Supplements. Phys. 201 Methods of Theoretical Physics 1. Dr. Nidal M. Ershaidat. Phys. 201 Methods of Theoretical Physics 1

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1 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics Ps. Metods of Teoeticl Psics Supplemets D. Nidl M. Esidt

2 Tble of Cotets Vecto Spce... Vecto Idetities... Te Cul... Comple Numbes...7 Comple Logitms... Detemits... Odi Diffeetil Equtios (ODE s)...6 ODE A Emple...9 Ect Diffeetil... Metod of Udetemied Coefficiet o Guessig Metod... Metod of Vitio of Pmetes... Fouie Seies...7 Cuvilie Coodites...5

3 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. Vecto Spce A. Te Cocept Oe of te fudmetl cocepts of lie lgeb is te cocept of vecto spce. Fo emple, m fuctio sets studied i mtemticl lsis e wit espect to tei lgebic popeties vecto spces. I lsis te otio lie spce is used isted of te otio vecto spce. B. Defiitio A set X is clled vecto spce ove te umbe field K if to eve pi (,) of elemets of X tee coespods sum X, d to eve pi (α,) wee α K d X, tee coespods elemet α X, wit te popeties -8: ) (commutbilit of dditio); ) ( z) ( ) z (ssocitivit of dditio); ) X suc tt X (eistece of ull elemet); ) X - X: (-) (eistece of te ivese elemet); 5). (uitism); 6) α (β) (αβ) (ssocitivit wit espect to umbe multiplictio); 7) α ( ) α α (distibutivit wit espect to vecto dditio); 8) (α β ) α β (distibutivit wit espect to umbe dditio). Te popeties -8 e clled te vecto spce ioms. Aioms - sow tt X is commuttive goup o Abeli goup wit espect to vecto dditio. Te secod coespodece is clled multiplictio of te vecto b umbe, d it stisfies ioms 5-8. Elemets of vecto spce e clled vectos. If K R, te oe speks of el vecto spce, d if K C, te of comple vecto spce. Isted of te otio vecto spce we sll use te bbevitive spce.

4 Psics Deptmet, Ymouk Uivesit, Ibid Jod D. Nidl M. Esidt Ps. Mtemticl Psics Doc. A Vecto Idetities I te followig φ d ψ e scl fuctios, AdB e vectos. A B A B () A B A B () φ A φ Aφ A () A B B A A B () A B B A A B A B B A (5) A (6) φ φ φ φ (Lplci) (7) (8) A A A (9)

5 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. Te Cul A) Geometicl sigificce of te Cul Te divegece d cul of vecto field e two vecto opetos wose bsic popeties c be udestood geometicll b viewig vecto field s te flow of fluid o gs. Hee we give oveview of bsic popeties of cul t c be ituited fom fluid flow. Te cul of vecto field cptues te ide of ow fluid m otte. Imgie tt te below vecto field F epesets fluid flow (See Fig. ). Te vecto field idictes tt te fluid is cicultig oud cetl is. B) Mesuig te Cul I ode to mesue te desit of some mtte t poit, we mesue te mss (dm) of smll volume (dv) oud te poit, d te divide b te volume. Mtemticll, tis c be epessed b: dm ρ () dv Te smlle te volume, te bette te ppoimtio. Actull we defie te desit ρ s beig te limit: dm ρ lim () dv dv A simil pocedue is used to mesue te stegt of te ottio of fluid. If te vecto field is itepeted s velocit of fluid flow, te fluid ppes to flow i cicles. Tis mcoscopic cicultio of fluid oud cicles ctull is ot wt cul mesues. But, it tus out tt tis vecto field lso s cul, wic we migt tik of s micoscopic cicultio. To test fo cul, imgie tt ou immese smll spee ito te fluid flow, d ou fi te cete of te spee t some poit so tt te spee cot follow te fluid oud (See Fig. ).

6 () Time t (b) Time t Figue : Te ottig vecto field F t two diffeet times. Altoug ou fi te cete of te spee, ou llow te spee to otte i diectio oud its cete poit. Te ottio of te spee mesues te cul of te vecto field F t te poit i te cete of te spee. (Te spee sould ctull be ell ell smll, becuse, emembe, te cul is micoscopic cicultio.) () Time t (b) Time t Figue : A smll spee immesed ito te fluid flow. Te vecto field F detemies bot i wt diectio te spee ottes, d te speed t wic it ottes. We defie te cul of F, deoted cul F, b vecto tt poits log te is of te ottio d wose legt coespods to te speed of te ottio. We c dw te vecto coespodig to cul F s follows. We mke te legt of te vecto cul F popotiol to te speed of te spee's ottio. Te diectio of cul F poits log te is of ottio, but we eed to specif i wic diectio log tis is te vecto sould poit. We will (bitil?) set te diectio of te cul vecto b usig te followig

