Series FOURIER SERIES - cse.salford.ac.uk

1 SeriesFOURIER SERIESG raham S McDonaldA self-contained Tutorial Module for learningthe technique of FOURIER SERIES analysislTable of contentslBegin Tutorialc of trig on using solutionsFull worked solutionsSection 1: Theory31. TheorylA graph ofperiodicfunctionf(x) that has periodLexhibits thesame pattern everyLunits along thex-axis, so thatf(x+L) =f(x)for every value ofx. If we know what the function looks like over onecomplete period, we can thus sketch a graph of the function over awider interval ofx(that may contain many periods) f(x)xPERIOD = LTocJJIIJIBackSection 1: Theory4lThis property of repetition defines afundamental spatial fre-quencyk=2 Lthat can be used to give afirst approximationtothe periodic patternf(x).

2 F(x)'c1sin(kx+ 1) =a1cos(kx) +b1sin(kx),where symbols with subscript 1 are constants that determine the am-plitude and phase of this first approximationlA muchbetter approximationof the periodic patternf(x) canbe built up by adding an appropriate combination ofharmonicstothis fundamental (sine-wave) pattern. For example, addingc2sin(2kx+ 2) =a2cos(2kx) +b2sin(2kx) (the 2nd harmonic)c3sin(3kx+ 3) =a3cos(3kx) +b3sin(3kx) (the 3rd harmonic)Here, symbols with subscripts are constants that determine the am-plitude and phase of each harmonic contributionTocJJIIJIBackSection 1: Theory5 One can even approximate a square-wave pattern with a suitable sumthat involves a fundamental sine-wave plus a combination of harmon-ics of this fundamental frequency.

3 This sum is called aFourier seriesFundamental + 5 harmonicsFundamental + 20 harmonics xPERIOD = LFundamental Fundamental + 2 harmonicsTocJJIIJIBackSection 1: Theory6lIn this Tutorial, we consider working out FOURIER SERIES for func-tionsf(x) with periodL= 2 . Their fundamental frequency is thenk=2 L= 1, and their FOURIER SERIES representations involve terms likea1cosx,b1sinxa2cos 2x,b2sin 2xa3cos 3x,b3sin 3xWe also include a constant terma0/2 in the FOURIER SERIES . Thisallows us to represent functions that are, for example, entirely abovethex axis. With a sufficient number of harmonics included, our ap-proximate SERIES can exactly represent a given functionf(x)f(x) =a0/2 +a1cosx+a2cos 2x+a3cos 3x+.

4 +b1sinx+b2sin 2x+b3sin 3x+..TocJJIIJIBackSection 1: Theory7A more compact way of writing the FOURIER SERIES of a functionf(x),with period 2 , uses the variable subscriptn= 1,2,3,..f(x) =a02+ n=1[ancosnx+bnsinnx]lWe need to work out theFourier coefficients(a0,anandbn) forgiven functionsf(x). This process is broken down into three stepsSTEPONEa0=1 2 f(x)dxSTEPTWOan=1 2 f(x) cosnxdxSTEPTHREEbn=1 2 f(x) sinnxdxwhere integrations are over a single interval inxofL= 2 TocJJIIJIBackSection 1: Theory8lFinally, specifying a particular value ofx=x1in a FOURIER SERIES ,gives a SERIES of constants that should equalf(x1).

5 However, iff(x)is discontinuous at this value ofx, then the SERIES converges to a valuethat ishalf-waybetween the two possible function values f(x)xFourier SERIES converges to half-way point"Vertical jump"/discontinuityin the function representedTocJJIIJIBackSection 2: Exercises92. ExercisesClick onExerciselinks for full worked solutions (7 exercises in total).Exercise (x) be a function of period 2 such thatf(x) ={1, <x<00,0<x< .a) Sketch a graph off(x) in the interval 2 <x<2 b) Show that the FOURIER SERIES forf(x) in the interval <x< is12 2 [sinx+13sin 3x+15sin 5x+..]c) By giving an appropriate value tox, show that 4= 1 13+15 17+.}

6 LTheorylAnswerslIntegralslTriglNotationT ocJJIIJIBackSection 2: Exercises10 Exercise (x) be a function of period 2 such thatf(x) ={0, <x<0x,0<x< .a) Sketch a graph off(x) in the interval 3 <x<3 b) Show that the FOURIER SERIES forf(x) in the interval <x< is 4 2 [cosx+132cos 3x+152cos 5x+..]+[sinx 12sin 2x+13sin 3x ..]c) By giving appropriate values tox, show that(i) 4= 1 13+15 17+..and (ii) 28= 1 +132+152+172+..lTheorylAnswerslIntegrals lTriglNotationTocJJIIJIBackSection 2: Exercises11 Exercise (x) be a function of period 2 such thatf(x) ={x,0<x< , <x<2 .a) Sketch a graph off(x) in the interval 2 <x<2 b) Show that the FOURIER SERIES forf(x) in the interval 0<x<2 is3 4 2 [cosx+132cos 3x+152cos 5x+.]}}

