Integral Transforms. Chapter 6 Integral Transforms. Overview. Introduction. Inverse Transform. Physics Department Yarmouk University

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1 Ovrviw Phy. : Mhmicl Phyic Phyic Dprm Yrmouk Uivriy Chpr Igrl Trorm Dr. Nidl M. Erhid. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Igrl Trorm Iroducio I igrl o h orm b ( ) ( ) K( α, )d g α g(α) i clld h (igrl) rorm o () by h Krl* K(α,). Th ur o h krl di h yp o h rorm. Th vribl α d r clld cojug vribl. For mpl: rqucy d im r cojug vribl. I i lo h c o wvvcor d poiio (k d ) * Grm word or uclu Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Ivr Trorm Th ivr rorm i did by: ( ) ( ) ( ) α b g α K α, d Th imporc o h igrl rorm ppr by lookig crully quio d. Som problm r diicul o olv i hir origil rprio or i hir domi. Th id i o mp h problm i ohr domi, olvig i i h w domi d h by chooig h ppropri domi d uig h ivr rorm h oluio i h origil domi i mppd bck! Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5 Ivr Trorm Th procdur i ummrizd chmiclly i Fig. -: Figur -: Schmic igrl rorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

2 Fourir Alyi Fourir Sri d Igrl Th prcuror o h rorm wr h Fourir ri o pr ucio i ii irvl. r h Fourir rorm w dvlopd o rmov h rquirm o ii irvl. Fourir ri r bic ool or olvig ordiry diril quio (ODE) d pril diril quio (PDE) wih priodic boudry codiio. Fourir igrl or opriodic phom r dvlopd i hi chpr. Th commo m or h ild i Fourir lyi. Appdi -: Fourir Sri Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 8 Domi o Applicio Th Fourir rorm i o udml imporc i brod rg o pplicio, icludig boh ordiry d pril diril quio, probbiliy, quum mchic, wv, dircio d irromry, igl d img procig, d corol hory, c... Fourir Trorm Th ppropri krl i imply iω d i rl pr (co ω) or i imgiry pr (iω). Bcu h krl r h ucio ud o dcrib wv, Fourir rorm ppr rquly i udi o wv d h rcio o iormio rom wv, priculrly wh ph iormio i ivolvd (dircio or mpl). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Domi o Applicio Empl I opic, h dircio pr i h Fourir rorm o h "obcl" rpoibl o "dircig" h wv. Th Fruhor dircio pr i h Fourir rorm o h mpliud lvig h dircig prur. I quum mchic h phyicl origi o h Fourir rorm i h duliy wv-mr, i.. h wv ur o mr d our dcripio o mr i rm o wv (Scio ). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Domi o Applicio Empl Th oupu o llr irromr, or ic, ivolv Fourir rorm o h brigh cro llr dik. Th lcro diribuio i om my b obid rom Fourir rorm o h mpliud o crd X ry. Elcro crig prim wr ud, bck i h ' o h l cury, i ordr o di h hp o uclu. Th dircio pr" i ud i ordr o di h "rucur" o h crig uclu. By uig ivr Fourir rorm. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

3 Domi o Applicio Empl I img procig, Fourir rorm i crucil ool. Th ipu i "pil" (rl) img which i dcompod io i i d coi compo. Th oupu o h rormio, i.. h rul o pplyig h Fourir rorm, i h img i h Fourir or rqucy domi. I h Fourir domi img, ch poi rpr priculr rqucy coid i h pil domi img. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Empl - FOURIER TRANSFORM OF GAUSSIAN Th Fourir rorm o Gui ucio i: g ( ) i ω ω d ( ) K ( α, ) I ordr o clcul () w compl h qur i h po ollow: i ω ω i ω Thi yild: ω i ω g ( ω) 5 d Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5 Empl - FOURIER TRANSFORM OF GAUSSIAN A impl chg o vribl (hi o origi), giv: g ( ω) ω p ( ω ) i ω d d ω ξ d which i Gui, bu i h (Fourir) ω pc. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm dξ Th biggr i, h i, h rrowr h origil Gui i ω ~. i, h widr i i Fourir rorm Shiig h Origi Thi i juiid by pplicio o Cuchy horm o h rcgl wih vric T, T, T iω, T iω or T, oig h h igrd h o igulrii i hi rgio d h h igrl ovr h id rom ±T o ±T iω bcom gligibl or T. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7 Ohr uul Krl Thr ohr uul krl ch givig ri o priculr rorm r Krl α J ( ) α α g g g ( α) ( ) α d ( α) ( ) J ( α ) ( α) ( ) α Trorm d d plc Trorm 7 Hkl Trorm 8 Mlli Trorm 9 8 Mlli d plc W hv Mlli rorm or -. g α ( α) d Γ( α) ( α )! plc rorm o i g α ( α) d α! 9 9 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

4 iriy ir Opror All h prviou igrl rorm (Fourir, plc, Hkl d Mlli) r lir d w c wri g ( α) ( ) A ivr opror i pcd o i uch h ( ) g( α) I grl, h drmiio o h ivr rorm i h mi problm i uig igrl rorm. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Ovrviw. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm - Dvlopm o h Fourir Igrl Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Fourir Sri & Fourir Igrl Fourir ri r uul i rprig cri ucio () ovr limid rg [, ], [,], d o o, or () or h iii irvl (,), i h ucio i priodic. Fourir rorm i grlizio o Fourir ri rprig opriodic ucio ovr h iii rg. Phyiclly hi m rolvig igl pul or wv pck io iuoidl wv. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Drivig Fourir Igrl W r rom h diiio o h coici o Fourir ri. For picwi rgulr ucio, () iyig h Dirichl codiio did i h irvl [-,], w c wri: ( ) co, d b, h Fourir coici r giv by: Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm b i ( ) co d, ( ) i d. b 5

