Even and Odd Functions


 Maximillian McGee
 1 years ago
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1 Eve d Odd Fuctios Beore lookig t urther emples o Fourier series it is useul to distiguish two clsses o uctios or which the Euler Fourier ormuls or the coeiciets c be simpliied. The two clsses re eve d odd uctios which re chrcterized geometriclly by the property o symmetry with respect to the yis d the origi respectively. b d si d Deiitio o Eve d Odd Fuctios Alyticlly is eve uctio i its domi cotis the poit wheever it cotis d i  or ech i the domi o. See igure below. The uctio is odd uctio i its domi cotis the poit wheever it cotis d i   or ech i the domi o. See igure b below. Note tht or odd uctio. Emples o eve uctios re. Emples o odd uctios re 3 si.
2 Arithmetic Properties The ollowig rithmetic properties hold: The sum dierece o two eve uctios is eve. The product quotiet o two eve uctios is eve. The sum dierece o two odd uctios is odd. The product quotiet o two odd uctios is eve. These properties c be veriied directly rom the deiitios see tet or detils. Itegrl Properties I is eve uctio the d d I is odd uctio the d These properties c be veriied directly rom the deiitios see tet or detils.
3 Cosie Series Suppose tht d ' re piecewise cotiuous o [ d tht is eve periodic uctio with period. The / is eve d si/ is odd. Thus d K b K Hece the Fourier series o is Thus the Fourier series o eve uctio ists oly o the ie terms d tt term d is clled Fourier ie series. Sie Series Suppose tht d ' re piecewise cotiuous o [ d tht is odd periodic uctio with period. The / is odd d si/ is eve. Thus K b si d K It ollows tht the Fourier series o is b si Thus the Fourier series o odd uctio ists oly o the sie terms d is clled Fourier sie series. 3
4 4 Epd  s hlrge sie series over the itervl. weget si d b g si si d d b b si 3 [ ] 4 3 [ ] 3 si 4 Obti the ie series over pi d d d d d d
5 5 Emple : Swtooth Wve o 3 Cosider the uctio below. This uctio represets swtooth wve d is periodic with period T. See grph o below. Fid the Fourier series represettio or this uctio. ± Emple : Coeiciets o 3 Sice is odd periodic uctio with period we hve I ll h h F i i i K K si si d b It ollows tht the Fourier series o is si
6 Emple : Grph o Prtil Sum 3 o 3 The grphs o the prtil sum s 9 d re give below. Observe tht t is discotiuous t ± d t tthese poits the series coverges to the verge o the let d right limits s give by Theorem.3. which is zero. The Gibbs pheomeo gi occurs er the discotiuities. Eve Etesios It is ote useul to epd i Fourier series o period uctio origilly deied oly o [ ] s ollows. Deie uctio g o period so tht g g g The uctio g is the eve periodic etesio o. Its Fourier series which is ie series represets o [ ]. For emple the eve periodic etesio o o [ ] is the trigulr wve g give below. g 6
7 Odd Etesios As beore let be uctio deied oly o. Di Deie uctio h o period d so tht t h h h The uctio h is the odd periodic etesio o. Its Fourier series which is sie series represets o. For emple the odd periodic etesio o o [ is the swtooth wve h give below. h ± Geerl Etesios As beore let be uctio deied oly o [ ]. Di Deie uctio k o period d so tht t k k k m where m is uctio deied i y wy istet with Theorem.3.. For emple we my deie m. The Fourier series or k ivolves both sie d ie terms d represets o [ ] regrdless o how m is deied. Thus there re iiitely my such series ll o which coverge to o [ ]. 7
8 Emple Cosider the uctio below. As idicted previously we c represet either by ie series or sie series o [ ]. Here. The ie series or coverges to the eve periodic etesio o o period 4 d this grph is give below let. The sie series or coverges to the odd periodic etesio o o period 4 d this grph is give below right. 8
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