7 5 igt d ule. To see wee cul F sould poit, cul te figes of ou igt d i te diectio te spee is ottig; ou tumb will poit i te diectio of cul F. Fo ou emple, cul F is sow b te gee ow. Figue : Te diectio of cul F is covetioll cose usig igt-d ule. Te cul is tee-dimesiol vecto, d ec of its tee compoets tus out to be combitio of deivtives of te vecto field F. Oce ou ve te fomul, clcultig te cul of vecto field is simple mtte. Te cul is sometimes clled te ottio, o "ot". C) Te Cul i diffeet sstem of coodites Te cul of vecto fuctio is te vecto poduct of te del opeto wit tis vecto fuctio: Te Cul i Ctesi coodites k F F j F z F i z F F F z z ˆ ˆ ˆ () wee k j i ˆ,ˆ, ˆ e uit vectos i te,, z diectios. It c lso be epessed i detemit fom: z F F F z k j i ˆ ˆ ˆ () Te Cul i clidicl pol coodites

8 6 Te cul i clidicl pol coodites, epessed i detemit fom is: z θ θ F F F z θ k ˆ (5) Te Cul i speicl pol coodites Te cul i speicl pol coodites, epessed i detemit fom is: φ θ F θ F F φ θ θ θ F si si si (6) Refeece: - ttp://pepsics.p-st.gsu.edu/bse/cul.tml - (See te pplets i) ttp://mtisigt.og/cul_ide

9 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods fo Teoeticl Psics D. Nidl M. Esidt Doc. Comple Numbes A. Defiitio of comple umbe A comple umbe z is defied b z i wee d e el umbes d i. is clled te el pt of z d deoted Re(z) is clled te imgi pt of z d deoted Im(z) Te fom z i is clled te ectgul fom of te comple umbe z. Note: I tis documet te lette z efes to comple umbe d ote lette, i pticul, d, efe to el umbes o vibles. B. Te Comple (Numbes) Spce Comple umbes defie spce clled te spce of comple umbes o simpl te comple spce. Te eigt ioms ecess to defie spce i lie lsis e veified, s follows:. Commutbilit of dditio Fo two comple umbes z d z we defie z z z z ( z, z ). Associtivit of dditio Fo set of comple umbes z, z d z we defie z (z z ) (z z ) z. Eistece of ull elemet: C suc tt z z C. Eistece of te ivese elemet: z C - z C suc tt z (-z) 5. Uitism: z C. z z Ad fo (α, β d z C) we ve: 6. Associtivit wit espect to umbe multiplictio α (β z) (α β) z 7. Distibutivit wit espect to comple umbe dditio α (z z ) α z α z 8. Distibutivit wit espect to umbe dditio (α β ) z α z β z 7

10 C. Comple cojugte of comple umbe Te comple cojugte of comple umbe z (z i ) is defied b z* - i D. Bsic Opetios i te Comple Spce Coside te comple umbes z i d z i - Additio z z z ( ) i ( ) z z - Subtctio z z - z ( - ) i ( - ) z' z z ( ) i ( )# z - z -Multiplictio z z. z ( i ).( i ) i i - -Divisio z z z i i i i. i i i i i ( i ) E. Gpicl Repesettio of comple umbes A comple umbe z i is epeseted i D ple, clled te Agd Ple, b poit P wose coodites e ( d ) (Fig. ). Figue : Gpicl Repesettio of z 8

11 Te vecto OP epesets te comple umbe z. Fig. epesetig z is te so-clled Agd digm of z. Te comple cojugte z* is epeseted b te vecto OP * i Fig.. Figue : Gpicl Repesettio of z* Fom Fig. oe c see tt: θ t d θ e clled te pol coodites of P d ltetivel we ve: z c be witte i te fom: cos θ si θ z (cos θ i si θ) Usig Mclui s seies fo e iθ, cosθ d si θ we c wite: z c be ewitte i te fom: e iθ cos θ i si θ z e iθ Wic we cll te pol fomt of z. Tis fom is ve useful we defiig d usig fuctios of comple vibles. F. Modulus (Mgitude) d Agumet of z epesets te mgitude o modulus of z d θ is clled te gumet of z. Ad we wite: ( z) Mod θ g ( z) t 9

