Introductory Explorations of the Fourier Series by

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1 page Itroductory Exploratios of the Fourier Series by Theresa Julia Zieliski Departmet of Chemistry, Medical Techology, ad Physics Momouth Uiversity West Log Brach, NJ Copyright Theresa Julia Zieliski All rights reserved. You are welcome to use this documet i your ow classes but commercial use is ot allowed without the permissio of the author. Goal of this exercise: To give studets the opportuity to efficietly ad effectively practice usig a Fourier Series Expasio. This exercise serves as a foudatio for future work i quatum mechaics where liear combiatios of atomic orbitals ad expressios of solutios to the Schrodiger Equatio are writte as expasios usig sets of fuctios that are complete orthoormal sets. Prerequisites:. A basic uderstadig of Mathcad icludig drawig graphs of fuctios.. Itegral calculus, summatios, trigoometric fuctios. 3. Be able to distiguish betwee eve ad odd fuctios. 4. Ability to use a Mathcad documet as a template for extesio of the exercise. 5. Itroductory lectures o Fourier series or textbook readigs o the topic. (This documet is aotated to assist the ovice Mathcad user.) Objectives: At the ed of the lesso you will be able to:. express a simple fuctio as a Fourier Series.. compute the Fourier coefficiet for the first few terms i a expasio. 3. prepare graphical represetatios of the origial fuctio ad the Fourier fit to that fuctio. 4. discuss the priciples uderlyig a Foruier expasio. Creatio Date: July 4, 996 Modified: October, 3 page

2 page Backgroud: I physical scieces we ofte express fuctios as series. Power series are very commo, for example the Taylor series. Aother importat series is the Fourier series. The Fourier series is a trigoometric series, a sum of sie ad cosie terms. The Fourier series is importat because certai fuctios that caot be expaded as a Taylor series ca be expaded istead usig a Fourier series. The Fourier series is also the prefered series for represetig periodic fuctios. Examples of periodic fuctios iclude harmoic oscillators, rotors of various kids, pedulums, etc. Sice the Fourier series is a sum of sie ad cosie terms it is essetially a periodic series. More geerally we ca say that the Fourier series is a expasio of a periodic fuctio, f(x), with a uiformly coverget series, i.e. a sum of sie ad cosie terms, i a rage from 'a' to 'b' where x is greater tha or equal to 'a' ad less tha or equal to 'b'. For a rage [a,b] that is symmetric about zero, if f(x) is a eve fuctio, f(-x) = f(x), the oly cosie terms cotribute to the sum ad if f(x) is a odd fuctio, f(-x) = -f(x), the oly sie terms cotribute to the sum. A series is uiformly coverget for a fuctio f(x) if, i a give iterval, the series equals f(x) for every value of x i the iterval. Expasios of a fuctio as a ifiite series like this are oly possible because the sie ad cosie fuctios form a complete orthoormal set of fuctios that fully spa the space of the periodic fuctio that they are beig used to approximate. Mathematically the orthoormal property is expressed as: b sc i ( x) sc j ( x) := δ ij (ote: this equatio is toggled off) a where sc i is either the sie(ix) or cosie(ix) ad sc j is either sie(jx) or cosie(jx) ad δ i,j is the Kroecker delta. δ i,j = whe i is equal to = j = whe i is ot equal to j Creatio Date: July 4, 996 Modified: October, 3 page

3 page 3 The followig expressios are useful i derivatios of the coefficiets of the terms i a Fourier series. These expressios are toggled off i this part of the documet. si( x) cos( x) d x = =,,, 3,... d x = =,, 3,... = = cos( m x) si( x) d x = m =,,, 3,... ad =,,, 3,... cos( m x) cos( x) d x = m ot equal to = m = ot equal to si( m x) si( x) d x = = m = ot equal to m ot equal to Creatio Date: July 4, 996 Modified: October, 3 page 3

4 page 4 The geeral equatios for Fourier series expasios are show here. Notice the use of a ad b for the coefficiets for the cos ad si terms respectively. Here F(x) is the fuctio that is to be fit by the Fourier series. I the sectios that follow you will be led through a series of exercises that will put ito practice the Fourier series method for both odd ad eve fuctios. F(x) = some fuctio of x f(x) = a + a cos x + b si x a ( ) ( ) = = = F(x) cos( x) =,,,... May of the equatios i this documet were created with Mathtype ad thus are ot active equatios. Whe you click o such equatios a text box surroud will appear. b p = F(x) si ( x) L p =,, 3,... for eve fuctios we have: a = F( x)cos( x) =,,,... b = For odd fuctios we have: b = F( x)si( x) a = These expressios for a ad b assume that the itegrals exist ad that F(x) is a cotiuous fuctio. For piecewise regular fuctios, each cotiuous compoet of the fuctio must be determied separately. Creatio Date: July 4, 996 Modified: October, 3 page 4

