C Complex Integration

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1 Fourier Trasform Methods i Fiace By Umberto herubii Giovai Della Luga abria Muliacci ietro ossi opyright Joh Wiley & os Ltd omple Itegratio. DEFINITION Let t be a real parameter ragig from t A to t B, ad let (t) be a curve, or cotour i the comple plae with edpoits A (t A ), B (t B ). Now we mark off a umber of poits t i betwee t A ad t B ad approimate the curve by a series of straight lies draw from each (t i )to(t i ). To defie the itegral of a fuctio f of a comple variable, we form the quatity i i f ( i ) i where i (t i ) (t i ) ad f ( i ) is the fuctio evaluated at a poit i o betwee (t i ) ad (t i ). The sum is evaluated i the it of a arbitrarily fie partitio of the rage through which the real parameter t moves while geeratig the cotour from A to B: that is, as, or, what is the same thig, i the it of arbitrarily small i for all i. Writig u( y) i ( y) ad d i dy we have (u d dy) i We ca also write this i parametric form. If (u dy d) we have t A d (t)dt dy y (t)dt t B u d dt dy dt For a give cotour ruig from A to B, we defie the opposite cotour, writte as to be the same curve but traversed from B to A. The itegral of alog is clearly give by the above equatio but with t A ad t B iterchaged. Thus dt i t A t B u dy dt d dt dt It also follows that If is a closed curve that does ot itersect itself, we shall always iterpret itegral take couter-clockwise alog the closed cotour. to mea the Eample.. Let us itegrate the fuctio couter-clockwise aroud the uit circle cetred at the origi. The values of o this curve are give by e i.

2 86 Fourier Trasform Methods i Fiace Therefore I e i e i d i Eample.. osider the fuctio, ad let the cotour be the uit circle about, which ca be parameteried by e it, with t i [ ). ubstitutig, we fid e it i eit dt i e it e it dt i dt i( ) i No itegral aroud the closed cotour is ero. The reaso, as we shall see, is that is ot aalytic aywhere ad therefore ot withi, ad is ot aalytic at which is withi. Both these eamples are eplaied by the auchy Goursat theorem.. THE AUHY GOUAT THEOEM Defiitio.. A ope subset U of is said to be simply coected if U has o holes ; for istace, every ope disk U : r qualifies. The theorem is usually formulated for closed paths as follows: Theorem.. (hauchy) Let U be a ope subset of which is simply coected, let f : U be a aalytic fuctio with cotiuous throughout this regio, ad let be a cotour i U whose start poit is equal to its ed poit. The, roof. Let us cosider the followig idetity (u d dy) i (u dy d) to evaluate the two lie itegrals o the right, we use Gree s theorem for lie itegrals. It states that if the derivatives of ad Q are cotiuous fuctios withi ad o a closed cotour, the ( d Q dy) Q y d dy where is the surface bouded by. By hypothesis is cotiuous, so the first partial derivatives of u ad are also cotiuous; the Gree s theorem yields (u d dy) i u y (u dy d) d dy i u y d dy

3 : omple Itegratio 87 But sice the auchy iema equatios hold, the itegrads above all vaish, therefore The coditio that U be simply coected is crucial; cosider (QED) (t) e it t [ ] which traces out the uit circle ad the the cotour itegral As we have see i the previous eample, its cotour itegral is o-ero: the auchy itegral theorem does ot apply here sice is ot defied (ad certaily ot aalytic) at. Oe importat cosequece of the theorem is that cotour itegrals of aalytic fuctios o simply coected domais ca be computed i a maer familiar from the fudametal theorem of real calculus: let U be a simply coected ope subset of,let f : U be a holomorphic fuctio, ad let be a piecewise cotiuously differetiable cotour i U with start poit A ad ed poit B, the F(b) F(a) As was show by Goursat, auchy s itegral theorem ca be proved assumig oly that the comple derivative eists everywhere i U without requirig cotiuity. This is because ay fuctio which is aalytic i a regio ecessarily has a cotiuous derivative. I fact a aalytic fuctio has derivatives of all orders ad therefore all its derivatives are cotiuous, the cotiuity of the th derivative beig a cosequece of the eistece of the derivative of order. But it is possible to establish this result o higher derivatives oly after oe shows that the cotiuity of is ot eeded i the proof of auchy s theorem. The relaatio of this hypotheses is therefore of utmost importace, ad it is Goursat s result that really distiguishes the theory of itegratio of a fuctio of comple variable from the theory of lie itegrals i the real plae. Theorem.. (hauchy Goursat) Let U be a ope subset of which is simply coected, let f : U be a aalytic fuctio ad let be a cotour i U whose start poit is equal to its ed poit. The, The proof of the theorem is more ivolved tha the previous oe ad we refer the iterested reader to the literature..3 ONEQUENE OF AUHY THEOEM The auchy itegral theorem leads to the auchy itegral formula ad the residue theorem. Theorem.3. uppose U is a ope subset of the comple plae, ad as usual f : U is a aalytic fuctio, ad the disk D : r is completely cotaied

