Phys. 201 Mathematical Physics 1 Dr. Nidal M. Ershaidat Doc. 12

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1 Physics Departmet, Yarmouk Uiversity, Irbid Jorda Phys. Mathematical Physics Dr. Nidal M. Ershaidat Doc. Fourier Series Deiitio A Fourier series is a expasio o a periodic uctio (x) i terms o a iiite sum o sies ad cosies. Fourier series make use o the orthogoality relatioships o the sie ad cosie uctios. The computatio ad study o Fourier series is kow as harmoic aalysis ad is extremely useul as a way to break up a arbitrary periodic uctio ito a set o simple terms that ca be plugged i, solved idividually, ad the recombied to obtai the solutio to the origial problem or a approximatio to it to whatever accuracy is desired or practical. Illustratio Examples o successive approximatios to commo uctios usig Fourier series are illustrated below. Applicatio i the solutio o ordiary dieretial equatios I particular, sice the superpositio priciple holds or solutios o a liear homogeeous ordiary dieretial equatio, i such a equatio ca be solved i the case o a sigle siusoid, the solutio or a arbitrary uctio is immediately available by expressig the origial uctio as a Fourier series ad the pluggig i the solutio or each siusoidal compoet. I some special cases where the Fourier series ca be summed i closed orm, this techique ca eve yield aalytic solutios. Geeralized Fourier Series Ay set o uctios that orm a complete orthogoal system have a correspodig geeralized Fourier series aalogous to the Fourier series. For example, usig orthogoality o the roots o a Bessel uctio o the irst kid gives a so-called Fourier-Bessel series. Author s eidal@yu.edu.jo, Address: Physics Departmet, Yarmouk Uiversity 63 Irbid Jorda

2 Computatio o Fourier series The computatio o the (usual) Fourier series is based o the ollowig itegral idetities which represet the orthogoality relatioships o the sie ad the cosie uctios: ( m x) si( x) si δ m () ( m x) cos( x) cos δ m () ( m x) cos( x) si (3) ( m x) si (4) ( m x) cos (5) For m,, where δ m is the Kroecker delta. Usig the method or a geeralized Fourier series, the usual Fourier series ivolvig sies ad cosies is obtaied by takig (x) cos x ad (x) si x. Sice these uctios orm a complete orthogoal system over [-, ], the Fourier series o a uctio (x) is give by ( x) a a ( x) b si( x) cos (6) a, a a are called the Fourier coeiciets. These coeiciets are obtaied usig the ollowig relatios: a ( x) (7) a b ( x) cos( x) ( x) si( x),, 3,. Note that the coeiciet o the costat term a has bee writte i a special orm compared to the geeral orm or a geeralized Fourier series i order to preserve symmetry with the deiitios o a ad b. A Fourier series coverges to the uctio (equal to the origial uctio at poits o cotiuity or to the average o the two limits at poits o discotiuity) (8) (9)

3 lim ( x) lim ( x) x x x x lim ( x) lim ( x) x x i the uctio satisies so-called Dirichlet coditios. or or < x x <, () As a result, ear poits o discotiuity, a "rigig" kow as the Gibbs pheomeo, illustrated above, ca occur. For a uctio (x) periodic o a iterval [-, ] istead o [-, ], a simple chage o variables ca be used to trasorm the iterval o itegratio rom [-, ] to [-, ]. et x x () () Solvig or x gives x x, ad pluggig this i gives Thereore, ( ) x a a x cos b si x (3) a ( x ) (4) a b ( x ) ( x ) cos x si x (5) (6) Similarly, the uctio is istead deied o the iterval [,], the above equatios simply become a ( x ) (7) 3

