Unit_III Complex Numbers: Some Basic Results: 1. If z = x +iy is a complex number, then the complex number z = x iy is

Transcription

1 Unt_III Comple Nmbes: In the sstem o eal nmbes R we can sole all qadatc eqatons o the om a b c, a, and the dscmnant b 4ac. When the dscmnant b 4ac <, the solton o ths qadatc eqaton do not belong to the sstem o. In act, a smple qadatc eqaton o the om, does possesses solton n eal. Ths dclt was oecame b ntodcng the magna pat nt, whee. Ths the set o comple nmbes dened as. C {( :, R and }. Some Basc Reslts:. I s a comple nmbe, then the comple nmbe s called the comple conjgate o, and ( (.I s a comple nmbe, then the modls o, denoted b 3. A comple nmbe s epesented b a pont p(, n the Catesan plane wth abscssa and odnate. Then the -as s called eal as and the -as s called the magna as.the pont p(, s eeed to as the pont. Let OP and XOP. Then cos, sn Ee Comple nmbe can be epessed n the om as gen below (cos sn pola om e eponental om We obsee that the modls o and t epesents the dstance o the pont om the ogn. Also tan - the angle s called the ag ment o. - ( ( 4. Let then ( (. Now - ma be epesented as

2 - R (cos sn and R sees as the comple eqaton o the ccle C wth (, and ads R. In patcla epesents the ccle wth cente at the ogn and ads eqal to. Fnctons o a comple aable: Let C be a set comple nmbes. I to each comple nmbe n C thee coesponds a nqe comple nmbe w., then w s called a comple ncton o dened on C, and we wte w (. Hence,w has a eal pat, sa and an magna pat, sa. Then, w has the epesentaton W ( (, (, (Catesan om W (, (, (pola om Contnt : A comple aled ncton ( s sad to be contnos at a pont ( lm ( s dened at and. Note:. I a comple aled ncton ( s deentable at a pont, then t s contnos at.. The conese o the aboe eslt s not alwas te. The contnt o a comple ncton need not mpl ts deentablt. Deate o a comple ncton: The deate o a comple ncton at a pont, denoted b (, ( ( s dened as ( lm, poded ths lmt ests. Sbstttng -,we hae ( ( ( lm We shold emembe that b the denton o lmt ( s dened n a neghbohood o and ma appoach om an decton n the comple plane. The deate o a ncton at a pont s nqe t ests. Analtc Fnctons. Cach-Remann eqatons. In comple analss we ae nteested n the nctons, whch ae deentable n some doman, called the analtc nctons. A lage aet o nctons o comple aables whch ae sel o applcatons ppose ae analtc.

3 A Fncton ( s sad to be analtc at a pont, t s deentable at and, n addton, t s deentable thoghot some neghbohood o. Fthe a ncton ( s sad to be analtc n a doman D ( s dened and deentable at all ponts o D. In act, analtct s a global popet whle deentablt s a local popet. The tems egla and holomophc ae also sed n place o analtc. Cach-Remann Eqatons : Cach Remann eqatons pode a cteon o the analtct o a comple ncton W ( (, (,. Statement: Necessa condtons o a ncton to be analtc.: I ( (, (, s contnos n some neghbohood o a pont and s deentable at, then the st ode patal deates o (, and (, est and sats the Cach-Remann eqatons and. At the pont. Poo: Snce ( s deentable at, we hae ( ( ( lm { (, (, } { (. (, } ( lm Let s assme to wholl eal and wholl magna. Case I: When wholl eal, then, so that the ght sde o eqaton (I becomes, (I.The lmt on { (, - (. } { (, - (. } ( lm lm Case II: When wholl magna, then, so that lmt on the ght sde o eqaton (I becomes, (II.The

4 { } (. -, ( ( lm { } (. -, ( lm (III Snce ( s deentable the ale o the lmts obtaned om (II and (III mst be eqal. Compang the eal and magna pats, we get and at the (,. These ae known as the Cach-Remann eqatons. Satsacton o these eqatons s necessa o deentablt and analtct o the ncton ( at a gen pont. Ths, a ncton ( does not sats the Cach- Remann eqatons at a pont, t s not deentable and hence not analtc at that pont. E : I w log, nd d dw, and detemne whee w s not analtc. Let s consde n eponental om, sn (cos e ( tan, log( w ( tan log - Eqatng eal and magna pats ( tan : log - : :

