Fourier Series. with the period 2π, given that sin nx and cos nx are period functions with the period 2π. Then we.

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1 . Definition We c the trigonometric series the series of the form + cos x+ b sin x+ cos x+ b sin x+ or more briefy + ( ncos nx+ bnsin nx) () n The constnts, n nd b, n ( n,, ) re coefficients of the series (Fourier coefficients). If the series () converges, its sum is periodic function with the period, given tht sin nx nd cos nx re period functions with the period. Then we f x f x+. obtin ( ) ( ). Computtion of the coefficients f Wht conditions needed to impose on ( ) converging to f ( x )? x so tht there exists trigonometric series Suppose tht the periodic function with the period of cn be represented by trigonometric series which converges to in the interv(, ); tht is, + ( ncos nx+ bnsin nx) () n Integrting both sides from to yieds dx dx + ncos nxdx bnsin nxdx + n Since, dx cos nxdx cos nxdx then, n n nd bnsin nxdx bn sin nxdx ( ) f x dx dx Before computing ndb, you need to recognize tht for integer n nd, Prepred by Mong Mr nd Ngov Simrong

2 If n, we obtin cos nx cos xdx cos nxsin xdx sin nxsin xdx nd if n, we obtin cos xdx sin x cos xdx ( II ) sin xdx For, mutipying both sides of () by cos x yieds cos x cos x+ ( ncos nxcos x+ bnsin nxcos x) n Integrting from to yieds cos xdx cos xdx n cos nx cos xdx bn sin nx cos xdx + + n Resuting from ( I ) nd (II) we obtin ( ) cos cos f x xdx xdx cos xdx Once gin, mutipying both sides of () by sin x yieds sin x sin x+ ( ncos nxsin x+ bnsin nxsin x) n sin xdx sin xdx n cos nxsin xdx bn sin nxsin xdx + + n From (I) nd (II) we obtin ( ) sin sin f x xdx b xdx b b sin xdx ( I ) Prepred by Mong Mr nd Ngov Simrong

3 3. Expnding functions into Exmpe: Given the periodic function with the period defined s x, < x. Find its Fourier series. x xdx x cos xdx sinx x sin xdx cos x b sin cos ( ) + x xdx x xdx + Then we obtin sin x sin x sin 3x + sin x + ( ) + 3 Exmpe: Given the periodic function with the period defined s foow x,for x x,for < x Find its Fourier series. Determining coefficients of Fourier: dx ( x) dx xdx + ( x) cos xdx x cos xdx + xsin x xsin x + sin xdx + cosx cosx +,for even ( cos ) 4,for odd sin xdx Prepred by Mong Mr nd Ngov Simrong 3

4 b ( x) sin xdx xsin xdx + Hence, we obtin the series 4 cos x cos 3x cos x cos( p+ ) x ( p + ) 4. Remr on expnding functions into Fourier series For ny re λ nd the periodic function ψ ( x) with the period, we hve λ+ ( x) dx ψ ( ) ψ x dx λ (The proof of this property is omitted) Exmpe: Expnd into Fourier series the periodic( ) function xover the interv x. soution: dx xdx Therefore, cos nxdx xcos nxdx xsinnx cosnx + n n b sin nxdx xsin nxdx xcosnx sinnx + n n n + sin nx n n Prepred by Mong Mr nd Ngov Simrong 4

5 . Fourier series of the even nd odd functions ψ x is n even function, we obtin If ( ) ( x) dx ψ ( ) ψ nd if the function ϕ ( x) is odd, then ϕ ( x) dx. x dx In the expnsion of n odd function into Fourier series, the product nd sin xis n even function. Therefore, f x dx f x cos xdx ( ) ( ) b sin xdx sin xdx cos xis odd If we expnd n even function the product sin xis n odd function nd cos x is n even function. Therefore, ( ) ( ) f x dx f x cos xdx b sin xdx Exmpe: Expnd the periodic function with the period defined by x, x We notice tht the function is n even one. Then, b f x sin xdx ( ) 3 x dx x dx 3 3 cos xdx x cos xdx ( ) x sinx xcosx 4cos sin x 3 + Therefore, we obtin ( ) 4 x + cos x 3 Exmpe: A periodic function with the period is defined s 4 Prepred by Mong Mr nd Ngov Simrong

6 Answer: if x < if x 4 sin x sin 3x sin x Fourier series of the function with period Let be period function with the period, different from in gener. Let expnd this ind of function into Fourier series. Ting x t, we obtin f t + cos t + b sin t where ( ) f t dt f t cos tdt b f t sin tdt Going bc to the previous vribe x: x x t, t dt dx then dx cos xdx b ( ) f x sin xdx Therefore, + cos x+ b sin x Exmpe: Find the Fourier coefficients corresponding to the function, < x < 3, < x < with period. Write the corresponding Fourier series. Since the period, then. Therefore, ( ) f x dx 3dx 3 cos xdx 3cos xdx Prepred by Mong Mr nd Ngov Simrong 6

7 3 x 3 x cos dx sin Thus, 3 nx b 3sin dx sin dx f ( ) 3 x 3 cos cos,if iseven 6,if is odd ( x) sin x sin x sin x Exercises.Expnd x, x.expnd x( x), x 3.Expnd in Fourier series in the interv (, ) f ( x) < < in Fourier series if the period is. < < in Fourier series if the period is. the function x, x x, < x f x x, in sine series. 4.Expnd the function ( ) in the interv [ ].Expnd the function xin the interv [ ], in cosine series. Prepred by Mong Mr nd Ngov Simrong 7