12 Comple Logitms Te epoetil fuctio c be eteded to fuctio wic gives comple umbe s e fo bit comple umbe ; simpl use te ifiite seies wit comple. Tis epoetil fuctio c be iveted to fom comple logitm tt eibits most of te popeties of te odi logitm. Tee e two difficulties ivolved: o s e ; d it tus out tt e πi e. Sice te multiplictive popet still woks fo te comple epoetil fuctio, e z e z πi, fo ll comple z d itegl. So te logitm cot be defied fo te wole comple ple, d eve te it is multi-vlued comple logitm c be cged ito "equivlet" logitm b ddig πi t will. Te comple logitm c ol be sigle-vlued o te cut ple. Fo emple, l i ½ πi o 5/ π i o / π i, etc.; d ltoug i, log i c be defied s π i, o π i o 6 π i, d so o. z Re(l(i)) z Im(l(i)) z l(i) Plots of te tul logitm fuctio o te comple ple (picipl bc) Nidl M. Esidt

13 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods fo Teoeticl Psics D. Nidl M. Esidt Doc. 5 Detemits Coside te sque mti of ode : A b c d Te mti A is ivetible if d ol if d bc#. Tis umbe is clled te detemit of A. It is cle fom tis, tt we would like to ve simil esult fo bigge mtices (meig ige odes). So is tee simil otio of detemit fo sque mti, wic detemies wete sque mti is ivetible o ot? I ode to geelize suc otio to ige odes, we will eed to stud te detemit d see wt kid of popeties it stisfies. Fist let us use te followig ottio fo te detemit: Detemit of b b b c d det c d c d d bc Popeties of te Detemit. A mti A d its tspose ve te sme detemit, meig det A det A T Tis is iteestig sice it implies tt weeve we use ows, simil bevio will esult if we use colums. I pticul we will see ow ow elemet opetios e elpful i fidig te detemit. Teefoe, we ve simil coclusios fo elemet colum opetios.. Te detemit of tigul mti is te poduct of te eties o te digol, tt is d b c d d.. If we itecge two ows, te detemit of te ew mti is te opposite of te old oe, tt is b c d c d. b. If we multipl oe ow wit costt, te detemit of te ew mti is te detemit of te old oe multiplied b te costt, tt is

14 λ c λb d λ b b c d. λc λd I pticul, if ll te eties i oe ow e zeo, te te detemit is zeo. 5. If we dd oe ow to ote oe multiplied b costt, te detemit of te ew mti is te sme s te old oe, tt is λc c b λd d c b d c λ b. d λb Note tt weeve ou wt to eplce ow b sometig (toug elemet opetios), do ot multipl te ow itself b costt. Otewise, ou will esil mke eos (due to Popet ). 6. We ve det (AB) det(a) det (B) I pticul, if A is ivetible (wic ppes if d ol if det (A) #), te det - ( A ) det( A) If A d B e simil, te det (A) det(b). Let us look t emple, to see ow tese popeties wok. Emple. Evlute. Let us tsfom tis mti ito tigul oe toug elemet opetios. We will keep te fist ow d dd to te secod oe te fist multiplied b ½. We get 7. Usig te Popet, we get 7 Teefoe, we ve wic oe m ceck esil.

15 Detemits of Mtices of Hige Ode As we sid befoe, te ide is to ssume tt pevious popeties stisfied b te detemit of mtices of ode, e still vlid i geel. So let us see ow tis woks i cse of mti of ode. Emple. Evlute We ve If we subtct eve ow multiplied b te ppopite umbe fom te fist ow, we get We do ot touc te fist ow d wok wit te ote ows. We itecge te secod wit te tid to get If we subtct eve ow multiplied b te ppopite umbe fom te secod ow, we get Usig pevious popeties, we ve. If we multipl te tid ow b d dd it to te fout, we get

16 wic is equl to. Puttig ll te umbes togete, we get ( ) ( ) Tese clcultios seem to be te legt. We will see lte o tt geel fomul fo te detemit does eist. Emple. Evlute. I tis emple, we will ot give te detils of te elemet opetios. We ve 9. Emple. Evlute. We ve Geel Fomul fo te Detemit Let A be sque mti of ode. Wite A ( ij ), wee ij is te et o te ow umbe i d te colum umbe j, fo i,,, d j,,. Fo i d j, set C ij (clled te cofctos) to be te detemit of te sque mti of ode (-) obtied fom A b emovig te ow umbe i d te colum umbe j multiplied b (-) ij. We ve ( ) j j ij ij A det C fo fied i, d ( ) i i ij C ij A det

