Theoretical Physics. Course codes: Phys2325 Course Homepage:

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1 Theoretcal Phscs Course codes: Phs35 Course Homepage: Lecturer: Z.D.Wag, Oce: Rm58, Phscs Buldg Tel: E-mal: Studet Cosultato hours: :3-4:3pm Tuesda Tutor: Mss Lu Ja, Rm55

2 Tet Book: Lecture Notes Selected rom Mathematcal Methods or Phscsts Iteratoal Edto (4 th or 5 th or 6 th Edto) b George B. Arke ad Has J. Weber Ma Cotets: Applcato o comple varables, e.g. Cauch's tegral ormula, calculus o resdues. Partal deretal equatos. Propertes o specal uctos (e.g. Gamma uctos, Bessel uctos, etc.). Fourer Seres. Assessmet: Oe 3-hour wrtte eamato (8% weghtg) ad course assessmet (% weghtg)

3 Fuctos o A Comple Varables I Fuctos o a comple varable provde us some powerul ad wdel useul tools theoretcal phscs. Some mportat phscal quattes are comple varables (the wave-ucto Ψ) Evaluatg dete tegrals. Obtag asmptotc solutos o deretals equatos. Itegral trasorms Ma Phscal quattes that were orgall real become comple as smple theor s made more geeral. The eerg E E Γ ( the te le tme). / Γ 3

4 . Comple Algebra We here go through the comple algebra brel. A comple umber (,), Where. We wll see that the orderg o two real umbers (,) s sgcat,.e. geeral X: the real part, labeled b Re(); : the magar part, labeled b Im() Three requetl used represetatos: () Cartesa represetato: () polar represetato, we ma wrte r(cos θ sθ) or r r the modulus or magtude o θ - the argumet or phase o θ e 4

5 r the modulus or magtude o θ - the argumet or phase o The relato betwee Cartesa ad polar represetato: r θ ( ) t a / / The choce o polar represetato or Cartesa represetato s a matter o coveece. Addto ad subtracto o comple varables are easer the Cartesa represetato. Multplcato, dvso, powers, roots are easer to hadle polar orm, ± ( ( ± ) ( ± ) ( ) ) r r / / e ( θ θ ) ( θθ r r e ) r e θ 5

6 Usg the polar orm, arg( ) arg arg From, comple uctos () ma be costructed. The ca be wrtte () u(,) v(,) whch v ad u are real uctos. For eample, we have ( ) The relatoshp betwee ad () s best pctured as a mappg operato, we address t detal later. 6

7 Fucto: Mappg operato Z-plae v u The ucto w(,)u(,)v(,) maps pots the plae to pots the uv plae. Sce e e θ θ cosθ sθ (cosθ sθ ) We get a ot so obvous ormula cos θ sθ (cosθ sθ ) 7

8 Comple Cojugato: replacg b, whch s deoted b (*), * Hece Note: We the have * r ( *) θ re l s a mult-valued ucto. To avod ambgut, we usuall set ad lmt the phase to a terval o legth o π. The value o l wth s called the prcpal value o l. ( θπ re ) l l r θ l lr ( ) Specal eatures: sgle-valued ucto o a real varable ---- mult-valued ucto θ π 8

9 9 Aother possblt > ad eve cos, s possbl however, ; real or a cos, s Questo: e e sh cos cos sh s s (b) sh s cosh cos ) cos( sh cos cosh s ) s( (a) to show e s ; e cos Usg the dettes :

10 . Cauch Rema codtos, Havg establshed comple uctos, we ow proceed to deretate them. The dervatve o (), lke that o a real ucto, s deed b δ δ d lm lm δ δ δ δ d provded that the lmt s depedet o the partcular approach to the pot. For real varable, we requre that lm ( ) Now, wth (or o) some pot a plae, our requremet that the lmt be depedet o the drecto o approach s ver restrctve. lm o o o Cosder δ δ δ δu δ δv δ δu δv δ δ δ

11 Let us take lmt b the two deret approaches as the gure. Frst, wth δ, we let δ, Assumg the partal dervatves est. For a secod approach, we set δ ad the let δ. Ths leads to I we have a dervatve, the above two results must be detcal. So, v u δ δ δ δ δ δ δ δ lm lm v u v u δ δ δ lm v u, v u

12 These are the amous Cauch-Rema codtos. These Cauch- Rema codtos are ecessar or the estece o a dervatve, that s, ests, the C-R codtos must hold. Coversel, the C-R codtos are satsed ad the partal dervatves o u(,) ad v(,) are cotuous, ests. (see the proo the tet book).