7 ] [sinx+12sin 2x+13sin 3x+..]c) By giving appropriate values tox, show that(i) 4= 1 13+15 17+..and (ii) 28= 1 +132+152+172+..lTheorylAnswerslIntegrals lTriglNotationTocJJIIJIBackSection 2: Exercises12 Exercise (x) be a function of period 2 such thatf(x) =x2over the interval 0<x<2 .a) Sketch a graph off(x) in the interval 0<x<4 b) Show that the FOURIER SERIES forf(x) in the interval 0<x<2 is 2 [sinx+12sin 2x+13sin 3x+..]c) By giving an appropriate value tox, show that 4= 1 13+15 17+19 ..lTheorylAnswerslIntegralslTriglNotatio nTocJJIIJIBackSection 2: Exercises13 Exercise (x) be a function of period 2 such thatf(x) ={ x,0<x< 0, <x<2 a) Sketch a graph off(x) in the interval 2 <x<2 b) Show that the FOURIER SERIES forf(x) in the interval 0<x<2 is 4+2 [cosx+132cos 3x+152cos 5x+.}

8 ]+ sinx+12sin 2x+13sin 3x+14sin 4x+..c) By giving an appropriate value tox, show that 28= 1 +132+152+..lTheorylAnswerslIntegralslTri glNotationTocJJIIJIBackSection 2: Exercises14 Exercise (x) be a function of period 2 such thatf(x) =xin the range <x< .a) Sketch a graph off(x) in the interval 3 <x<3 b) Show that the FOURIER SERIES forf(x) in the interval <x< is2[sinx 12sin 2x+13sin 3x ..]c) By giving an appropriate value tox, show that 4= 1 13+15 17+..lTheorylAnswerslIntegralslTriglNota tionTocJJIIJIBackSection 2: Exercises15 Exercise (x) be a function of period 2 such thatf(x) =x2over the interval <x<.

9 A) Sketch a graph off(x) in the interval 3 <x<3 b) Show that the FOURIER SERIES forf(x) in the interval <x< is 23 4[cosx 122cos 2x+132cos 3x ..]c) By giving an appropriate value tox, show that 26= 1 +122+132+142+..lTheorylAnswerslIntegrals lTriglNotationTocJJIIJIBackSection 3: Answers163. AnswersThe sketches asked for in part (a) of each exercise are given withinthe full worked solutions click on theExerciselinks to see thesesolutionsThe answers below are suggested values ofxto get the SERIES ofconstants quoted in part (c) of each 2,2.(i)x= 2, (ii)x= 0,3.(i)x= 2, (ii)x= 0, 2, 0, 2, .TocJJIIJIBackSection 4: Integrals174.

10 IntegralsFormula for integration by parts: baudvdxdx= [uv]ba badudxvdxf(x) f(x)dxf(x) f(x)dxxnxn+1n+1(n6= 1)[g(x)]ng (x)[g(x)]n+1n+1(n6= 1)1xln|x|g (x)g(x)ln|g(x)|exexaxaxlna(a>0)sinx cosxsinhxcoshxcosxsinxcoshxsinhxtanx ln|cosx|tanhxln coshxcosecxln tanx2 cosechxln tanhx2 secxln|secx+ tanx|sechx2 tan 1exsec2xtanxsech2xtanhxcotxln|sinx|cothx ln|sinhx|sin2xx2 sin 2x4sinh2xsinh 2x4 x2cos2xx2+sin 2x4cosh2xsinh 2x4+x2 TocJJIIJIBackSection 4: Integrals18f(x) f(x) dxf(x) f(x) dx1a2+x21atan 1xa1a2 x212aln a+xa x (0<|x|<a)(a>0)1x2 a212aln x ax+a (|x|>a>0)1 a2 x2sin 1xa1 a2+x2ln x+ a2+x2a (a>0)( a<x<a)1 x2 a2ln x+ x2 a2a (x>a>0) a2 x2a22[sin 1(xa) a2+x2a22[sinh 1(xa)+x a2+x2a2]+x a2 x2a2] x2 a2a22[ cosh 1(xa)+x x2 a2a2]TocJJIIJIBackSection 5: Useful trig results195.