17 5 fo fied j. I ote wods, we ve two tpes of fomuls: log ow (umbe i) o log colum (umbe j). A ow o colum will do. Te tick is to use ow o colum wic s lot of zeos. I pticul, we ve log te ows g e d c k g f d b k f e k g f e d c b o g b f k g c e k c b d k g f e d c b, o e d b k f d c f e c b g k g f e d c b. As eecise wite te fomuls log te colums. Emple 5. Evlute. We will use te geel fomul log te tid ow. We ve ( ) ( ) 9 6 Wic tecique to evlute detemit is esie? Te swe depeds o te peso wo is evlutig te detemit. Some like te elemet ow opetios d some like te geel fomul. All tt mttes is to get te coect swe. Note tt ll of te bove popeties e still vlid i te geel cse. Also ou sould emembe tt te cocept of detemit ol eists fo sque mtices. ttp:// Auto: M.A. Kmsi

18 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. 6 Odi Diffeetil Equtios (ODE s) A) Defiitio A odi diffeetil equtio is equtio wic s ect diffeetils o ect deivtives i bot sides. Te ode of te igest deivtive is clled te ode of te diffeetil equtios. Te solutio of ODE is eltiosip betwee te ukow fuctio wit its vible wic veifies te ODE. B) Fist Ode Diffeetil Equtios Sepble st ode DE A st ode diffeetil of te fom: Is clled sepble st ode DE. ( ) d G( ) d F () Te geel solutio of Eq. is obtied b itegtig bot sides, i.e. Lie st ode DE ( ) d G( ) d C F () d d ( ) Q( ) If Q() te Eq. c be witte s: d P () ( ) P d wic is sepble st ode ODE (See Eq. ), wee ( ) P( ) G ( ). - Homogeeous Lie st ode DE Eq. is lso clled omogeeous st ode DE. d d ( ) P Te geel solutio of Eq. is obtied b itegtig bot sides, i.e. () F d Auto s emil: eidl@u.edu.jo Addess: Psics Deptmet, Ymouk Uivesit 6 Ibid Jod 6

19 { } ( ) A P( ) - Iomogeeous Lie st ode DE ep d (5) d d ( ) Q( ) Te geel solutio of Eq. 6 is give b: P (6) wee I( ) I( ) I( ) ( ) e e Q( ) d Ce (7) Ect Equtios I I( ) ( ) e P( )d (8) A fist ode odi diffeetil is clled ect equtio if its ls c be epessed s ect diffeetil du of fuctio U(,) suc tt: Tis is veified if d ol of: (, ) d N(, ) du M d (9) M N M is te ptil deivtive of te fuctio M(,) wit espect to. Tis mes tt we deive te fuctio M(,) cosideig s costt (wic we smbolize b te subscipt. N is te ptil deivtive of te fuctio N(,) wit espect to. Tis mes tt we deive te fuctio N(,) cosideig s costt (wic we smbolize b te subscipt. () Emple: if ( ) M, te M 6 M d. Homogeeous Fist Ode DE A fist ode odi diffeetil is clled ect equtio if te ls c be epessed s ect diffeetil du of fuctio U(,) suc tt: d d F () Auto s emil: eidl@u.edu.jo Addess: Psics Deptmet, Ymouk Uivesit 6 Ibid Jod 7

20 Te geel solutio is: Wee d dv F v C ( v) ( ) Secod Ode Diffeetil Equtios () v () Lie d ode DE A d ode DE equtio is lie if d ol of it c be witte i te fom: d P d d d ( ) Q( ) R( ) Wee P(), Q() d R() e geel fuctios of te idepedet vible, Vious metods e used to solve suc equtios. Tese metods deped o te tue of P(), Q() d R(). () Refeece: Ps. Metods of Teoeticl Psics Tetbook Itoductio to Mtemticl Psics, d Editio, Autos: N. Lm d N. Aoub Nidl M. Esidt 8

21 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. 8 ODE A Emple Emple : 5 Solve Tis is iomogeeous lie d ode diffeetil equtio. wit: ( ) b R (), b d R ( ) 5 costt Its geel solutio () is te sum C () P () wee C () is te complemet solutio i.e. te solutio of te coespodig omogeeous equtio ( ) d P () is te (uique) pticul solutio obtied usig te metod of udetemied coefficiets. wit d F ( ) ( ) ( ) P () C β ( ) ( αβ) e e F( )d P () α ( ) e R( )d () Complemet Solutio C ( ) C e α β C e (5) Wee α d β e te oots of te uili equtio ( ), i.e. α d β (6) Pticul Solutio α 5 α F( ) e ( 5 ) d e (7) α wic gives: P β ( ) ( αβ e e ) 5 e α β e β 5 α e d α 5 5 d αβ 5 ( ) C e Ce (9) (8) Auto s emil: eidl@u.edu.jo, Addess: Psics Deptmet, Ymouk Uivesit 6 Ibid Jod Nidl M. Esidt 9