13 Aaltc uctos I () s deretable at ad some small rego aroud, we sa that () s aaltc at Deretable: Cauth-Rema codtos are satsed the partal dervatves o u ad v are cotuous Aaltc ucto: Propert : u v Propert : establshed a relato betwee u ad v Eample: Fd the aaltc uctos w( ) u(, ) v(, ) ( a) u(, ) 3 3 ( b) v(, ) e s 3

14 .3 Cauch s tegral Theorem We ow tur to tegrato. close aalog to the tegral o a real ucto ' The cotour s dvded to tervals.let wth or j. The lm j j whereζ s a ad. j j j j ζ adζ, d provded that the lmt ests ad s depedet o choosg the pots j j j the detals o j pot o the curve bewtee j The rght-had sde o the above equato s called the cotour (path) tegral o () 4

15 As a alteratve, the cotour ma be deed b c d [ u(, ) v(, ) ][ d d ] c c [ udvd] [ vd ud] c wth the path C speced. Ths reduces the comple tegral to the comple sum o real tegrals. It s somewhat aalogous to the case o the vector tegral. A mportat eample d c where C s a crcle o radus r> aroud the org the drecto o couterclockwse. 5

16 I polar coordates, we parametere ad d re θ dθ, ad have re θ π π r π [ ( ) θ] d ep c or - { or - dθ whch s depedet o r. Cauch s tegral theorem I a ucto () s aaltcal (thereore sgle-valued) [ad ts partal dervatves are cotuous] through some smpl coected rego R, or ever closed path C R, d c 6

17 Stokes s theorem proo Proo: (uder relatvel restrctve codto: the partal dervatve o u, v are cotuous, whch are actuall ot requred but usuall satsed phscal problems) These two le tegrals ca be coverted to surace tegrals b Stokes s theorem d ( udvd) ( vd ud) c c c A dl A ds c s ) ) Usg A A A ad ) ds dd We have ( Ad Ad) A dl A c c s ds A A dd 7

18 For the real part, I we let u A, ad v -A, the v u ( udvd) dd c v u [sce C-R codtos ] For the magar part, settg u A ad v A, we have u v ( vd ud) dd d As or a proo wthout usg the cotut codto, see the tet book. The cosequece o the theorem s that or aaltc uctos the le tegral s a ucto ol o ts ed pots, depedet o the path o tegrato, d F( ) F( ) d 8

19 Multpl coected regos The orgal statemet o our theorem demaded a smpl coected rego. Ths restrcto ma easl be relaed b the creato o a barrer, a cotour le. Cosder the multpl coected rego o Fg..6 I whch () s ot deed or the teror R.6 Fg. Cauch s tegral theorem s ot vald or the cotour C, but we ca costruct a C or whch the theorem holds. I le segmets DE ad GA arbtrarl close together, the A G d E D d 9

20 d ABD DE GA C ABDEFGA EFG d ABD EFG d d C C d ABD C ' ' EFG C

21 .4 Cauch s Itegral Formula Cauch s tegral ormula: I () s aaltc o ad wth a closed cotour C the d π ( ) C whch s some pot the teror rego bouded b C. Note that here - ad the tegral s well deed. Although () s assumed aaltc, the tegrad (()/-) s ot aaltc at uless (). I the cotour s deormed as Fg..8 Cauch s tegral theorem apples. So we have d d C C

22 re θ Let, here r s small ad wll evetuall be made to approach ero C ( θ) d re θ d re dθ θ C re (r ) ( ) dθ π( ) C Here s a remarkable result. The value o a aaltc ucto s gve at a teror pot at oce the values o the boudar C are speced. What happes s eteror to C? I ths case the etre tegral s aaltc o ad wth C, so the tegral vashes.