5 page 5 Let us start with a simple Fourier expasio for a periodic fuctio, a simple step fuctio. Whe you work with this documet do ot repeat the exercises usig the same variable ames. This will cofuse Mathcad ad give you a headache tryig to debug your Mathcad code. - - =,< x < = < x <,= Fx ( ) Defiig the step fuctio (iactive here but active below). Plottig the step fuctio. Fx ( ) := x Gx ( ) := if( x, F( ), ) GG( x) := if( x, F ( ), ) Gx ( ) + GG( x) Settig up the Fourier expasio. x :=, 3... j :=.. N x is defiig the rage for the calculatio. N ca equal up to ay large iteger. Notice the relatioship betwee N ad i x Notice the way the rage of x is defied. fx ( ) := 4 j N = j + si[ ( j + ) x] Values of N are etered below just above the graph. The etry poit for this iteger was placed there so as to make exploratio ad graphical isights possible without excessive scrollig of the scree view. N is a Mathcad Global variable i this documet. Depedig o how you have your worksheet optios set a chage i N will immediately result i a ew figure or you may eed to press F9 to do do the calculatios that will show a ew figure. This ca also be accomplished by pressig the = sig i the first tool bar of the widow. Creatio Date: July 4, 996 Modified: October, 3 page 5

6 page 6 N fx ( ) GG( x) + G( x) Vary N, the umber of terms i the expasio ad describe the cosequeces as N is icreased to a large umber. Record your observatios i your otebook. Note how the Fourier series behaves relative to the step fuctio it is desiged to match. Cocisely express the fuctio that is beig expressed by the series expasio show above. This meas to write out explicitly the first few terms i the series for the fittig fuctio. x Creatio Date: July 4, 996 Modified: October, 3 page 6

7 page 7 Sample Problem: Expad the fuctio F(x)=x i the iterval - to by a Fourier series. Solutio: This fuctio is a odd fuctio. Oly sie terms will show up i the expasio. F( x) := x x :=, 3... :=.. M. First we write the fuctio ad set the iterval for the expasio.. The we choose a upper limit for the umber of terms i the expasio (the value for M). Exploratios o the effect of chagig the upper limit are easily implemeted by just chagig the value for M. M is a global variable ad is placed below just above the graph. B := Fx ( ) si( x) 3. Next we have the itegral for B. I this itegral ote the presece of the fuctio F(x). Ay odd fuctio ca be used here. is the idex used to idetify the itegrals i the expasio for the fittig process. Usig M ad makes possible a more geeral exploratio of the series i the area below. Ask Mathcad to show several of the B coefficiets i the f( x) := M = ( B si( x) ) Fourier expasio for the fuctio give here. Commet o their magitude ad the sigificace of the magitude as icreases. Write out the expressio that would correspod to the coeficiets for the cosie terms i a Fourier series expasio of this fuctio. Show that several of these A coefficiet values are i fact zero. 4. We wrote f(x) here istead of F(x) so as to prevet Mathcad from overwritig our origial fuctio. We will compare the fourier series fit, f(x), of the fuctio to the fuctio, F(x), itself as a exercise. Now write out by had f(x) as the sum of several terms. Use at least four terms i your aswer. Creatio Date: July 4, 996 Modified: October, 3 page 7

8 page 8 Notice M is a upper limit i the sum show above. Varyig M allows rapid ad efficiet exploratio of the effect of icreasig umber of terms i a Fourier expasio. M 5 4 fx ( ) Fx ( ) x Notice the dips ad humps i the Fourier curve. The Fourier fit overshoots the fuctio f(x)=x. This is kow as the Gibbs Pheomeo. The Gibbs pheomeo remais eve at higher values of N ad is most oticeable at the eds of the rage for the fit. Explore the fittig process by varyig N ad recordig your observatios. Follow this example to examie aother odd fuctio, f(x)=x 3. This should take o more tha a few miutes to implemet. If you choose to retype the various parts of the template, do ot use the same ames for fuctios. This makes Mathcad very uhappy ad the results are upredictable. Creatio Date: July 4, 996 Modified: October, 3 page 8

9 page 9 Exercise: For several values of M compute ad compare the umerical value of F(x) to the f(x) predicted by the Fourier series at x = pi ad at x = zero. record your observatios. What will the calculated Fourier series value for f(x) always be at x = + pi or x = - pi? What will the value of F(x) always be at at x = + pi or x = - pi? (Hit: Type f() = ad see what you get. Also Look at the graphs ad the calculated values you obtaied.). Mastery Level Exercise Demostrate that you uderstad this material by expressig the fuctio f(x) = x with a Fourier series over the iterval x. Test your model by usig it with oe other eve fuctio. This is a eve fuctio so be careful about what you use i the Fourier expasio. Create space below to complete this assigmet. Ackowledgmet: Partial support for this work was provided to TJZ by the Natioal Sciece Foudatio's Divisio of Udergraduate Educatio through grat DUE # ad by the New Traditios project at the Uiversity of Wiscosi - Madiso through the Natioal Sciece Foudatio's Divisio of Udergraduate Educatio through grat DUE # TJZ also thaks George Hardgrove of St. Olaf College for the square wave expasio that lead to the developmet of this documet. Creatio Date: July 4, 996 Modified: October, 3 page 9