5 Sp Th rulig Fourir ri i: ( ) ( ) d co ( ) i ( ) i d, Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm uig h rigoomric idiy: co α β coαcoβ i αi w hv: ( ) β co d 5- ( ) ( ) d ( ) co ( ) d 5- Sp Th p i o l pproch, rormig h irvl [-,] io [-,]. W lo di w vribl ω ω, ω, wih. Th w hv: or ( ) ω ( ) coω( ) d, ( ) dω ( ) coω( ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm d, 7 7 Fourir Igrl No h rm h vihd umig h ( ) d i. Eq. 7 i k h Fourir igrl, udr h ollowig codiio: ) () i picwi diribl ) () i picwi coiuou ) () i boluly igrbl, i.. i ii ( ) d 8 Fourir Igrl Epoil Form Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Fourir Igrl Thorm Eq. 7 c b wri : uig h c h: ( ) dω ( ) coω( ) d, ( ) dω ( ) iω( ) d i ω ( ) dω ( ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm bcu i ω(-) i odd ucio o ω. i ω d, Eq. i clld h Fourir igrl. 8 9 Th vribl ω Th vribl ω iroducd hr i rbirry mhmicl vribl. I my phyicl problm, howvr, i corrpod o h gulr rqucy ω. W my h irpr Eq. 8 or Eq. rprio o () i rm o diribuio o iiily log iuoidl wv ri o gulr rqucy ω,, i which hi rqucy i coiuou vribl. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5

6 Impor Applicio Drivio o Dirc Dl Fucio A Uul Rprio o δ Uig h Fourir igrl w c di Dirc dl ucio δ i ω ( ) ( ) i ω( ) dω dω Appdi - Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Ovrviw. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm - Fourir Trorm Ivr Thorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Uig h Epoil Trorm u di g(ω),, h Fourir rorm o h ucio (),, by g i ω ( ω) ( ) d. Epoil Trorm Th, rom Eq.,, w hv h ivr rlio, i ω dω. ( ) g( ω) Impor Rmrk Eq. d Eq. r lmo ymmricl, dirig i h ig o i. Th ymmry i mr o choic or covic. Thi cor i omim rplcd by i o quio d h ir cor ½ i h ohr. I phyic w r mor ird i h Fourir rorm (Eq.( d Eq. ) ) rhr h Eq. (Fourir igrl) 7 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

7 Th D Form - Phyic Movig h Fourir rorm pir (Fourir rorm d i ivr) o hr- dimiol pc, h pir bcom: g ( k ) ( ) i k r ( r ) d, r i k r ( r ) ( ). b ( ) g k d k Th igrl r ovr ll pc. Eq. b my b irprd pio o Erci: Vriy Eq. d Eq. b by ubiuig h (r) l-hd i id o o coiuum quio io o h igrd pl wv o h igucio; ohr quio d uig g(k) h h hr-dimiol bcom dl h mpliud ucio. o h wv p( - i k. r). 8 Fourir Coi d Si Trorm C o Ev d Odd Fucio I () i v h Eq. d Eq.,, rpcivly, c b wri : g c c ( ω) c( ) coω d, ( ) gc( ω) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm coω dω. I () i odd h Eq. d Eq.,, rpcivly, c b wri : g ( ω) ( ) iω d, ( ) g ( ω) iω dω. 5 7 Priy Th pir, Eq. d Eq. 5 r clld h Fourir coi rorm. Th ohr pir, Eq. d Eq. 7 r h i Fourir i rorm. Th Fourir coi rorm d h Fourir i rorm ch ivolv oly poiiv vlu (d zro) Th priy o () i ud o blih h rorm; bu oc h rorm r blihd, h bhvior o h ucio d g or giv rgum i irrlv. Th rorm quio impo dii priy: v or h Fourir coi rorm d odd or h Fourir i rorm. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Phyicl Mig I Eq. 7, () i big dcribd by coiuum o i wv. Th mpliud o iω i giv by ( ) g ( ω), i which g (ω) i h Fourir i rorm o (). ( ) g( ω) iω dω. I h ollowig mpl h impor pplicio o h Fourir rorm i h roluio o ii pul o iuoidl wv, i dild. Empl FINITE WAVE TRAIN Imgi h iii wv ri iω i clippd by Krr cll or urbl dy cll hur (Fig. -) o h w hv N iω, <, ω ( ) 8 N, >. ω I Fig. -, N, which rpr h umbr o cycl o h wv ri, i qul o 5. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7

8 Ampliud i h Fourir Spc Sigl-Sli Dircio Pr 5 () i odd, hu g N ω ( ω) i ω iω d. Igrig, w id h mpliud ucio: 9 Fig. - how h ir rm. g ( ω) i ( ω ω)( N ω ) ( ω ω) i ( ω ω)( N ω ) ( ω ω). Dpdc o g(ω) o rqucy. For lrg ω d ω ω, h ir rm will domi bcu o h domior (ω-ω ). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Thi i h mpliud curv or h igl-li dircio pr! Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm N v. Sprd i rqucy ( ω ω) For lrg N, g(ω) my lo b irprd Dirc Dl diribuio. Th coribuio ouid h crl mimum big mll i hi c, ω N c b k good mur o h prd i rqucy o our wv pul. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm ω Th lrgr N i, h mllr i h rqucy prd. For mll N, h prd i lrg d h codry mim bcom mor impor. Ucriy Pricipl I w r dlig wih m wv w hv E ħ ω E ħ ω E rpr ucriy i h rgy o our pul. Thr i lo ucriy i h im bcu our wv o N cycl rquir N/ω cod o p. N Tkig ω N ω N Th produc E ħ ω h h ω N ω Th Hibrg ucriy pricipl h: ħ h E, 5 which i clrly iid i hi mpl. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7 Ovrviw 8. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm Fourir Trorm o Driviv Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 8