4 88 Fourier Trasform Methods i Fiace y L L r Figure. hauchy itegral theorem i U. Let be the circle formig the boudary of D. The for every a i the iterior of D we have: f (a) i a where the cotour itegral is to be take couter-clockwise. The proof of this statemet uses the auchy itegral theorem ad, just like that theorem, oly eeds f to be comple differetiable. It is worth followig the proof i order to become acquaited with comple itegral calculus. roof. Let us cosider Figure.: iside the cotour we draw a circle of radius r about ad cosider the cotour formed by the circle, the lie ad the two straight lie segmets L ad L, which lie arbitrarily close to each other. Let us call this etire cotour. Now cosider Iside, L is aalytic, so by the auchy Goursat theorem L Now, as we brig the lie segmets L ad L arbitrarily close together, L L

5 : omple Itegratio 89 sice the lies are traversed i opposite directios. Thus, i this it we have so that At this poit we ote that is traversed i a clockwise directio, sice it is cosidered as a cotour i its ow right i.e. ot just as a part of. Let us therefore defie so that is a couter-clockwise cotour, the we may write f ( ) f ( ) We ow use the fact that is a circle to write r e i o, thus the first itegral o the right becomes for all r ir e i r e i d i withi. A auchy formula will therefore be established if we ca show that f ( ) for some choice of the cotour. The cotiuity of at tells us that, for all, there eists a such that if, the f ( ). o, by takig r, we satisfy the coditio which i tur implies that f ( ) f ( ) ( ) Thus by takig r small eough but still greater tha ero, the absolute value of the itegral ca be made smaller tha ay pre-assiged umber, implyig that: if( ) This result meas, amog the other thigs, that if a fuctio is aalytic withi ad o a cotour, its value at every poit iside is determied by its values o the boudig curve. Oe may replace the circle with ay closed rectifiable curve i U which does t have ay self-itersectios ad which is orieted couter-clockwise. The formulas remai valid for ay poit from the regio eclosed by this path. Oe ca the deduce from the formula that f must actually be ifiitely ofte cotiuously differetiable, with f () ( )! i ( )

6 9 Fourier Trasform Methods i Fiace ome call this idetity auchy s differetiatio formula. A proof of this last idetity is a by-product of the proof that holomorphic fuctios are aalytic. A importat cosequece of the auchy s itegral formula is the followig: Theorem.3. (Liouville s theorem) If is etire ad is bouded for all values of, the is a costat. roof. From auchy s itegral formula, takig the derivative of both members, we have that f ( ) i ( ) if we take to be the circle r, the f ( ) i ( ) r M r where M withi ad o. Therefore f ( ) M r, ad we may take r as large as we like because is etire. o takig r large eough, we ca make f ( ) for ay pre-assiged. That is f ( ), which implies that f ( ) for all so f ( ) costat. I particular, from Liouville s theorem we ca coclude that if we have a fuctio that is aalytic i the etire comple plae ad is such that as i the etire comple plae, the this fuctio is idetically ero i the etire plae. M r.4 INIAL VALUE Let us begi by cosiderig a fuctio that is aalytic i the upper half of the comple plae ad is such that as i the upper half plae. Now cosider the cotour itegral where is the cotour show i Figure. ad is real. By assumptio, is aalytic withi ad o ;sois ( ). Thus Let us break this itegral as follows: f () d f () d Here is the radius of the small semicircle cetred at ad is the radius of the large semicircle cetred at the origi, as show i Figure.. The radius ca be chose as small as we please, ad ca be chose as large as we like. I the it of arbitrarily small, the quatity f () d f () d

7 : omple Itegratio 9 δ δ α δ α α + δ + Figure. The cotour,, used to obtai equatio (.). The radius,, of the semicircle,,may be made as large as ecessary ad the radius,, of the semicircle,, may be made as small as we please is called the pricipal-value itegral of f () ( ) ad is deoted by f () d Now alog the large semicircle we set e i, so that i f ( e i ) e i e i d But so we ca write e i [ cos ] [ ] f ( e i ) d But as ad ( ). Therefore the itegral over the semicircle of radius ca be made arbitrarily small by choosig sufficietly large. Thus we may write: f () d f ( ) f ( ) where we have added ad subtracted the term f ( ) ettig e i