22 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. 8 Ect Diffeetil A diffeetil of te fom (, ) d Q( )d df P, () is ect (lso clled totl diffeetil) if df is pt-idepedet. Tis will be tue if df So P d Q must be of te fom f f d d. () But P f f, () ( ) d Q(, ) P f, () Q f. (5) wic ields P Q. (6) Te otio of ect diffeetils pls impott ole i psics. I temodmics, popeties of temodmic sstem (fo emple, te pessue, volume d tempetue fo gs) sould be ect diffeetils. Auto s emil: eidl@u.edu.jo, Addess: Psics Deptmet, Ymouk Uivesit 6 Ibid Jod Nidl M. Esidt

23 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. 9 Metod of Udetemied Coefficiet o Guessig Metod Piciple Tis metod is bsed o guessig tecique. Tt is, we will guess te fom of P d te plug it i te equtio to fid it. Howeve, it woks ol ude te followig two coditios: Coditio : te ssocited omogeeous equtios s costt coefficiets Coditio : te oomogeeous tem R() is specil fom R ( ) P( ) e ( β) α cos () O R ( ) L( ) e ( β) α si () wee P() d L() e polomil fuctios. Note tt we m ssume tt R() is sum of suc fuctios (see te emk below fo moe o tis). Assume tt te two coditios e stisfied. Coside te equtio wee, b d c e costts d R ( ) b c R () ( ) P ( ) e α cos ( β) o R( ) P ( ) e ( β) α si () wee P () is polomil fuctio wit degee. Te pticul solutio P is give b Wee Ad P ( ) s α α ( ) T ( ) e ( β) U ( ) e si( β) T U cos (5) ( ) A A A A (6) ( ) B B B B Wee te costts A k d B k e to be detemied. Te vlue of te powe s depeds o α iβ i te followig me: if α iβ is ot oot of te ccteistic equtio te s. if α iβ is simple oot of te ccteistic equtio te s. if α iβ is double oot of te ccteistic equtio te s.

24 Remk: If te oomogeeous tem R() stisfies te followig R i N i ( ) R i ( ) wee R i () e of te foms cited bove, te we split te oigil equtio ito N equtios: ( ) ( i,, N) b c Ri, (8) Te fid pticul solutio i. A pticul solutio to te oigil equtio is give b (7) Summ i ( ) i N P i. (8) Let us summize te steps to follow i pplig tis metod: () Fist, ceck tt te two coditios metioed bove e stisfied; () If te equtio is give s R b c k N k ( ) R k ( ) P ( ) e α cos ( β) o R( ) P ( ) e ( β), (9) α si () wee P is polomil fuctio wit degee, te split tis equtio ito N equtios ( ) b c R, () wee () Wite dow te ccteistic equtio b c d fid its oots; () Wite dow te umbe α k i β k. Compe tis umbe to te oots of te ccteistic equtio foud i te pevious step. (.) If α k iβ k is ot oe of te oots, te set s ; (.) If α k iβ k is oe of te two distict oots, te set s ; (.) If α k iβ k is equl to bot oots (wic mes tt te ccteistic equtio s double oot, te set s ; I ote wods, s mesues ow m times α k i β k is oot of te ccteistic equtio; (5) Wite dow te fom of te pticul solutio k Nidl M. Esidt

25 wee d P ( ) s α α ( ) T ( ) e ( β) U ( ) e si( β) T cos, () ( ) A A A A () U ( ) B B B B. () (6) Fid te costts A i d B i b pluggig R k ito te equtio ( ) b c R, (7) Oce ll te pticul solutios k e foud, te te pticul solutio of te oigil equtio is te seies k k ( ) k N P k (5) Refeece: ttp:// Nidl M. Esidt