9 Trorm o h Driviv Srig rom h diiio o h Fourir rorm or () d or d/d d ( ω) ( ) Igrig by pr: g ( ω) g i ω d d ( ) u dv i ω Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm ( ) d d v d i ω d. du i ω ( ) i ω 7 5 Diriio bcom muliplicio W hv g i ω d. g( ω) i ω ( ω) ( ) ( i ω) ( ) W u h c h, pr rom om c, () mu vih ± i ordr or h Fourir rorm o () o i. Th ir rm o h rh vih d w obi: g( ω) i ω g( ω). 9 i.. h rorm o h driviv i (iω)( im h rorm o h origil ucio. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 8 5 Trorm o h d Driviv Coidr Igrig by pr: g g ( ω) i ω ( ω) ( i ω) g ( ω) ( i ω) g( ω) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm d d ( ) i ω u du i ω d ( ) d ( ) dv v d d d ( ) ( ) i ω d ( ) i ω g ω i ω. d d d g ( ω) d i ω 5 Grlizio h Driviv Thi my rdily b grlizd o h h driviv o yild g ( ω) ( i ω) g( ω), providd ll h igrd pr vih ±. Thi i h powr o h Fourir rorm, h ro i i o uul i olvig (pril) diril quio. Th oprio o diriio h b rplcd by muliplicio i ω-pc. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5 A impor Empl H Flow PDE Fourir' w H Coducio I h rr, coducio i h rr o h rgy by microcopic diuio d colliio o pricl or qui-pricl wihi body du o mprur grdi. Th lw o h coducio, lo kow Fourir' lw, h h im r o h rr q hrough mril i proporiol o h giv grdi i h mprur (T) d o h r, righ gl o h grdi, hrough which h h i lowig. Th diril orm o Fourir' lw, i which w look h low r or lu o rgy loclly, i giv by q k T k i h hrml coduciviy (MKS ui W.m -.K - ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 55 9

10 H Equio I ordr o id h mprur ild ψ(,) (Tmprur i ) or giv ym, o h o olv h h quio (which c b drivd rom Fourir' lw) ψ(, ) ψ, i co clld h hrml diuiviy, lo kow Fourir co. i rld o h hrml coduciviy o mril o diy ρ by h impl rlio: k c P ρ whr c P i h h cpciy co prur. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5 Tmprur Fild Diig h Fourir rorm o ψ(,) i ω Ψ( ω, ) ψ(, ) d 5 Tkig h Fourir rorm o boh id o Eq. w g ( ) Ψ ω Th prviou quio i ODE or h Fourir rorm Ψ o ψ i h im vribl. Igrig w obi (,) ω lc l Ψ ω, ω Ψ ω (, ). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm or ω Ψ( ω, ) C 57 Th Igrio Co Th igrio co C my ill dpd o ω d, i grl, i drmid by iiil codiio. I c, CΨ(ω,) i h iiil pil diribuio o Ψ, o i i giv by h rorm (i ) o h iiil diribuio o ψ, mly, ψ(, ). Puig hi oluio bck io our ivr Fourir rorm, hi yild ψ (, ) C( ω) ω i ω dω. 7 Ψ( ω, ) 58 δ Fucio Iiil Tmprur Diribuio Tkig ψ(ω,) δ(ω,), C i ω-idpd. Igrig Eq. 7 by complig h qur w did i Empl (whr w clculd h Fourir rorm o Gui*), w g ψ ( ) p,, C *ψ(,) i h ivr Fourir rorm o C p (- ω ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Ovrviw. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm 5 - Covoluio Thorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

11 Covoluio - Diiio Covoluio i ud o olv diril quio, o ormliz momum wv ucio ( cio), d o ivig rr ucio. u coidr wo ucio () d g() wih Fourir rorm F() d G(),, rpcivly. W di h oprio * g Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm g( y) ( y) dy h covoluio o h wo ucio d g ovr h irvl (,).. Som uhor u h Grm word Flug (which m oldig) id o covoluio. 9 Covoluio - U Thi orm o igrl ppr i probbiliy hory i h drmiio o h probbiliy diy o wo rdom, idpd vribl Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Covoluio Grphicl Illurio ) y, (y) d For (y) d ( y) r plod i Fig. -.. Clrly, (y) d ( y) r mirror img o ch ohr i rlio o h vricl li y /, i..,, w could gr (y) by oldig ovr (y) o h li y /. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Bck o h Elcroic Alog Th oluio o Poio' quio (Chpr( Eq. 9), i.. ( ) ( ). ρ r ψ r dτ 5 ε r r c b wri ψ ( r ) ρ( r ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm ε r r τ. d ( ) g r ( r r ) which w my irpr h covoluio o chrg diribuio ρ( r ) d wighig ucio, ε r r 5 5 Fourir Trorm d Covoluio ' rorm Eq. by iroducig h Fourir rorm i ( ) ( ) ( ) ( ) ( y) g y y dy g y F F F( ) g( y) i y [ dy] G( ) ( ) G( ) i d * g d dy i d 5 Fourir Trorm d Covoluio For h pcil c, Eq. 5 giv F ( ) G( ) d ( y) g( y) dy Th miu ig i y ugg h modiicio b rid. W ow do hi wih g id o g uig dir chiqu. 5 7 Thi rul my b irprd ollow: Th Fourir ivr rorm o produc o Fourir rorm i h covoluio o h origil ucio, g. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