23 π C Dervatves Cauch s tegral ormula ma be used to obta a epresso or the dervato o () d ( ) d, Moreover, or the -th order o dervatve, d π ( ) ( ) d teror eteror d d π d π! π ( ) d d ( ) 3

24 We ow see that, the requremet that () be aaltc ot ol guaratees a rst dervatve but dervatves o all orders as well! The dervatves o () are automatcall aaltc. Here, t s worth to dcate that the coverse o Cauch s tegral theorem holds as well Morera s theorem: I a ucto () s cotuous a smpl coected rego R ad C d or ever closed C wth R, the () s aaltc throught R (see the tet book). 4

25 5. d a crcle about the org, s aaltc o ad wth a ) ( I. Eamples a { } j j j j a a j! j j a j!! d a π

26 .I the above case, M o a crcle o radus r about the org, the ar M (Cauch s equalt) Proo: a π d πr M πr r M ( r) r where M ( r) Ma ( r) r 3. Louvlle s theorem: I () s aaltc ad bouded the comple plae, t s a costat. Proo: For a, costruct a crcle o radus R aroud, π ( ) M R R d M π ( ) π R R 6

27 R Sce R s arbtrar, let, we have,.e, ( ) cost. Coversel, the slghtest devato o a aaltc ucto rom a costat value mples that there must be at least oe sgulart somewhere the te comple plae. Apart rom the trval costat uctos, the, sgulartes are a act o le, ad we must lear to lve wth them, ad to use them urther. 7

28 .5 Lauret Epaso Talor Epaso Suppose we are trg to epad () about,.e., a ( ) ad we have as the earest pot or whch () s ot aaltc. We costruct a crcle C cetered at wth radus < From the Cauch tegral ormula, π C π ( ) d ( ) d π ( ) ( ) C ( ) d ( )[ ( ) ( )] C 8

29 9 Here s a pot o C ad s a pot teror to C. For t <, we ote the dett So we ma wrte whch s our desred Talor epaso, just as or real varable power seres, ths epaso s uque or a gve. t t t t L C d π C d π!

30 Schwar relecto prcple From the bomal epaso o or teger (as a assgmet), t s eas to see, or real [ ] * * * ( ) g( ) ( ) * g Schwar relecto prcple: I a ucto () s () aaltc over some rego cludg the real as ad () real whe s real, the * * ( ) We epad () about some pot (osgular) pot o the real as because () s aaltc at. ( )! Sce () s real whe s real, the -th dervate must be real. * * ( ) g ( ) ( ) * ( )! 3

31 Lauret Seres We requetl ecouter uctos that are aaltc aular rego 3

32 3 Drawg a magar cotour le to covert our rego to a smpl coected rego, we appl Cauch s tegral ormula or C ad C, wth rad r ad r, ad obta We let r r ad r R, so or C, whle or C,. We epad two deomators as we dd beore (Lauret Seres) d C C π > < [ ] [ ] C C d d π d d C C π π a

33 where a π ( ) ( ) d C Here C ma be a cotour wth the aular rego r < < R ecrclg oce a couterclockwse sese. Lauret Seres eed ot to come rom evaluato o cotour tegrals. Other techques such as ordar seres epaso ma provde the coecets. Numerous eamples o Lauret seres appear the et chapter. 33

34 34, m m δ π π < - or - or a 3 L The Lauret epaso becomes m m m e r d re a θ θ θ π m m d d a π π [ ] Eample: () Fd Talor epaso l() at pot () d Lauret seres o the ucto I we emplo the polar orm ) ( ) l(

35 35 For eample whch has a smple pole at - ad s aaltc elsewhere. For <, the geometrc seres epaso, whle epadg t about leads to, ; ) ( ; ) ( Aaltc cotuato

36 36 Suppose we epad t about, so that coverges or (Fg..) The above three equatos are deret represetatos o the same ucto. Each represetato has ts ow doma o covergece. L [ ] L < A beautul theor: I two aaltc uctos cocde a rego, such as the overlap o s ad s, o cocde o a le segmet, the are the same ucto the sese that the wll cocde everwhere as log as the are well-deed.