Complex Variables. For ECON 397 Macroeconometrics Steve Cunningham

Transcription

1 Comple Variables For ECON 397 Macroeconometrics Steve Cnningham

2 Open Disks or Neighborhoods Deinition. The set o all points which satis the ineqalit <ρ where ρ is a positive real nmber is called an open disk or neighborhood o. Remark. The nit disk i.e. the neighborhood < < 1 is o particlar signiicance. 1

3 Interior Point Deinition. A point S is called an interior point o S i and onl i there eists at least one neighborhood o which is completel contained in S. S

4 Open Set. Closed Set. Deinition. I ever point o a set S is an interior point o S we sa that S is an open set. Deinition. I BS S i.e. i S contains all o its bondar points then it is called a closed set. Sets ma be neither open nor closed. Open Closed Neither

5 Connected An open set S is said to be connected i ever pair o points 1 and in S can be joined b a polgonal line that lies entirel in S.. Roghl speaking this means that S consists o a single piece althogh it ma contain holes. S 1

6 Domain Region Closre Bonded Compact An open connected set is called a domain. A region is a domain together with some none or all o its bondar points. The closre o a set S denoted S is the set o S together with all o its bondar. Ths S S BS. A set o points S is bonded i there eists a positive real nmber R sch that < <R or ever S. A region which is both closed and bonded is said to be compact.

7 Review: Real Fnctions o Real Variables Deinition. Let Γ R.. A nction is a rle which assigns to each element a Γ one and onl one element b Ω Ω R.. We write : Γ Ω or in the speciic case b a and call b the image o a nder. We call Γ the domain o deinition o or simpl the domain o. We call Ω the range o. We call the set o all the images o Γ denoted Γ the image o the nction. We alternatel call a mapping rom Γ to Ω.

8 Real Fnction In eect a nction o a real variable maps rom one real line to another. Γ Ω

9 Comple Fnction Deinition. Comple nction o a comple variable. Let Φ C. A nction deined on Φ is a rle which assigns to each Φ a comple nmber w. The nmber w is called a vale o at and is denoted b i.e. w. The set Φ is called the domain o deinition o. Althogh the domain o deinition is oten a domain it need not be.

10 Remark Properties o a real-valed nction o a real variable are oten ehibited b the graph o the nction. Bt when w where and w are comple no sch convenient graphical representation is available becase each o the nmbers and w is located in a plane rather than a line. We can displa some inormation abot the nction b indicating pairs o corresponding points and w v. To do this it is sall easiest to draw the and w planes separatel.

11 Graph o Comple Fnction w v -plane domain o deinition w-plane range

12 Eample 1 Describe the range o the nction i deined on the domain is the nit disk 1. Soltion: We have and v. Ths as varies over the closed nit disk varies between and 1 and v is constant. Thereore w iv i is a line segment rom w i to w 1 i. domain v range

13 Eample Describe the nction 3 or in the semidisk given b Im. Soltion: We know that the points in the sector o the semidisk rom Arg to Arg π/3 when cbed cover the entire disk w 8 becase i π i e 3 8e The cbes o the remaining points o also all into this disk overlapping it in the pper halplane as depicted on the net screen. 3 π

14 w 3 v

15 Seqence Deinition. A seqence o comple nmbers denoted { n} k 1 is a nction sch that : N C i.e it is a nction whose domain is the set o natral nmbers between 1 and k and whose range is a sbset o the comple nmbers. I k then the seqence is called ininite and is denoted b { } n 1 or more oten n. The notation n is eqivalent. Having deined seqences and a means or measring the distance between points we proceed to deine the limit o a seqence.

16 Limit o a Seqence { } n 1 Deinition. A seqence o comple nmbers is said to have the limit or to converge to i or an ε > there eists an integer N sch that n < ε or all n > N.. We denote this b lim or n n n as n. Geometricall this amonts to the act that is the onl point o n sch that an neighborhood abot it no matter how small contains an ininite nmber o points n.

17 Limit o a Fnction We sa that the comple nmber w is the limit o the nction as approaches i stas close to w whenever is sicientl near. Formall we state: Deinition. Limit o a Comple Seqence. Let be a nction deined in some neighborhood o ecept with the possible eception o the point is the nmber w i or an real nmber ε > there eists a positive real nmber δ > sch that w < ε whenever < - < δ.

18 Limits: Interpretation We can interpret this to mean that i we observe points within a radis δ o we can ind a corresponding disk abot w sch that all the points in the disk abot are mapped into it. That is an neighborhood o w contains all the vales assmed b in some ll neighborhood o ecept possibl. v w δ w ε -plane w-plane

19 I as lim then Properties o Limits lim A and lim g B lim [ ± g ] A ± B lim g AB and lim /g A/B. i B.

20 Continit Deinition. Let be a nction sch that : C C. We call continos at i: F is deined in a neighborhood o The limit eists and lim A nction is said to be continos on a set S i it is continos at each point o S. I a nction is not continos at a point then it is said to be singlar at the point.

21 Note on Continit One can show that approaches a limit precisel when its real and imaginar parts approach limits and the continit o is eqivalent to the continit o its real and imaginar parts.