12 Prvl' Rlio Uiriy o Fourir Trorm Prvl' Rlio Eq. 5 d h corrpodig i d coi covoluio r o lbld Prvl rlio by logy wih Prvl horm or Fourir ri (Ark( Chpr 9- h, Chpr i h 7 h diio). Th Prvl' rlio F * * ( ω) G ( ω) dω ( ) g ( ) rl h produc o h ucio d g* o hir rpciv Fourir rorm (F( d G*) (i h rorm (Fourir) pc) d 5 9 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Drivio o h Prvl' Rlio Uig dl ucio rprio, w c wri * i ω * ( ) g ( ) d F ( ω) dω G ( ) Igrig ovr d uig δ w hv i ( ) ( ω ) i ( ω ω d ) i d d Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm * * ( ) g ( ) d F ( ω) G ( ) δ( ω) F * ( ω) G ( ω) dω, d d dω Spcil c () g() I h vry impor c whr () ) g(),, h igrl o boh id o Eq. 5 r ohig l bu ormlizio igrl. * * d F ω F ω dω 57 ( ) ( ) ( ) ( ) Thi impor rlio gur h i () i ormlizd i h "-pc",, h i Fourir rorm F(ω) (i h rorm (rqucy) pc) i ormlizd oo! Thi i wh w cll h uiriy o Fourir rorm which h lrg imporc i quum phyic. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7 7 Uiriy o Fourir Trorm I my b how h h Fourir rorm i uiry oprio (i h Hilbr pc, qur igrbl ucio). Th Prvl' rlio i rlcio o hi uiry propry. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Applicio I Fruhor dircio opic h dircio pr (mpliud) ppr h rorm o h ucio dcribig h prur. Wih iiy proporiol o h qur o h mpliud h Prvl rlio impli h h rgy pig hrough h prur m o b omwhr i h dircio pr m o h corvio o rgy. Prvl rlio my b dvlopd idpdly o h ivr Fourir rorm d h ud rigorouly o driv h ivr rorm. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7

13 Empl Sigl Sli Dircio A rcgulr pul i dcribd by, <, ( ), >. ) Th Fourir poil rorm i F ( ) ( ) i i i i d i i Thi i h igl-li dircio problm. Th li i dcribd by (). Th dircio pr mpliud i giv by h Fourir rorm F(). d 7 Empl Sigl Sli Dircio ) b) ) Nrrow li ucio o i d o high d b) i Fourir Trorm 77 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Empl Sigl Sli Dircio 78 Ovrviw 79 b) U Prvl' rlio o vlu i d ( ) F W lv hi clculio rci.. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm - Momum Rprio Cojug Vribl W I dvcd dymic d i quum mchic, lir momum d pil poiio r pir o vribl did i uch wy h hy bcom Fourir rorm dul o o ohr. Spil poiio d lir momum (or wvumbr pħk) ) r h EXAMPE o cojug vribl. Aohr pir i h im, rqucy pir. 8 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

14 Empl - Thrmodymic I hrmodymic mprur T d ropy S r cojug vribl. Prur (P) d volum (V) r lo cojug vribl. Th pir (T,S) or h pir (P,V) r ud o di ll h propri o hrmodymic ym uch h irl rgy. I c ll hrmodymic poil r prd i rm o cojug vribl. 8 Impor Coquc i Phyic Th duliy ld urlly o ucriy pricipl i phyic clld h Hibrg ucriy pricipl. 8 I iicl phyic, pir o iv d iiv propri o giv ym orm pir o cojug vribl Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Rl Spc I hi cio w hll r wih h uul pc diribuio d driv h corrpodig momum diribuio. For h o dimiol c our wv ucio ψ() h h ollowig propri: ) ψ * () ψ() d i h probbiliy o idig h quum ym bw d d. ) ψ() d i ormlizd (ol probbilii ) ) Th pcio vlu, i.. h vrg poiio o h pricl log h -i i * * P( ) d ψ ( ) ψ( ) d 58 ψ ( ) ψ( ) d 59 8 Momum Spc W w ucio g(p) h will giv h m iormio bou h momum: ) g * (p) g(p) dp i h probbiliy h h pricl h momum bw p d pdp. ) g(p) i ormlizd (ol probbilii ) ( p) g( p) dp. ) Th pcio vlu, momum o h pricl i * g g ( p) p g( p). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm * 85 i.. h vrg p d Momum (Fourir) Spc Such ucio i giv by Fourir rorm o h pc ucio ψ(), i.. i p ħ ( p) ψ( ) d, g g ħ Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm * i p ħ ( p) ψ ( ) d. * ħ Prvl' rlio gur h ormlizio o g(p) i ψ() i ormlizd. Th corrpodig D momum ucio i g ( p) ψ ħ ( ħ) ( r ) i r p d r. 8 Epcio Vlu Chckig propry () m howig h * * ħ d p g ( p) p g( p) d ψ ( ) ψ( ) d i d whr p i h momum opror i h pc rprio, W rplc h momum ucio by Fourir rormd pc ucio, d h ir igrl bcom i p * ħ p Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm p ( ) ħ ( ) ( ). ψ ψ dpd d 5 87

15 Epcio Vlu Uig h pl wv idiy i p( ) ħ d ħ ( ) i p ħ p, 7 d i Hr p i co, o opror. Subiuig io Eq. d igrig by pr, holdig d p co, w obi ( ) [ ] i p ħ ψ * ħ d p dp ( ) ψ( ) d d. ħ i d Hr w um ψ() vih ±, limiig h igrd pr. Uig h Dirc dl ucio, Eq. 8 rduc o Eq. 5 o vriy our momum rprio (qd( qd). Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 8 88 Ovrviw. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8. plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Fourir Trr Fucio 9 Ovrviw 9 Sl Rdig Ark h Ediio Scio 5.7, pg: 9-9 Ark 7h Ediio Scio.5, pg: 997- S boh cio.. Igrl Trorm - Fourir. Dvlopm o h Fourir Igrl. Fourir Trorm Ivr Thorm. Fourir Trorm o Driviv 5. Covoluio Thorm. Momum Rprio 7. Trr Fucio 8.plc Trorm 9. plc Trorm o Driviv.Ohr Propri.Covoluio (Flug) Thorm.Ivr plc Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 8- plc Trorm Diiio Th plc rorm () or o ucio F() i did by ( ) { F ( ) } lim F ( ) d F ( ) d. Th igrl 9 F ( ) d, d o i! For ic, F() my divrg poilly or lrg.. Howvr, i hr i om co uch h F ( ) M, 7 9 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 5