26 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. Metod of Vitio of Pmetes Piciple Tis metod s o pio coditios to be stisfied. Teefoe, it m soud moe geel t te metod of udetemied coefficiet o guessig metod. We will see tt tis metod depeds o itegtio wile te ote oe cited is puel lgebic wic, fo some t lest, is dvtge. Coside te equtio ( ) Q( ) R( ) P () I ode to use te metod of vitio of pmetes we eed to kow tt {, } is set of fudmetl solutios of te ssocited omogeeous equtio P( ) Q( ). We kow tt, i tis cse, te geel solutio of te ssocited omogeeous equtio is c c. Te ide beid te metod of vitio of pmetes is to look fo pticul solutio suc s ( ) u ( ) ( ) u ( ) ( ) P, () wee u d u e fuctios. Fom tis, te metod got its me. Te fuctios u d u e solutios to te sstem wic implies u u u u R Wee ( )( ) ( ) ( ) R( ) (, )( ) ( ) R( ) (, )( ) u ( ) d, W u( ) d. W W, is te woski of d. Teefoe, we ve ( ) ( ) ( ) R( ) (, )( ) ( ) R( ) (, )( ) d ( ) d W W P () () (5)

27 Summ Let us summize te steps to follow i pplig tis metod: () Fid {, } set of fudmetl solutios of te ssocited ; omogeeous equtio P( ) Q( ) () Wite dow te fom of te pticul solutio ( ) u ( ) ( ) u ( ) ( ) P (6) () Wite dow te sstem: u u (7) u u R( ) () Solve it. Tt is, fid u d u ; (5) Plug u d u ito te equtio givig te pticul solutio. Emple Fid te pticul solutio to t( ) Solutio ; Let us follow te steps i te summ: π π < < () A set of fudmetl solutios of te ssocited omogeeous equtio is { cos(), si()}; () Te pticul solutio is give s ( ) u ( ) ( ) u ( ) si( ). P cos () We, tus, ve te sstem: u cos u si ( ) u si( ) ( ) u cos( ) t( ) () We solve fo u d u d get: u u si si( ) ( t( ) ) ( ) cos( ) ( t( ) ) Usig Itegtio teciques, we get: u ( ), u ( ) cos( ) si( ) l sec( ) t( ) ( ) si( ) cos( ). (5) Te pticul solutio is o P cos ( ) ( cos( ) si( ) l( sec( ) t( ) )) si( ) ( si( ) cos( ) ) 5

28 P cos( ) l( sec( ) t( ) ). Remk: Note tt sice te equtio is lie, we m still split if ecess. Fo emple, we m split te equtio t ( ) ito te two equtios (R) d t( ) (R ) te, fid te pticul solutios fo (R) d fo (R), to geete pticul solutio fo te oigil equtio b P Tee e o estictios o te metod to be used to fid o. Fo emple, we c use te metod of udetemied coefficiets to fid, wile fo, we e ol left wit te vitio of pmetes. Refeece: ttp:// 6

29 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. Fouie Seies A) Defiitio A Fouie seies is epsio of peiodic fuctio f() i tems of ifiite sum of sies d cosies. Fouie seies mke use of te otogolit eltiosips of te sie d cosie fuctios. Te computtio d stud of Fouie seies is kow s moic lsis d is etemel useful s w to bek up bit peiodic fuctio ito set of simple tems tt c be plugged i, solved idividull, d te ecombied to obti te solutio to te oigil poblem o ppoimtio to it to wteve ccuc is desied o pcticl. B) Illusttio Emples of successive ppoimtios to commo fuctios usig Fouie seies e illustted below. () (b) (c) (d) Figue : Usig Fouie seies i ode to ppoimte some fuctios. ) Te sque wve, b) Te swtoot wve, c) te tigle wve d d) te semicicle. 7

30 C) Applictio - solutio of odi diffeetil equtios I pticul, sice te supepositio piciple olds fo solutios of lie omogeeous odi diffeetil equtio, if suc equtio c be solved i te cse of sigle siusoid, te solutio fo bit fuctio is immeditel vilble b epessig te oigil fuctio s Fouie seies d te pluggig i te solutio fo ec siusoidl compoet. I some specil cses wee te Fouie seies c be summed i closed fom, tis tecique c eve ield ltic solutios. D) Geelized Fouie Seies A set of fuctios tt fom complete otogol sstem ve coespodig geelized Fouie seies logous to te Fouie seies. Fo emple, usig otogolit of te oots of Bessel fuctio of te fist kid gives so-clled Fouie-Bessel seies. E) Computtio of Fouie seies Te computtio of te (usul) Fouie seies is bsed o te followig itegl idetities wic epeset te otogolit eltiosips of te sie d te cosie fuctios: π π π π π π π π π π ( m) si( ) si d πδm () ( m) cos( ) cos d πδm () ( ) cos( ) si m d () ( ) si m d () ( ) cos m d (5) Fo m,, wee δ m is te Koecke delt. Usig te metod fo geelized Fouie seies, te usul Fouie seies ivolvig sies d cosies is obtied b tkig f () cos d f () si. Sice tese fuctios fom complete otogol sstem ove [- π, π], te 8