5 Now om- C-R eqatons and Ths w log the C-R eqatons holds good o ( Fthe, dw d dw d d d - ( ( ( Ths ee pont othe than ogn (.e. wlog s deentable and the ncton log s analtc ee whee ecept at ogn. E: Show that the ncton w sn s analtc and nd the deate. w sn( sn cos cos sn ( - e e e e Now sn and cos sn snh : cos cosh Usng these n eqaton ( W ( sn cosh cos snh Eqatng eal and magna pats, we get sn cosh : cos snh cos cosh, sn snh ( sn snh, cos cosh The C-R eqatons ae satsed and ( sn s analtc. d ( cos cosh ( - sn snh d cos cos sn sn

6 ( cos cos Conseqences o C-R Eqatons:. I ( s an analtc ncton then and both sats the two dmensonal Laplace eqaton. φ φ Ths eqaton s also wtten as φ. Hee s the two- dmensonal Laplacan. Snce ( s analtc we hae Cach-Remann eqatons ( I and ( II Deentatng (I w,,t. and (II w..t patall we get and Bt s alwas te and hence we hae o,ths mples s hamonc. Smlal Deentatng (I w..t. and (II w..t. patall we get o, ths mples hamonc. I ( s an analtc ncton, then and ae hamonc nctons. Hee, and ae called hamonc conjgates o each othe. Conseqence II: I ( s an analtc ncton, then the eqatons (, c And (, c epesent othogonal amles o ces. Soln:, c ( ( (, c ( Deentatng eqn ( patall w..t

7 d d ( d o m (I d Deentatng eqn ( patall w..t. ( d d o m (II d d The two amles ae othogonal to each othe, then m m, And sng C-R eqatons ( ( m m ( ( ( ( ( ( Hence the ces ntesect othogonall at ee pont o ntesecton. Note: The conese o the aboe eslt s not te. The ollowng eample eeals the popet. : c : c d d ( ( m o ce c (- ( ( d m o ce c d ( 4 m m. The ntesect othogonall. Bt C-R Eqatons ae not satsed

8 and. Some deent oms o C-R Eqatons: I w (, s analtc, then the ollowng eslts ollows.. ( w w. ( ( sng C-R Eqatons. Based on the eslts aboe mentoned the ollowng eslts ae ald, a ( ( ( b ψ s an deental ncton o and then ψ ψ ψ ψ (. c ( [ ] Re (

9 d ( 4 ( Constcton o An Analtc Fncton When eal o Imagna pat s Gen (Pttng n Eact deental M d N d The Cach-Remann eqatons pode a method o constctng ± s gen. an analtc ncton ( when o o Sppose s gen, we detemne the deental d, snce (,, d d d sng C-R eqatons,ths becomes d d d M.d N d And t s clea that N M Becase s hamonc. Ths shows that M d N d s an eact deental. Conseqentl, can be obtaned b ntegatng M w..t. b teatng as a constant and ntegatng w..t. onl those tems n N that do not contan, and addng the eslts. Smlal, s gen then b sng d d d d d. Followng the pocede eplaned aboe we nd, and hence ( can be obtaned. Analogos pocede s adopted to nd when ± s gen. Mlne-Thomson Method: An altenate method o ndng s gen. Sppose we ae eqed to nd an analtc ncton ( when s gen. We ecall that ± when o o ±

10 ( (I φ ( φ (, (, Let s we set (, and (, (II φ Then (III φ Replacng b and b, ths becomes (IV ( φ (, φ (, Fom whch the eqed analtc ncton ( can be got. Smlal, s gen we can nd the analtc ncton ( b statng wth ( Analogos pocede s sed when ± s gen. Applcatons to low poblems: As the eal and magna pats o an analtc ncton ae the soltons o the Laplace s eqaton n two aable. The conjgate nctons pode soltons to a nmbe o eld and low poblems. Let be the eloct o a two dmensonal ncompessble ld wth gatonal moton, V j ( Snce the moton s otatonal cl V. Hence V can be wtten as φ φ φ j (II Theeoe, φ s the eloct component whch s called the eloct potental. Fom (I and (II we hae