16 Courmpl whr M i poiiv co or uicily lrg, >, h plc rorm (Eq.( ) ) will i or > ; F() i id o b o poil ordr. A courmpl, ( ) 9 do o iy h codiio giv by Eq. 7 d i o o poil ordr. do o i. F { } "Filur" i Th plc rorm my lo il o i bcu o uicily rog igulriy i h ucio F() ; ; h i, divrg h origi or.. Th plc rorm { } do o i or. d 7 95 Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm iriy Sic, or wo ucio F() d G(),, or which h igrl i { F ( ) bg( ) } { F ( ) } b { G( ) } 7 Th oprio dod by i lir. 9 Elmry Fucio Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Elmry Fucio 98 Sl Rdig Ark h Ediio pg: 95-9 Ark 7 h Ediio pg: 8- Ivr Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm

17 Diiio Th plc rorm o F() i (Eq.( 9) ( ) { F( ) } F ( ) d. Th ivr i, by diiio, F 7 ( ) { ( ) } ( ) d. Thi ivr rorm i o uiqu. Uiciy o - rch' Thorm Two ucio F () d F () my hv h m rorm, (). Howvr, i hi c F () F () N() whr N() i ull ucio, idicig h ( ) d, For ll poiiv. (rch' horm) I phyic, w d - hror w k N(). N 7 o b uiqu d Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Drmiio o - Svrl Mhod i: () A bl o rorm c b buil up d ud o crry ou h ivr rormio, cly bl o logrihm c b ud o look up ilogrihm, ) A grl chiqu or uig h clculu o ridu, (Ark i Scio 5. ), Pril Frcio Epio ) Numricl ivrio. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Uig Tbl Uilizio o bl o rorm (or ivr rorm) i cilid by pdig () i pril rcio. O, h plc rorm () occur i h orm o rcio g()/h(), whr g() d h() r polyomil wih o commo cor, g() big o lowr dgr h h(). I h cor o h() r ll lir d diic, h by h mhod o pril rcio w my wri ( ) c c c Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm whr h c i r idpd o. Th i r h roo o h(). 75 Pril Frcio Epio I y o o h roo, y,, i mulipl (occurrig m im), h () h h orm c, m, m, ( ) c i ci Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm Filly, i o o h cor i qudric, ( p q), h h umror, id o big impl co, will hv h orm (S Empl ) b c p q i

18 Empl Pril Frcio Epio ( ) k 78 ( k ) Th pril rcio mhod coi i wriig h prviou rcio h um o wo rcio. (No h dgr o h polyomil i h umror i h rh) c b ( ) 79 ( k ) Th lh i dvlopd d lik powr o r qud, i.. k c ( k ) ( b) 8 c, ; b ; c k k which giv, or, c, b d -, Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm. Empl Pril Frcio Epio W illy hv ( ) 8 ( ), k F() i obid by clculig h plc rorm o h wo rcio i h rh. W hv (S Ark, Elmry Fucio, pg 95-9). { } d { cok } 8 k d coquly: { ( ) } cok 8 ( k ) Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm 7 9- plc Trorm o Driviv 9 9- plc Trorm o Driviv Sl Rdig S ) cur by Dr. Rd Almomi ) Ark h Ediio Scio 5.9 pg: h Ediio Scio.8 pg: -8. Dr. Nidl M. Erhid - Mhmicl Phyic - Phy. - Chpr Igrl Trorm لقققةةةة الا لا لا لا خخخخييييررررةةةة cur اللللحححح Dr. Nidl M. Erhid 8

19 Chpr Igrl Trorm Appdi - Fourir Sri Iroducio Picwi rgulr ucio Diiio: A picwi rgulr ucio i ucio () which h ii umbr o dicoiuii d ii umbr o rm vlu ovr igl irvl. or }, { Empl: ( ) or Aohr Empl Th wooh ucio ( ) or < < or < < Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dirichl Codiio A picwi ucio () which i ) priodic o priod, b) igl vlud ovr h irvl [, ] d ( ) c) d i ii. i id o iy h Dirichl codiio Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Fourir Sri A picwi ucio () which vrii h Dirichl codiio c b prd i Fourir ri o i d coi, i.. co b i ( ) Whr,, d b coici. r clld h Fourir A Fourir ri i pio o priodic ucio () i rm o iii um o i d coi. Fourir ri mk u o h orhogoliy rliohip o h i d coi ucio. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

20 Orhogoliy Th ollowig igrl idii (or m,#): i co i ( m ) i( ) ( m ) co( ) ( m ) co ( ) d δ d δ d m m i ( m ) d co ( m ) δ m Krockr ymbol d rpr h orhogoliy rliohip o h i d coi ucio. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Hrmoic Alyi Compuio o Fourir Sri Th compuio d udy o Fourir ri i kow hrmoic lyi d i rmly uul wy o brk up rbirry priodic ucio io o impl rm h c b pluggd i, olvd idividully, d h rcombid o obi h oluio o h origil problm or pproimio o i o whvr ccurcy i dird or prcicl. Clculio o Fourir Coici Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Coici o Fourir Sri For hi purpo w igr boh id o h Fourir pio, i..: d d co d b i d ( ) Compuig Accordig o h orhogoliy rlio w hv: which giv: ( ) d ( ) Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri d Clculio o For hi purpo w ir muliply boh id o h Fourir pio by co m d igr ovr h priod [, ], i..: ( ) co m d co m d co m co d δ b co m i d m Accordig o h orhogoliy rlio w hv: ( ) co m d δ m m Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

