Introduction to Fourier Series - Purdue University

1 The BasicsFourier seriesExamplesIntroduction to Fourier SeriesMA 16021 October 15, 2014 The BasicsFourier seriesExamplesEven and odd functionsDefinitionA functionf(x) is said to beeveniff( x) =f(x).The functionf(x) is said to beoddiff( x) = f(x).Graphically, even functions have symmetry about they-axis,whereas odd functions have symmetry around the BasicsFourier seriesExamplesEven and odd functionsExamples:ISums of odd powers ofxare odd: 5x3 3xISums of even powers ofxare even: x6+ 4x4+x2 3 Isinxis odd, and cosxis evensinx(odd)cosx(even)IThe product of two odd functions is even:xsinxis evenIThe product of two even functions is even:x2cosxis evenIThe product of an even function and an odd function isodd: sinxcosxis oddThe BasicsFourier seriesExamplesIntegrating odd functions over symmetric domainsLetp >0 be any fixed number. Iff(x) is an odd function, then p pf(x)dx= : The area beneath the curve on [ p,0] is the same asthe area under the curve on [0,p], but opposite in sign.

2 So, theycancel each other out!A AThe BasicsFourier seriesExamplesIntegrating even functions over symmetric domainsLetp >0 be any fixed number. Iff(x) is an even function, then p pf(x)dx= 2 p0f(x) : The area beneath the curve on [ p,0] is the same asthe area under the curve on [0,p], but this time with the samesign. So, you can just find the area under the curve on [0,p] anddouble it!AAThe BasicsFourier seriesExamplesPeriodic functionsDefinitionA functionf(x) is said to beperiodicif there exists a numberT >0 such thatf(x+T) =f(x) for everyx. The smallest suchTis called theperiodoff(x).Intiutively, periodic functions have repetitive periodic function can be defined on a finite interval, thencopied and pasted so that it repeats cosxare periodic with period 2 Isin( x) and cos( x) are periodic with period 2 IIfLis a fixed number, then sin(2 xL) and cos(2 xL) haveperiodLSine and cosine are the most basic periodic functions!

3 The BasicsFourier seriesExamplesFourier seriesLetp >0 be a fixed number andf(x) be a periodic functionwith period 2p, defined on ( p,p). The Fourier Series off(x) isa way of expanding the functionf(x) into an infinite seriesinvolving sines and cosines:f(x) =a02+ n=1ancos(n xp) + n=1bnsin(n xp)( )wherea0,an, andbnare called the Fourier coefficients off(x),and are given by the formulasa0=1p p pf(x)dx, an=1p p pf(x) cos(n xp)dx,( )bn=1p p pf(x) sin(n xp)dx,The BasicsFourier seriesExamplesFourier SeriesRemarks:ITo find a Fourier Series , it is sufficient to calculate theintegrals that give the coefficientsa0,an, andbnand plugthem in to the big Series formula, equation ( ) ,f(x) will be piecewise advantage that Fourier Series have over Taylor Series :the functionf(x) can have discontinuities!Useful identities for Fourier Series : ifnis an integer, thenIsin(n ) = sin( ) = sin(2 ) = sin(3 ) = sin(20 ) = 0 Icos(n ) = ( 1)n={1neven cos( ) = cos(3 ) = cos(5 ) = 1,but cos(0 ) = cos(2 ) = cos(4 ) = BasicsFourier seriesExamplesFourier coefficients of an even functionIff(x) is an even function, then the formulas for the coefficientssimplify.}

4 Specifically, sincef(x) is even,f(x) sin(n xp) is an oddfunction, and thusbn=1p p podd f(x) evensin(n xp) odddx= 0 Therefore, for even functions, you can automatically conclude(no computations necessary!) that thebncoefficients are all BasicsFourier seriesExamplesFourier coefficients for an odd functionIff(x) is odd, then we get two freebies:a0=1p p podd f(x)dx= 0an=1p p podd f(x) oddcos(n xp) evendx= 0 Note: In general, your function may be neither even nor odd. Inthose cases, you should use the original formulas for computingFourier coefficients, given in equation ( ).The BasicsFourier seriesExamplesDisclaimerThe following examples are just meant to give you an idea ofwhat sorts of computations are involved in finding a Fourierseries. You re not meant to be able to carry out thesecomputations yet. So just sit back, relax, and enjoy the ride!

5 The BasicsFourier seriesExamplesExample 1 Letf(x) be periodic and defined on one period by the formulaf(x) ={ 1 2< x <010< x <2 Graph off(x) (original part in green): 4 224 The BasicsFourier seriesExamplesExample 1 Sincef(x) is an odd function, we conclude thata0=an= 0 foreachn. A bit of computation revealsbn=12 2 2f(x) sin(n x2)dx=2n (1 cos(n )) =2n (1 ( 1)n)Thereforef(x) = n=12n (1 ( 1)n) sin(n x2)=4 sin( x2) n=1+43 sin(3 x2) n=3+ Notice: The evenbnterms are all 0 since 1 ( 1)n= 1 1 = 0whennis BasicsFourier seriesExamplesExample 1If we plot the firstNnon-zero terms, we get approximations off(x):N= 1N= 2N= 3N= 4 2 112 2 112 2 112 2 112N= 10N= 20N= 30N= 40 2 112 2 112 2 112 2 112 The BasicsFourier seriesExamplesExample 1 2 112 Observations:IAs the numberof terms used increases, theapproximation gets closer andcloser to the original functionIThe original function hasa discontinuity atx= 0.}

6 Theapproximation converges to0 there, which is the average ofthe right- and left-hand limitsasx general, iff(x)has a discontinuity atx0, then theFourier Series converges to the average oflimx x+0f(x)andlimx x 0f(x).The BasicsFourier seriesExamplesExample 2 Letf(x) be periodic and defined on one period by the formulaf(x) ={0 < x <0x20< x < Graph off(x) (original part in green): 2 2 2 The function is neither even nor odd since it has no BasicsFourier seriesExamplesExample 2 After some calculations (which are very tedious and involve lotsof IBP),a0=13 2, an=2( 1)nn2, bn=( 1)n(2 2n2) 2n3 Thus,f(x) =16 2 a02+ n=1 2( 1)nn2 ancos(nx) +( 1)n(2 n2 2) 2n3 bnsin(nx) The BasicsFourier seriesExamplesExample 2 Plot of Fourier Series (first 20 terms): 2 2 22 2 Notice: Atx= , the Series converges to12( 2+ 0) = BasicsFourier seriesExamplesExample 2By plugging inx= into the Fourier Series forf(x) and usingthe fact that the Series converges to 22, 22= 26+ n=1(2( 1)nn2cos(n ) +( 1)n(2 2n2) 2n3 sin(n ))Because sin(n ) = 0 and ( 1)ncos(n ) = ( 1)n( 1)n= 1, onecan derive the following formula ( example from lecture 14) n=11n2= BasicsFourier seriesExamplesThat s all for now!}

7 Reminders:IReview FridayINext office hours: Thursday, 6:00 7:00 pm (Math 609)IExam 2: Monday, 6:30 pm in Elliot