4 a b ( ) x x d x cos (8) ( ) x x d x si (9) I act, or (x) periodic with period, ay iterval [x, x ] ca be used, with the choice beig oe o coveiece or persoal preerece (Arke 985, p. 769). The coeiciets or Fourier series expasios o a ew commo uctios are give i Beyer (987, pp. 4-4) ad Byerly (959, p. 5). Oe o the most commo uctios usually aalyzed by this techique is the square wave. The Fourier series or a ew commo uctios are summarized i the table below. The coeiciets or Fourier series expasios o a ew commo uctios are give i Beyer (987, pp. 4-4) ad Byerly (959, p. 5). Oe o the most commo uctios usually aalyzed by this techique is the square wave. The Fourier series or a ew commo uctios are summarized i the table below. Examples Fuctio (x) Fourier series sawtooth wave x si x x square wave x 4 H H, triagle wave T(x), 3,5, K 3, 5, K ( ) si x 8 si x I a uctio is eve so that (x) (- x), the (x) si(x) is odd. (This ollows sice si(x) is odd ad a eve uctio times a odd uctio is a odd uctio.) Thereore, b or all. Similarly, i a uctio is odd so that (x) - (-x), the (x) cos(x) is odd. (This ollows sice cos(x) is eve ad a eve uctio times a odd uctio is a odd uctio.) Thereore, a or all. Complex Coeiciets The otio o a Fourier series ca also be exteded to complex coeiciets. Cosider a real-valued uctio (x). Write Now examie ( ) x A e i x () i m x ( x) e A e i x e i m x () i ( m ) x e () A [ ( m) i si ( m) ] A cos (3) 4

5 So A A m δ (4) m, (5) A ( x) e i x The coeiciets ca be expressed i terms o those i the Fourier series A ( x) [ cos ( x) i si ( x) ] ( )[ ( ) ( )] x cos x i si x ( ) x ( )[ ( ) ( )] x cos x i si x ( a i b ) or < a or < ( a i b ) or < For a uctio periodic i [- /, /], these become ( x i ( x ) ) A e i ( x ) ( x) e < > (6) (7) (8) (9) (3) A (3) These equatios are the basis or the extremely importat Fourier trasorm, which is obtaied by trasormig A rom a discrete variable to a cotiuous oe as the legth. 5

6 Reereces. Arke, G. "Fourier Series." Ch. 4 i Mathematical Methods or Physicists, 3rd ed. Orlado, F: Academic Press, pp , Askey, R. ad Haimo, D. T. "Similarities betwee Fourier ad Power Series." Amer. Math. Mothly 3, 97-34, Beyer, W. H. (Ed.). CRC Stadard Mathematical Tables, 8th ed. Boca Rato, F: CRC Press, Brow, J. W. ad Churchill, R. V. Fourier Series ad Boudary Value Problems, 5th ed. New York: McGraw-Hill, Byerly, W. E. A Elemetary Treatise o Fourier's Series, ad Spherical, Cylidrical, ad Ellipsoidal Harmoics, with Applicatios to Problems i Mathematical Physics. New York: Dover, Carslaw, H. S. Itroductio to the Theory o Fourier's Series ad Itegrals, 3rd ed., rev. ad el. New York: Dover, Davis, H. F. Fourier Series ad Orthogoal Fuctios. New York: Dover, Dym, H. ad McKea, H. P. Fourier Series ad Itegrals. New York: Academic Press, Follad, G. B. Fourier Aalysis ad Its Applicatios. Paciic Grove, CA: Brooks/Cole, 99.. Groemer, H. Geometric Applicatios o Fourier Series ad Spherical Harmoics. New York: Cambridge Uiversity Press, Körer, T. W. Fourier Aalysis. Cambridge, Eglad: Cambridge Uiversity Press, Körer, T. W. Exercises or Fourier Aalysis. New York: Cambridge Uiversity Press, Kratz, S. G. "Fourier Series." 5. i Hadbook o Complex Variables. Bosto, MA: Birkhäuser, pp. 95-, ighthill, M. J. Itroductio to Fourier Aalysis ad Geeralised Fuctios. Cambridge, Eglad: Cambridge Uiversity Press, Morriso, N. Itroductio to Fourier Aalysis. New York: Wiley, Sasoe, G. "Expasios i Fourier Series." Ch. i Orthogoal Fuctios, rev. Eglish ed. New York: Dover, pp , Weisstei, E. W. "Books about Fourier Trasorms." 8. Whittaker, E. T. ad Robiso, G. "Practical Fourier Aalysis." Ch. i The Calculus o Observatios: A Treatise o Numerical Mathematics, 4th ed. New York: Dover, pp. 6-84, 967. Source iteret Weisstei, Eric W. "Fourier Series." From MathWorld - A Wolram Web Resource. Documet CRC Press C, Wolram Research, Ic. Terms o Use 6