21 Clculio o b Now w ir muliply boh id o h Fourir pio by im d igr ovr h priod [, ], i..: ( ) i md i md i m co d b i m i d δ m Accordig o h orhogoliy rlio w hv: ( ) i m d b δ bm m m Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Orhogoliy Th orhogoliy rlio or priodic ucio o h irvl [-, ] bcom: i co i ( m ) i( ) ( m ) co( ) ( m ) co( ) d d δ m δ m d i ( m ) d co( m ) δ m Krockr ymbol d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Coici o Fourir Sri Th FS coici r hu giv by h ollowig rlio: d ( ) co( ) d ( ) ( ) i( ) d,,,. No h w diiguih h coici o h co rm by wriig i i pcil orm i ordr o prrv ymmry wih h diiio o d b. b Covrgc o Fourir Sri A Fourir ri covrg o h ucio qul o h origil ucio poi o coiuiy or o h vrg o h wo limi poi o dicoiuiy) lim lim ( ) lim ( ) ( ) lim ( ) I ii h Dirichl codiio. or < < or, Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Fourir Sri or y priodic Fucio Grlizio For ucio () priodic o irvl [-, ] id o [-, ], impl chg o vribl c b ud o rorm h irvl o igrio rom [-, ] o [-, ]. d d ( ) co b i Thror ( ) d, ( ) co d, b ( ) i d. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8

22 Grl Form o Fourir Sri Uig h diiio o h coici, h Fourir ri i wri : ( ) ( u) du [ co ( u) i ( u) co udu] co udu Applicio Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Empl Empl: Fid h Fourir pio or h priodic ucio: - ( ) - - or or Soluio compu h Fourir coici d ( ) ( ) ( ) co( ) d or or d d co ( ) d ( ) i ( ) b d i ( ) d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Soluio W d o compu h igrl ( ) ( ) ( ) co d i i d u v v du u dv whr w ud h igrio by pr: u, dv co( ) d v co( ) d i ( ) i( ) i [ ] ( ) d [ co( ) co( ) ] ( ) Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Thu w hv: ( ) d ( ) I i v I i odd [ ] co Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

23 b Now w compu b, W procd or, i.. b i( ) d ( ) co ( ) co d u v v du u dv co co ( ) co( ) ( ) ( ) ( ) d Soluio Th Fourir ri or h ucio i ( ) ( ) or or [( ) ] ( ) co i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Aohr Empl Th wooh ucio or < < ( ) or < < - - Soluio compu h Fourir coici ( ) ( ) d ( ) d ( ) d ( ) co ( ) d b ( ) i( ) d or < < or < < ( ) co ( ) d ( ) co( ) ( ) i( ) d ( ) i( ) d d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Soluio W d o compu h ollowig igrl: co co ( ) d ( ) d ( ) i ( ) d co( ) ( ) ( ) i ( ) d co ( ) ( ) Soluio d h igrl co co ( ) d i( ) i( ) ( ( ) ) ( ) d i( ) i( ) ( ( ) ) d d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

24 Soluio d h igrl i ( ) d co( ) ( ) co d i co ( ) ( ) d co ( ) co( ) co ( ) d Uul Igrl co co i ( ) d ( ) d ( ) ( ) d ( ) ( ) i ( ) d ( ) ( ) co co i i ( ) d ( ) ( ) ( ) d ( ) ( ) d ( ) ( ) d ( ) Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri ( ) co ( ) d ( ) co( ) d ( ) co ( ) co co ( ) d co ( ) ( ) d ( ) co ( ) ( ( ) ) ( ) ( ) d d d b Now w compu b, W procd or, i.. b ( ) i( ) d ( ) i( ) d ( ) i( ) i i ( ) d i( ) ( ) d ( ) i( ) ( ) ( ) ( ) ( ) d ( ) d d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Th Sri Th Fourir ri or h ucio i ( ) ( ) or < < or < < ( ) i Eio o Compl Coici - Applicio Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5

25 Compl Form o Fourir Sri Uig h diiio: i i i ( ), co( ), i Subiuig i h diiio o Fourir ri co b i w g: i i ( ) i b i ( ) [ ] [ ] i i i [ ] i i b [ i b ] i Compl Form o Fourir Sri Th prviou quio i i b i ( ) [ ] [ ] i c b wri i h orm i [ ] i i b [ i b ] ( ) i Th limi - k io ccou h rm -i c i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Th Compl Coici For >, w hv c c i b c * i b c I ordr o id c w muliply boh id o h quio: i c by m i ( ) d igr ovr h irvl [, ], i.. i m i ( m ) ( ) d c c m ( ) d i m d c δm Som Propri d U o Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Covrgc Th problm, w hv, i o drmi h umbr o coici o hould clcul i ordr o g clo poibl o h hp o h priodic ucio (). Th r o covrgc o h ri giv id wh o op. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Covrgc - Propri - I h ucio () i dicoiuou, h om o h Fourir coici will vry /. Th covrgc i grl i low d my rm r dd o giv h hp o () S Empl. - I h ucio () i coiuou, h h Fourir coici vry /. Th covrgc i grl i d w rm r dd o giv h hp o (). S Empl. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

26 Igrio o Fourir Sri Igrio o Fourir Sri Coidr om priodic ucio () o priod l. Th Fourir ri i did by: ( ) co b i l l Igrig h prviou quio bw d giv: ( ) d co b i d l l i.. ( ) l bl d i co l l Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Igrio o Fourir Sri ( ) d ( ) l bl i co l l l b i i l l co co l l Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Applicio (), - < < Coidr h ucio () o priod. Compu h igrio o h Fourir ri o () ovr h limi d. ( ) d d ( ) ( ( ) ) l i i l l b co co l l b i co b i ( co ) Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Applicio (), - < < W c ily h, W hv: i.. which giv: i.. ( ) [ co ] b ( ) ( ) ( ) ( ) ( ) co co Diriio o Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7

27 Diriio o Fourir Sri Coidr om priodic ucio () o priod l. Th FS i did by: ( ) co b i l l Th driviv i, i pricipl, giv by: ( ) d d i Hr o hould b crul wh () i dicoiuou, ic h driviv i o did dicoiuii. l l b co l l Th driviv i lowly covrg Fourir ri i () i coiuou bcu ppr cor i h Fourir coici Chpr Igrl Trorm Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Empl Th Fourir ri or h ucio ( ) or or Fourir Sri or Dicoiuou Fucio i ( ) ( ) ] ( ) co [ i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Coici () i priodic o priod : i v ucio ( ) d d Coici co( ) d co( ) d co( ) d i( ) [ ] i( ) [ co( ) ] ( co( ) ) ( ) ) d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5