22 Properties o Continos Fnctions I and g are continos at then so are ± g and g. The qotient /g is also continos at provided that g. Also continos nctions map compact sets into compact sets.

23 Derivatives Dierentiation o comple-valed nctions is completel analogos to the real case: Deinition. Derivative. Let be a comple- valed nction deined in a neighborhood o. Then the derivative o at is given b lim Provided this limit eists. F is said to be dierentiable at.

24 Properties o Derivatives Properties o Derivatives [ ] [ ] [ ] Rle. Chain ' '. i ' ' ' ' ' '. constant or an ' ' ' ' ' g g g d d g g g g g g g g c c c g g ± ±

25 Analtic. Holomorphic. Deinition. A comple-valed nction is said to be analtic or eqivalentl holomorphic on an open set Ω i it has a derivative at ever point o Ω.. The term reglar is also sed. It is important that a nction ma be dierentiable at a single point onl. Analticit implies dierentiabilit within a neighborhood o the point. This permits epansion o the nction b a Talor series abot the point. I is analtic on the whole comple plane then it is said to be an entire nction.

26 Rational Fnction. Deinition. I and g are polnomials in then h /g g is called a rational nction. Remarks. All polnomial nctions o are entire. A rational nction o is analtic at ever point or which its denominator is nonero. I a nction can be redced to a polnomial nction which does not involve then it is analtic.

27 Eample 1 Eample i i i i let i Ths 1 is analtic at all points ecept 1.

28 Eample Eample i i i i let i Ths is nowhere analtic.

29 Testing or Analticit Determining the analticit o a nction b searching or in its epression that cannot be removed is at best awkward. Observe: It wold be diiclt and time consming to tr to redce this epression to a orm in which o cold be sre that the cold not be removed. The method cannot be sed when anthing bt algebraic nctions are sed.

30 Cach-Riemann Eqations 1 I the nction iv is dierentiable at i then the limit lim can be evalated b allowing to approach ero rom an direction in the comple plane.

31 Cach Cach-Riemann Eqations Riemann Eqations I it approaches along the -ais then and we obtain [ ] [ ] iv iv lim ' v v i lim lim ' Bt the limits o the bracketed epression are jst the irst partial derivatives o and v with respect to so that:. ' v i

32 Cach Cach-Riemann Eqations 3 Riemann Eqations 3 I it approaches along the -ais then and we obtain lim ' And thereore. ' v i v v i lim

33 Cach-Riemann Eqations 4 B deinition a limit eists onl i it is niqe. Thereore these two epressions mst be eqivalent. Eqating real and imaginar parts we have that v v and mst hold at i. These eqations are called the Cach-Riemann Eqations. Their importance is made clear in the ollowing theorem.

34 Cach-Riemann Eqations 5 Theorem. Let iv be deined in some open set Γ containing the point. I the irst partial derivatives o and v eist in Γ and are continos at and satis the Cach-Riemann eqations at then is dierentiable at. Conseqentl i the irst partial derivatives are continos and satis the Cach-Riemann eqations at all points o Γ then is analtic in Γ.

35 Eample 1 Eample v v i Hence the Cach-Riemann eqations are satisied onl on the line and thereore in no open disk. Ths b the theorem is nowhere analtic.

36 Eample Eample Prove that is entire and ind its derivative. e v e e v e ie e sin sin cos cos Soltion : sin cos The irst partials are continos and satis the Cach-Riemann eqations at ever point.. sin cos ' ie e v i

37 Harmonic Fnctions Deinition. Harmonic. A real-valed nction ϕ is said to be harmonic in a domain D i all o its second-order order partial derivatives are continos in D and i each point o D satisies ϕ ϕ Theorem. I iv is analtic in a domain D then each o the nctions and v is harmonic in D..

38 Harmonic Conjgate Given a nction harmonic in sa an open disk then we can ind another harmonic nction v so that iv is an analtic nction o in the disk. Sch a nction v is called a harmonic conjgate o.

39 Eample Eample Constrct an analtic nction whose real part is:. 3 3 Soltion: First veri that this nction is harmonic and 6 6 and 3 3 and

40 Eample Eample Contined Contined v and v Integrate 1 with respect to : v v v ξ

41 Eample Contined Now take the derivative o v with respect to : v 6 ξ'. According to eqation this eqals 6 1. Ths 6 and So ξ' ξ' ξ Eqivalentl and ξ ξ 1. C. And v 3 3 C.

42 Eample Contined The desired analtic nction iv is: i 3 C

43 Comple Eponential We wold like the comple eponential to be a natral etension o the real case with e entire. We begin b eamining e i e e i e i. e i cos i sin b Eler s and DeMoivre s relations. Deinition. Comple Eponential Fnction. I i then e e cos i sin. That is e e and arg e.

44 More on Eponentials Recall that a nction is one-to to-oneone on a set S i the eqation 1 where 1 S implies that 1. The comple eponential nction is not one-to to-one one on the whole plane. Theorem. A necessar and sicient condition that e 1 is that kπi where k is an integer. Also a necessar and sicient condition that 1 e e is that 1 kπi where k is an integer. Ths is a periodic nction. integer. Ths e is a periodic nction.