P such that: f ( x) f ( x+ P) ; P period. A Fourier series is a represetatio employed to express periodic fuctio f ( x ) defied i a iterval, say (, ) a liear relatio betwee the sies ad cosies of the same period. 5. Fourier series for fuctios of period is give by, f ( x) a + acosx+ acosx+ a3cos3x+ + a cosx + + b six+ b six+ b si3x+ + b six + 3 a + { a cosx+ b si x} where the value of the coefficiets a, a ad b are determied by, a f ( xdx ) mea value of f ( x ) over a period. a f ( x)cos xdx mea value of f ( x)cos x over a period b f( x)sixdx mea value of f ( x)six over a period

2 It ca also be expressed i compoud sie terms f ( x) a + c si( x α ) + a where c a + b ad α arcta b 6. he Fourier series expasio ad the determiatio of Fourier s costats is valid uder the assumptio that the give periodic fuctio f ( x ) satisfy the Dirichlet s coditios. his is to esure that the sum o the R.H.S. has a limit as ad, is equal to f ( x ) at ay poit x whe f ( x ) beig cotiuous, ad is equal to ( ) ( ) f x f x + whe there is a fiite discotiuity at the poit x. 7. Dirichlet coditios (a) he fuctio f ( x ) must be defied ad sigle-valued. (b) f ( x ) must be cotiuous or have a fiite umber of fiite discotiuities withi a period iterval. (c) f ( x ) ad f '( x ) must be piecewise cotiuous i the periodic iterval. 8. Sum of Fourier series (a) at ay cotiuous poit, x Let f ( x) y, the at x x, Fourier series of f ( x ) coverge to the value f ( x) y. (b) at a fiite discotiuity, x + + Let f ( x ) y ad f ( x+ ) y, he at x x, Fourier series for f ( x ) coverges to the value { f( x ) + + f( x + )} + ( y + y).

3 9. Odd ad eve fuctios (a) Eve fuctio: (i) f ( x) f ( x) (ii) the graph is symmetric about y axis (iii) (b) Odd fuctio: a a a f( xdx ) f( xdx ) (i) f ( x) f ( x) (ii) the graph is symmetric about origi. a (iii) f ( xdx ) a Product of odd ad eve fuctios (eve) (eve) (eve) (eve) (odd) (odd) (odd) (eve) (odd) (odd) (odd) (eve).. Sie series ad cosie series If f ( x ) is eve, the series cotais cosie terms oly (icludig a ) For example, give a periodic eve fuctio with period, the Fourier series is as followig, f ( x) a + a cosx b six + where a f ( x) dt a f( x)cos tdt b

4 If f ( x ) is odd, the series cotais sie terms oly. For example, give a periodic odd fuctio with period, the Fourier series is as followig, f ( x) a + a cosx b six + where a a b f( x)sixdx. Half-rage series period For a fuctio f ( x ) of period, defied over the rage to, ca be cosidered as half of a eve fuctio, or half of a odd fuctio. (a) Whe we cosidered it as half of a eve fuctio, the it ca be represeted as halfrage cosie series, f ( x) a a cosx + where a f( x) dx a f( x)cos xdx

5 (b) Whe we cosidered it as half of a odd fuctio, the it ca be represeted as half-rage sie series, f ( x) b six where b f( x)sixdx. Series cotaiig oly odd harmoics or oly eve harmoics If f ( x) f ( x+ ) the Fourier series cotais eve harmoics oly. If f ( x) f( x+ ) the Fourier series cotais odd harmoics oly. 3. Liearity Property If f ( x) gx ( ) + hx ( ), where gx ( ) ad hx ( ) are periodic fuctios of period, with Fourier expasio: gx ( ) a cos si + a x + b x hx ( ) α cosx six + α + β the Fourier series for f ( x ) is give by, f ( x) ( a + α ) + ( a α )cos x ( b β )six + + +

6 . Fourier series for periodic fuctios f (t) with period f () t a + { a cosωt b si ωt} + a f () t dt ω ω f() t dt a f ()cos t ωtdt ω ω f()cos t ω tdt b f()si t ω tdt ω ω f()si t ω tdt where ω i.e. ω 5. Fourier series for eve fuctio ad odd fuctio. Half-rage series period f () t a + { a cosωt b si ωt} + (a) For Eve fuctio or half-rage cosie series a / f () t dt / a f ()cos t ωtdt b

7 (b) For odd fuctio or half-rage sie series a a / b f()si t ω tdt

8 Example (Fourier Series) Example Let f ( x ) be a periodic fuctio with period defie as followig: ( ) ( ) ( + ) f x x,if < x<, f x f x. Expad f ( x ) i a Fourier series. Solutio Fourier series, [figure] f ( x) a + a cosx+ bsi x Let fid the a first, d + ( ) a f x dx, where i this case, we choose d. (Why?) d f ( x) dx xdx x. he, fid a, where,,3,. d + a f ( x) cosxdx xcos xdx d xsix si x d x, use itegratio by parts here xsix cos x + ( ) ( ) ( ) ( ) ( ) si cos si cos +