28 Coici b b i i ( ) ( ) ( ) [ co( ) ] co( ) [ co] d d d Coici b b b ( ) i d i( ) d odd Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 55 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 57 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri w 58 Th Fir Thr Trm ( ) [ co ] i ( ) [ co ] i i ( ) ( ) i ( ) co co i i i 9 ( ) ( ) co i 9 Fourh d Fih Trm ( ) co co i i i i 9 ( ) ( ) i 5 5( ) co co co 5 i i i i i ( ) ( ) co i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 59 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

29 ( ) i i co i co 9 i co 5 5 i 5 5 i 5( ) co co co 5 i i i i i ( ) ( ) 5 i 5 ( ) ( ) 9 8 i7 7 co 7 i8 8 co 9 i9 9 co i i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Soluio Th Fourir ri or h ucio Th Swooh Fucio i ( ) ( ) or < < or < < ( ) i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri Th Fir Trm ( ) i ( ) i i ( ) ( ) i ( ) i i i ( ) ( ) i Th Fir Trm 5 ( ) i i i i ( ) ( ) i 5 ( ) i i i i i ( ) ( ) i 5 5 ( ) ( ) i7 i8 i i i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 5 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

30 Fourir Sri or Coiuou Fucio Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Empl Th Fourir ri or h ucio ( ) or Fourir Sri or ( ) or - i ( ) ( ) co Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Th Fir Four Trm ( ) co ( ) co co ( ) ( ) co co co co 9 ( ) ( ) co 9 ( ) 9 ( ) ( ) co ( ) co co co co Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7

31 Ohr Trm ( ) ( ) co 5 co 8 co 9 co co 8 co Fourir Sri or h Trigulr Fucio ( ) or Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Empl Th Fourir ri or h ucio or i ( ) d ( ) ( ) ( ) () ( k ) (( k ) ) co Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 75 Coici () i priodic o priod : i v ucio ( ) d d d d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 7 Coici co( ) d co co ( ) ( ) co( ) d ( ) co( ) ( ) d [ i( ) ] i( ) d [ co( ) ] ( co( ) ) ( ) ) d d Coici b b b ( ) i d i( ) d odd Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 77 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 78

32 Empl 5 () Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 79 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Fourir Sri or ( ) or < < or < < Empl : Fourir Si Sri Th Fourir ri or h ucio or < < ( ) or < < () i ( ) odd i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Fourir Coici Th Fourir ri or h ucio l ( ) d b ( ) i d i d l odd odd co ( ( ) ) b odd v, ( ) co d l odd v ( ) odd i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Th Fir Trm i odd ( ) i i.7 5 ( ) ( ) i i ( ) i ( ) i i i 5 ( ) i 5 7 ( ) ( ) i Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8

33 Fourir Sri or h Squr Wv Squr Wv Th Fourir ri or h ucio h or ( ) h or i h ( ) i, odd Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 8 Coici d Th ucio h or < < ( ) h or < < i priodic o priod : ( ) d co d, ( ) Coici b b ( ) b h co h odd v i d ( ) i d h ( ) h [ ( ) ] i d Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 87 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 88 5 Th Fir Trm ( ) i ( ) i i ( ) ( ) i 5 5( ) i i i 5 5 5( ) ( ) i 5 5 ( ) i, odd Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9

34 Uul ik hp:// hp:// Chpr Igrl Trorm Appdi - Dirc Dl Fucio Drivio Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 9 Ivrig Fourir Igrl Th Fourir igrl horm giv i ω ( ) dω ( ) I h ordr i rvrd w my rwri i : i ω ( ) ( ) ( ). dω d Apprly h quiy i curly brck bhv dl ucio δ(). W migh k Eq. prig u wih rprio o h Dirc dl ucio. Alrivly, w k i clu o w drivio o h Fourir igrl horm. Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 95 i ω d, Drivig Fourir Trorm uig δ Shiig h igulriy rom o, w hv ( ) ( ) δ ( ) d, lim whr δ (-) i quc diig h diribuio δ (-). Eq. um h () i coiuou. W k δ (-) o b: i ( ) ( ) δ ( ) Phy. Chpr : Igrl Trorm Dr. Nidl M. Erhid - Appdi - Clculig Fourir Sri 9 i ω( ) dω, Subiuig Eq. io Eq., w hv: i ω ( ) lim ( ) ( ). dωd 5 Drivig Fourir Trorm uig δ Irchgig h ordr o igrio i Eq. 5 d kig h limi w rriv h Fourir igrl horm. dωd. i ω ( ) ( ) ( ) lim Th idiicio δ( ) i ω( ) dω provid vry uul rprio o h dl ucio. i ω ( ) dω ( ) i ω d 7 Chpr Igrl Trorm Appdi - Mlli Trorm Th Pl wv mpl Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 97

35 Mlli Trorm o ik ' clcul h Mlli rorm d how h α α i k i d α k α! W orc h igrl io rcbl orm by irig covrgc cor -b α i k b α ( b i k d ) d α α α ( b i k) d ( b i k) Γ( α ) whr w ud chg o vribl (b-ik) ( ) Soluio Tkig h limi b, w hv uig i k d Im co k d R d k α α i k α α ( α )!i α α i k α α α α i i α i k w g d i d k ( α )!co S Ark h Ediio - Erci 5.. pg 95 α α k ( α )! 5 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri 99 Dr. Nidl M. Erhid Phy. Chpr : Igrl Trorm - Appdi - Clculig Fourir Sri