31 Fouie seies of fuctio f() is give b f ( ) cos( ) b si( ) (6), e clled te Fouie coefficiets. Tese coefficiets e obtied usig te followig eltios: π π f ( ) d (7) π π π f ( ) cos( ) d π (8) b π π f ( ) si( ) d π,,,. Note tt te coefficiet of te costt tem s bee witte i specil fom comped to te geel fom fo geelized Fouie seies i ode to peseve smmet wit te defiitios of d b. A Fouie seies coveges to te fuctio f (equl to te oigil fuctio t poits of cotiuit o to te vege of te two limits t poits of discotiuit) (9) lim f lim π f f ( ) lim f ( ) ( ) lim f( ) π fo fo π< <π π, π if te fuctio stisfies so-clled Diiclet coditios. () As esult, e poits of discotiuit, "igig" kow s te Gibbs peomeo, illustted bove, c occu. 9

32 Fo fuctio f() peiodic o itevl [- L, L] isted of [- π, π], simple cge of vibles c be used to tsfom te itevl of itegtio fom [- π, π] to [- L, L]. Let π L π d d L Solvig fo gives L, d pluggig tis i Eq. 6 gives π () () Teefoe, π π f ( ) cos b si () L L L L f ( ) d () L L L L f ( ) π cos L d (5) b L L L f ( ) π si d L Simill, te fuctio is isted defied o te itevl [,L], te bove equtios simpl become (6) L f ( ) d L (7) b L π f ( ) cos L L d L π f ( ) si L L d (8) (9) I fct, fo f() peiodic wit peiod L, itevl [, L ] c be used, wit te coice beig oe of coveiece o pesol pefeece (Afke 985, p. 769). Te coefficiets fo Fouie seies epsios of few commo fuctios e

33 give i Bee (987, pp. -) d Bel (959, p. 5). Oe of te most commo fuctios usull lzed b tis tecique is te sque wve. Te Fouie seies fo few commo fuctios e summized i te tble below. F) Emples Fuctio f() Fouie seies swtoot wve L π si L sque wve H H L L π tigle wve T() 8 π,, 5,,, 5, π si L ( ) π si L If fuctio is eve so tt f() f(- ), te f() si() is odd. (Tis follows sice si() is odd d eve fuctio times odd fuctio is odd fuctio.) Teefoe, b fo ll. Simill, if fuctio is odd so tt f() - f(-), te f() cos() is odd. (Tis follows sice cos() is eve d eve fuctio times odd fuctio is odd fuctio.) Teefoe, fo ll. G) Comple Coefficiets Te otio of Fouie seies c lso be eteded to comple coefficiets. Coside el-vlued fuctio f(). Wite Now emie ( ) i f A e () π π f π im ( ) e d i im A e e d π () π π i( m A e ) d () π A π [ ( m) isi( m) ] cos d ()

34 A π δ () m So π, (5) A m A π π i f ( ) e d π Te coefficiets c be epessed i tems of tose i te Fouie seies (6) A π π f ( ) [ cos( ) isi( ) ] d π (7) π π π π π π π π π π π π f f f ( ) [ cos( ) isi( ) ] ( ) d ( ) [ cos( ) isi( ) ] ( ib ) ( ib ) fo fo fo d d < < < Fo fuctio peiodic i [- L/, L/], tese become f i ( ) ( π L) A e < > (8) (9) () L L i ( ) ( π L e ) A f d () L Tese equtios e te bsis fo te etemel impott Fouie tsfom, wic is obtied b tsfomig A fom discete vible to cotiuous oe s te legt L.