9 he, fid b, where,,3, d + b f ( x) sixdx xsi xdx d cosx cos x x d x, use itegratio by parts here xcosx si x + ( ) ( ) ( ) ( ) cos si cos si + +. ( ) Fially, substitute a,, a b ito the Fourier series, f ( x) a + a cosx+ bsi x ( ) + ( cos ) x+ si x si x six+ six+ si3x+ six+ 3 Example Fid Fourier series for f ( ) Solutio ( ) ( + ) x, where x,if < x<, f ( x) + x,if < x <, f x f x. [figure] From the graph, ca be observed that the graph is symmetric respected to y -axis, so f ( x ) is a eve fuctio. he, the Fourier series expasio of f ( ) x oly cotai

10 costat ad cosie terms (sice it is a eve fuctio), which is havig the followig structure, f ( x) a + a cos x. Let fid the a first, a f ( x) dx f ( x) dx ( x) dx x x. he, fid a, where,,3,. a f ( x) cosxdx ( x) cosxdx ( x) si x si x d x, use itegratio by parts here ( ) x si x cos x + cos ( ) ( ) ( ) ( ) si( ) cos( ) si( ) cos( ) ( ( )),if is odd ( ( + )),if is eve,if is odd a,if is eve Fially, substitute a, a ito the Fourier series,

11 f ( x) a + a cos x a + acosx+ acosx+ a3cos3x+ acosx+ + cosx+ ( cos ) x+ cos3x+ ( cos ) x+ 3 cosx+ cos3x+ cos5x+ cos7x Example 3 Give a periodic fuctio f ( x ) defied as Solutio,if < x <, f ( x),if < x <, f x f x. ( ) ( + ) [figure] From the graph, ca be observed that the graph is symmetric respected to origi, so f ( x ) is a odd fuctio. he, the Fourier series expasio of f ( x ) oly cotai sie terms (sice it is a odd fuctio), which is havig the followig structure, ( ) f x b si x, where,,3,. b f ( x) sixdx ( si ) xdx cosx cos cos + ( ) ( ) cos ( ) ( ( )),if is odd ( ( + )),if is eve

12 ,if is odd b,if is eve Fially, substitute b ito the Fourier series, ( ) f x b si x b six+ b six+ b si3x+ b six+ 3 six+ ( si ) x+ si3x+ ( si ) x+ 3 six+ si3x+ si5x+ si7x Example Fid Fourier series for f ( x ), where Solutio,if < x <, f ( x),if < x<,,if < x <, f x f x. ( ) ( + ) [figure] From the graph, ca be observed that the graph is symmetric respected to y -axis, so f ( x ) is a eve fuctio. he, the Fourier series expasio of f ( ) x oly cotai costat ad cosie terms (sice it is a eve fuctio), which is havig the followig structure, f ( x) a + a cos x. First, fid the a,

13 a f ( x) dx ( d ) x ( d ) x +. he, fid a, where,,3,. a f ( x) cosxdx ( cos ) xdx ( cos ) xdx + 8 si x 8 si si ( ) 8 ( ),if,5,9,3, 8 ( ),if 3,7,,5, 8 ( ),if is eve 8,if,5,9,3, 8 a,if 3,7,,5,,if is eve Fially, substitute a, a ito the Fourier series, f ( x) a + a cos x a + acosx+ acosx+ a3cos3x+ acosx+ a5cos5x ( ) + cosx+ ( cos ) x+ cos3x+ ( cos ) x+ cos5x cosx cos3x+ cos5x cos7x+ cos9x cosx

14 Example 5 (Liearity property of Fourier Series) Use the aswer from example ad the liearity property of Fourier series, write the g x is a periodic Fourier series expasio of the periodic fuctio g( x ). Give that ( ) fuctio with period, ad it s graph from < x < is as followig figure, Solutio he graph of f ( ) [figure] x i example is as followig: [figure] From the graph of g( x ) ad f ( x ), ca be observed that ( ) shifted dow uit of f ( x ) vertically. Mathematical, this meas that, g( x) f ( x) he, Fourier series of g( x ) ca be obtaied by the liearity properties, F { g( x) } F { f ( x) } F f ( x) + F { } { } g x ca be obtaied by 8 + cosx cos3x+ cos5x cos7x cosx cos3x+ cos5x cos7x Example 6 (Liearity Property) Give a periodic fuctio f ( x ) defie by,,< x < f ( x), < x < f x f x ( ) ( + ) (a) Sketch f ( x ) for < x <. (b) By subtractig the fuctio f ( x ) by uit (that is, by shiftig the graph of f ( x ) vertically dowward by uit), a odd fuctio is obtaied. Deoted the shirted g x. fuctio as ( ) Expad the shifted fuctio, g( x ) i a Fourier series.