36 Phyic Dprm, Yrmouk Uivriy, Irbid Jord Phy. Mhmicl Phyic Dr. Nidl M. Erhid Doc. APACE TRANSFORM ECTURE BY DR. RAID AMOMANI DEPARTMENT OF MATHEMATICS YARMOUK UNIVERSITY A) INTEGRA TRANSFORMS Th igrl rorm or y ucio ( ) c b wri whr ( ) β α { ( ) } F( ) K( ) ( ) T, d () K, i h krl (uclu) o h rormio. W lo di h ivr (igrl) rorm by: β α ( ) T { F( ) } K ( ) F( ), d () B) APACE TRANSFORM Wh h krl ( ) K,, α d β h h rorm (Eq. ) i clld h plc rorm (ymbol ) d w wri { ( ) } F( ) ( ) Applicio Elmry Fucio d () d > () { }, No h or < h krl which do o i (5)

37 ( ) d ( ), >. (7) i (8) { } { }. Th ivr plc rorm o Eq. 8 i -, >. (9) A priculr mpl { i } i d u dv i co d I ordr o compu h igrl o h lh o Eq. w igr by pr (wic) ollow i I i d d co u dv u dv co J co d i u dv Thi yild J u d dv J I I I I () d icidlly J co d ( ) Uig h diiio o h compl poil i i plc rormig i, w hv { co ii } liriy o h igrl rorm, w c wri { i } { co } i { i }. I () () () () co ii d. Thk o h (5) Combiig Eq. 8, Eq. d Eq. ld o h compl idiy: i i ()

38 C) APACE TRANSFORM OF DERIVATIVES clcul h plc rorm or h driviv y ( ); Igrio by pr; { y ( )} { y ( ) d. (7) u dv { y ( )} y( ) y( ) d { y( )} y( ) ow clcul h plc rorm or h cod driviv y ( ) Igrig, gi, by pr w hv ; (8) { y ( )} { y ( ) d. (9) u dv { y ( )} y ( ) y ( ) d { y ( )} y ( ) { y( )} y( ) y ( ). () W rwri h prviou quio (Eq. 7 d Eq. 9) bw h plc rorm o ucio d h o i driviv, uig h oio ( ) { y( )} Y, { y ( ) } Y( ) y ( ) { y ( )} Y( ) y( ) y ( ) Grlizio o h h driviv, y ( ) (igrio by pr i ud) d w hv (), i lmo righ orwrd { y ( )} Y( ) y( ) y( ) y ( ) y ( ). () W u h oio whr U d or u U { U (, ) } U (, ) d. () ( ) U,. { U (, ) } { U (, ) d U (, ) U (, ) dv ( ) U(, ). d ()

39 Th ucio i h ir rm o h rh dpd ow o o vribl oly. All h prviou rlio (Eq., Eq. d Eq. ) r il i h oluio o pril diril quio. Applicio: Th D H Equio Empl: Solv h quio U α U or < < d < <. Th boudry codiio or giv problm r: U (, ) BC U (,) BC Thi i h h low quio i D. U (, ) (5) Soluio: W w o id h mprur ild. For hi purpo w k h plc rorm o boh id o Eq. 5, i.. { U } α { } U ( ) U (,) ( ). W hv U d U α () d W rwri hi quio (uig h boudry codiio (BC)) d U d ( ) α U ( ) which i homogou cod-ordr ordiry diril quio. Th oluio i o h orm r r ( ) C C U (7) (8) Whr r d r r h roo o h uiliry quio D. α Hc U ( ) C α C α Th igrio co C mu b zro bcu o h diiio o h rg o h mprur ild, (h cod rm go o ). I ddiio h boudry codiio BC giv C d w hv d Eq. 9 c b wri ollow (9) C ()

40 U ( ) α () Th mprur ild i h ivr plc rorm o h prviou ucio. W hu hv U (, ) α. () No: Evidly, w c lo olv hi mpl uig Fourir rorm. D) FIRST TRANSATION THEOREM Th ir rlio horm i did { ( ) } F( ). () I i uul i plc oprio o orm rm h c b ir o rorm. Noio Th vribl my hv o b "ubiud" or om mipulio o. I hi c, vricl br i plcd r h plc oio wih wih rrow poiig o h mipulio o. For mpl, { ( ) } { ( ) } Th i, w olv plc rorm or ( ) ubiu -, io h rul or. Empl { i } ( ) ( ), h { } F( ). (). (b) { i } F( ) Empl: Solv h quio y y i ( ) giv h ( ) y (5) Soluio: Tkig h plc rorm, or boh id w g { y } { y} { i }, () Y y( ) Y ( ) Y 5 (7) 5

41 which giv ( ) ( )( ) 5 Y (8) uig h pril rcio w hv ( ) C B A Y (9) Equig h powr coici i h umror o h prviou wo quio w obi ( ) ( )( ) ( ) ( ) ( ) 5.,, 5 C A C B B A C B A () Solvig or A, B d C w hv.,, 8 C B A Filly w hv ( ). 8 Y () Coquly, h grl oluio o Eq. 5 i ( ) ( ) { } Y y i co () Ohr mpl ( ) { } ( ) ( ) ( ) ( ) y y y Y Y () { }! d E) USEFUNESS OF APACE TRANSFORM IN SOVING PDE S W c u i or olvig diril quio o h -ordr. Th u o plc rorm ld o h grl oluio, i.. hr i o d o id y priculr oluio whovr. Coquly, hr i o d o id h roo o chrcriic quio or h co or h grl oluio

42 7 F) INTEGRA EQUATIONS Empl: Solv h igrl quio ( ) ( ). τ τ τ d () Th igrl ( ) τ τ τ d I i h covoluio o ( ) τ d τ. Th chiqu i lwy h m. W k h plc rorm o boh id o h prviou quio d w g ( ) { } { } { } ( ) { } { } ( ) { } { } d τ τ τ τ (5) ( ) ( ) F F () Or ( ) ( ) F F (7) ( ) ( ) ( ) ( ) ( ) F (8) () i h ivr plc rorm o F(), i.. ( ) ( ) { } ( ) ( ) F (9) W d o clcul ( ). For hi w u h pril rcio mhod: ( ) ( ) B A B A (5) ( ) ( ) (5) Hc ( ) ] [ (5) Erci: Chck hi rul!