11 φ φ, (III Snce the ld s ncompessble d V. φ φ (IV φ φ Ths ndcates that φ s hamonc. The ncton φ (, s called the eloct potental, and the ces, ae known as eq -potental lnes. φ ( c ψ so that Note : The estence o conjgate hamonc ncton (, (, ψ (, w( φ s Analtc. d ψ The slope s Gen b d ψ φ φ Ths shows that the eloct o the ld patcle s along the tangent to the ce ψ (, c, the patcle moes along the ce. ψ (, c - s called steam lnes φ(, c - called eqpotental lnes. As the eqpotental lnes and steam lnes ct othogonall. w( φ (, ψ (, dw d φ ψ φ φ

12 The magntde o the ld eloct ( dw d The low patten s epesented b ncton w( known as comple potental. Comple potental w( can be taken to epesent othe two-dmensonal poblems. (stead low. In electostatcs (, c (, c φ --- ntepeted as eqpotental lnes. ψ --- ntepeted as Lnes o oce. In heat low poblems: φ(, c --- Intepeted as Isothemal lnes ψ, c --- ntepeted as heat low lnes. ( Cach Remann eqatons n pola om: Let ( (e (, (, be analtc at a pont, then thee ests o contnos st ode patal deates,,,, :,. and sats the eqatons Poo: The ncton s analtc at a pont ( lm ( ( e. ests and t s nqe. Now ( (, (,. Let be the ncement n, coespondng ncements ae, n and.

13 ( lm ( { (, (, } { (. (, } { (, - (. } { (, - (. } lm Now lm (I e and s a ncton two aables and, then we hae. ( e ( e e e When tends to eo, we hae the two ollowng possbltes. (I. Let, so that e And Z, mples ( { (, - (. } { (, - (. } lm lm e e The lmt ests, ( e (I. Let, so that e And, mpl { (, - (. } { (, - (. } ( lm e lm e

14 e e e ( (II Fom (I and(ii we hae o, Whch ae the C-R Eqatons n pola om. Hamonc Fncton: A ncton φ -s sad to be hamonc ncton t satses Laplace s eqaton φ Let (, (, (e ( be analtc. We shall show that and sats Laplace s eqaton n pola om. φ φ φ The C-R eqatons n pola om ae gen b, (I (II Deentatng (I w..t and (II w.., patall, we get : And we hae

15 Ddng b we, get Hence -satses Laplace s eqaton n pola om. The ncton s hamonc. Smlal - s hamonc. Othogonal Sstem: d Let ( and tanφ, φ beng the angle between d the ads ecto and tangent. The angle between the tangents at the pont o ntesecton o the ces s φ φ. Tanφ Tanφ,s the condton o othogonal. Consde (, c. Deentatng w..t, teatng as a ncton o. d d Ths Tan d d d φ ( d ( (, c Smlal o the ce ( ( (I - ( (II Tanφ (

16 T anφ T anφ B C-R Eqatons ( ( ( (, The eqaton edces to T anφtanφ ( (- ( ( Hence the pola aml o ces and, ntesect othogonall. (, c (, c Constcton o An Analtc Fncton When eal o Imagna pat s Gen(Pola om. The method de to Eact deental and Mlne-Thomson s eplaned n eale secton. E: Ve that ( cos ncton. s hamonc. nd also an analtc Soln: cos : sn 6 cos : 4 cos 4. Then the Laplace eqaton n pola om s gen b, 6 cos 4 cos 4 cos 4

17 Hence -satses the laplace eqaton and hence s hamonc. Let s nd eqed analtc ncton (. We note that om the theo o deentals, d d d Usng C-R eqatons - d d sn d cos d 3 d- sn, Fom ths - sn c ( cos - sn c [ cos - sn ] c - e c c ( e ( c. E :Fnd an analtc ncton ( gen that - sn

18 Soln: sn : cos To nd sng the deentals d d d Usng C-R eqatons, d - d - cos d - sn d cos d - sn d d cos cos c ( cos c - sn (cos sn cos - sn - ( e e c ( c E: Constcton an analtc ncton gen (Mlne Thomson Method cos.

19 cos (I cos ( e Usng C-R eqatons, sn - ( e cos (- sn - [ cos ( sn ] e - e - [ cos sn ] Now pt, and. ( ( c on ntegatng. COMPLETION OF UNIT-I

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