15 (c) Hece, write the Fourier series for f ( x ) based o the liearity property of Solutio Fourier series. (a) (b) [figure] [figure] Sice the graph of g( x ) is symmetric respected to origi, this mea that ( ) a odd fuctio, therefore, the Fourier series expasio of g( x ) oly cotais sie terms oly ad havig the followig structure, ( ) g x b si x, where,,3,. b g( x) sixdx ( si ) xdx ( ) ( ) cosx cos cos + cos ( ) ( ( )),if is odd ( ( + )),if is eve,if is odd b,if is eve Fially, substitute b ito the Fourier series, ( ) g x b si x b six+ b six+ b si3x+ b six+ 3 six+ ( si ) x+ si3x+ ( si ) x+ 3 g( x) six+ si3x+ si5x+ si7x g x is

16 (b) Sice the graph of g( x ) ca be obtaied from f ( ) dowward, this mea, mathematically, the relatio for f ( x ) ad ( ) ( ) ( ) ( ) ( ) g x f x f x g x + By the liearity property, F { f ( x) } F g( x) F g( x) { } { } F { } + + x by shifted uit vertically g x is, six+ si3x+ si5x+ si7x f ( x) + six+ si3x+ si5x+ si7x Example: A fuctio f ( ) f ( x) x is defied by x,< x< 8 x, < x < 8 Show that the half-rage sie series for f ( ) f 3 x si si 8 ( x) x over the iterval [,8] is Solutio Half rage sie series of f ( x ), ( ) si ω, where ( ) f x b x b f x siω xdx, ω he half period, 8, 6, so, ω x b f ( x) si xdx f ( x) si dx ω 6 8 x 8 x xsi dx ( 8 x) si dx I + J

17 8 x 8 x I x cos cos dx x 8 8 x x cos si cos + si + ( )( ) ( ) si ( ) 8 6 cos + si ( ) 8 8 x 8 8 x J ( 8 x) cos cos ( dx) x 8 8 x ( 8 x) cos si ( )( ) ( ) si( ) cos + si ( ) 8 6 cos + si b I + J 3 si ( ) ( ) cos + si cos si + + ( ) ( ) Half rage sie series of f ( x ), ( ) f x b si ω x 3 si si ωx, ω 8 3 ( ) x si si ( Show ). 8 8

18 Example Give a periodic fuctio defied by si x, < x<, f ( x) si x, < x <, f x f x+. ( ) ( ) Expad f ( x ) i a Fourier series. Solutio [figure] Observed that the graph is symmetric respected to y -axis, so f ( x ) is a eve fuctio. he, the Fourier series expasio of f ( x ) oly cotai costat ad cosie terms (sice it is a eve fuctio), which is havig the followig structure, f ( x) a + a cos x. First, fid the a, a f ( x) dx sixdx [ cos x]. he, fid a, where,,3,. a f ( x) cosxdx sixcosxdx six si x six ( cosxdx) cosx cos x ( ) + cosx ( sixdx) + ( cos ) + sixcosxdx cos ( ) + a

19 a ( cos ) a si x ( cos ) ( ),if is eve, ( )( ) + ( ( ) ),if is odd,, ( + )( ),if is eve, a ( )( ) +,if is odd,, a sixcosxdx Fially, substitute a, a, a (where ) ito the Fourier series, f ( x) a + a cos x a + acosx+ acosx+ a3cos3x+ acosx+ a5cos5x+ + ( ) + cosx+ ( ) + cosx+ ( cos5 ) x cosx+ cosx+ cos6x+ cos8x

20 Fourier Series Exercises. Determie the Fourier series for the fuctio defied by f(x) x ; for ( < x < ) f(x) f(x + ).. State whether each of the followig products is odd, eve, or either. Assume that all the fuctios are defied over < x <. (a) x 3 cos x (b) x si 3x (c) si x si 3x (d) x e x (e) (x + 5) cos x (f) si x cos x 3. A fuctio f(x) is defied by f(x) x f(x) f(x + ). ; for < x < Express the fuctio (a) as a half-rage cosie series, (b) as a half-rage sie series.. A fuctio f(t) is defied by f(t) ; for < t < f(t) t ; for < t < f(t) f(t + ). Determie its Fourier series. 5. A periodic fuctio f(x) is defied by f(x) x ; for ( < x < ) f(x) f(x + ). Determie the Fourier series up to ad icludig the third harmoics.

21 6. Determie the Fourier series represetatio of the fuctio f(x) defied by f(t) 3 ; for ( < t < ) f(t) 5 ; for ( < t < ) f(t) f(t + ). 7. Determie the half-rage cosie series for the fuctio f(x) si x defied i the rage < x <. 8. A fuctio is defied by f(x) + x ; for ( < x < ) f(x) x ; for ( < x < ) f(x) f(x + ). Obtai the Fourier series. 9. A periodic fuctio is defied by f(x) A si x ; for ( < x < ) f(x) A si x ; for ( < x < ) f(x) f(x + ). Determie its Fourier series up to ad icludig the fourth harmoic.. Determie the Fourier series to represet a half-wave rectifier output curret, i amperes, defied by i f(t) A si ωt ; for ( < t < ) f(t) ; for ( < t < ) f(t) f(t + ).

Fourier Series and their Applications

Fourier Series and their Applications Fourier Series ad their Applicatios The fuctios, cos x, si x, cos x, si x, are orthogoal over (, ). m cos mx cos xdx = m = m = = cos mx si xdx = for all m, { m si mx si xdx = m = I fact the fuctios satisfy