35 H) Refeeces Afke, G. "Fouie Seies." C. i Mtemticl Metods fo Psicists, d ed. Oldo, FL: Acdemic Pess, pp , 985. Aske, R. d Himo, D. T. "Similities betwee Fouie d Powe Seies." Ame. Mt. Motl, 97-, 996. Bee, W. H. (Ed.). CRC Stdd Mtemticl Tbles, 8t ed. Boc Rto, FL: CRC Pess, 987. Bow, J. W. d Cucill, R. V. Fouie Seies d Boud Vlue Poblems, 5t ed. New Yok: McGw-Hill, 99. Bel, W. E. A Elemet Tetise o Fouie's Seies, d Speicl, Clidicl, d Ellipsoidl Hmoics, wit Applictios to Poblems i Mtemticl Psics. New Yok: Dove, 959. Cslw, H. S. Itoductio to te Teo of Fouie's Seies d Itegls, d ed., ev. d el. New Yok: Dove, 95. Dvis, H. F. Fouie Seies d Otogol Fuctios. New Yok: Dove, 96. Dm, H. d McKe, H. P. Fouie Seies d Itegls. New Yok: Acdemic Pess, 97. Folld, G. B. Fouie Alsis d Its Applictios. Pcific Gove, CA: Books/Cole, 99. Goeme, H. Geometic Applictios of Fouie Seies d Speicl Hmoics. New Yok: Cmbidge Uivesit Pess, 996. Köe, T. W. Fouie Alsis. Cmbidge, Egld: Cmbidge Uivesit Pess, 988. Köe, T. W. Eecises fo Fouie Alsis. New Yok: Cmbidge Uivesit Pess, 99. Ktz, S. G. "Fouie Seies." 5. i Hdbook of Comple Vibles. Bosto, MA: Bikäuse, pp. 95-, 999. Ligtill, M. J. Itoductio to Fouie Alsis d Geelised Fuctios. Cmbidge, Egld: Cmbidge Uivesit Pess, 958. Moiso, N. Itoductio to Fouie Alsis. New Yok: Wile, 99. Ssoe, G. "Epsios i Fouie Seies." C. i Otogol Fuctios, ev. Eglis ed. New Yok: Dove, pp. 9-68, 99.

36 Weisstei, E. W. "Books bout Fouie Tsfoms." ttp:// Wittke, E. T. d Robiso, G. "Pcticl Fouie Alsis." C. i Te Clculus of Obsevtios: A Tetise o Numeicl Mtemtics, t ed. New Yok: Dove, pp. 6-8, 967. Souce iteet Weisstei, Eic W. "Fouie Seies." Fom MtWold - A Wolfm Web Resouce. ttp://mtwold.wolfm.com/fouieseies.tml 999 CRC Pess LLC, Wolfm Resec, Ic. Tems of Use

37 Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. 5 Cuvilie Coodites Wt e cuvilie coodites? A cuvilie coodite sstem is composed of itesectig sufces. If te itesectios e ll t igt gles, te te cuvilie coodites e sid to fom otogol coodite sstem. If ot, te fom skew coodite sstem. Cuvilie q q q Ctesi z î ĵ kˆ Speicl θ φ si θ ˆ θˆ φˆ Clidicl (o ρ) φ z ˆ o ρˆ φˆ kˆ dq dq dq l d φ φ φ i i i i i i i q l ( ) ( ) ( ) q A q A q A A A A A q q q A φ φ φ φ φ q q q q q q

38 dl ψ Psics Deptmet, Ymouk Uivesit, Ibid Jod Ps. Metods of Teoeticl Psics D. Nidl M. Esidt Doc. Ctesi to Clidicl Clidicl to Ctesi Ctesi to Speicl Speicl to Ctesi i ˆ ˆ cosφφ ˆ siφ φ ˆ ˆ iˆcos jsiφ i ˆ ˆ siθcosφθˆ cosθcosφφˆ siφ θ φ θ φ ˆ ˆ iˆsi cos ˆsi j si kcosθ ˆj ˆ siφφ ˆ cosφ φˆ i ˆsiφ ˆj cosφ ˆj ˆ siθsiφθˆ cosθsiφφˆ cosφ θˆ i ˆcosθcosφ ˆcos j θsiφkˆ siθ kˆ kˆ k ˆ k ˆ k ˆ ˆ cos θθ ˆ siθ φˆ i ˆ siφ ˆj cosφ A A Ctesi Speicl Clidicl dl diˆ d ˆj dzkˆ dl d dθθ ˆ siθdφφˆ dl d dφφˆ iˆ ψ ψ ψ ˆj kˆ z A A Az z ψ ˆ ˆ ψ θˆ ( ) ( ) ( ) iˆ A ˆj A kˆ z A ψ z ψ ψ ψ ψ z ψ θ si φˆ θ ψ φ A siθa θa siθ θ si A O ψ ψ siθ siθ ˆ A ψ ψ siθ θˆ θ A θ siθ θ θ θ siθφˆ φ siθa φ φ ψ siθ θ ψ siθ θ si A φ si ψ θ φ ψ θ φ ψ ˆ A A ˆ ψ φˆ ψ φ dzkˆ ψ kˆ z ( A ) Aφ ( A ) ˆ A φˆ φ A φ kˆ z φ A z ψ ψ ψ ψ ψ φ z z ψ ψ φ z ψ φ ψ z z 6