8 9 Fourier Trasform Methods i Fiace i the first itegral o the right-had side of this equatio, we fid that Thus f ( ) f () if ( ) d i f ( ) d i f ( ) f ( ) ice is cotiuous at, the argumet used i derivig auchy s itegral formula tell us that this last itegral over vaishes. Hece f () For the sake of brevity we write this simply as d i f ( ) f () d i f ( ) (.) where f () is a comple-valued fuctio of a real variable. The pricipal-value itegral ca be see as a way to avoid sigularities o a path of itegratio: oe itegrates to withi of the sigularity i questio, skips over the sigularity ad begis itegratig agai a distace beyod the sigularity. This prescriptio is also very useful i the oe-dimesioal real aalysis where it eables oe to make sese of such itegrals as: Oe would like this itegral to be ero, sice we are itegratig a odd fuctio over a symmetric domai. However, uless we isert a i frot of this itegral, the sigularity at the origi makes the itegral meaigless. Followig the prescriptio for pricipal-value itegrals we ca easily evaluate the above itegral, we have d d I the first itegral o the right-had side, set d y. The d d dy y d The sum of the two itegrals iside the bracket is obviously ero sice b a thus a d b

9 : omple Itegratio 93 Eample.4. Let us evaluate the followig itegral d a where a. Aswer: First of all we write the itegral i the form d a a d a a d a ettig y i the first itegral o the right-had side, we fid that d a dy y a l( a) l [l l( a) l( a) l ] thus d a l a a a.5 LAUENT EIE We ow come to oe of the most importat applicatios of the auchy Goursat theorem, amely the possibility of epadig a aalytic fuctio i a power series. The mai result may be stated as follows: Theorem.5. If is aalytic throughout the aular regio betwee ad o the cocetric circles ad cetred at a ad of radii r ad r r respectively, the there eists a uique series epasio i terms of positive ad egative powers of ( a), a k ( a) k b k ( k k a) k where a k i ( a) k b k i ( a) k roof. Let there be two circular cotours ad, with the radius of larger tha that of.let be at the cetre of ad, ad be betwee ad. Now create a cut lie c betwee ad, ad itegrate aroud the path c c, so that the plus ad mius cotributios of c cacel oe aother, as illustrated i Figure.3. ice

10 94 Fourier Trasform Methods i Fiace c c Figure.3 omple itegral cotour used for the proof of uicity of Lauret eries is aalytic withi ad o, from the auchy itegral formula, i f ( ) f ( ) i f ( ) i c f ( ) i f ( ) i c f ( ) i f ( ) (.) i sice cotributios from the cut lie i opposite directios cacel out. Now f ( ) i ( ) ( ) f ( ) i ( ) ( ) f ( ) i f ( ) i f ( ) i f ( ) i (.3)

11 : omple Itegratio 95 For the first itegral,. For the secod,. Now use the Taylor epasio (valid for t ) t t to obtai f ( ) i f ( ) i ( ) f ( ) i ( ) ( ) ( ) f ( ) i ( ) f ( ) i ( ) ( ) ( ) f ( ) i where the secod term has bee re-ideed. e-ideig agai, (.4) i ( ) f ( ) ( ) i ( ) f ( ) (.5) ( ) ice the itegrads, icludig the fuctio, are aalytic i the aular regio defied by ad, the itegrals are idepedet of the path of itegratio i that regio. If we replace paths of itegratio ad by a circle of radius r with r r r, the ( ) f ( ) i ( ) i ( ) f ( ) ( ) ( ) f ( ) i ( ) a ( ) (.6) Geerally, the path of itegratio ca be ay path that lies i the aular regio ad ecircles oce i the positive (couter-clockwise) directio. The comple residues a are therefore defied by a i f ( ) ( )

12 96 Fourier Trasform Methods i Fiace The costat a i the Lauret series.6 OMLEX EIDUE a ( ) of about a poit is called the residue of. If f is aalytic at, its residue is ero, but the coverse is ot always true (for eample, has residue at but is ot aalytic at ). The residue of a fuctio f at a poit may be deoted es ( ). Two basic eamples of residues are give by es ad es for. The residue of a fuctio f aroud a poit is also defied by es f i f where is a couter-clockwise simple closed cotour, small eough to avoid ay other poles of f. I fact, ay couter-clockwise path with cotour-widig umber which does ot cotai ay other pole gives the same result by the auchy itegral formula. Figure.4 shows a suitable cotour for which to defie the residue of fuctio, where the poles are idicated as black dots. The residues of a fuctio may be foud without eplicitly epadig ito a Lauret series as follows. If has a pole of order m at, the a for m ad a m. Therefore, a ( ) a m ( ) ( m ) m ( ) m a m ( ) es f() = i = 3 +i es f() = = es f() = = γ es f() = 5 = i es f() = = i Figure.4 omple itegral cotour for the eample i sectio.7

13 : omple Itegratio 97 d ( ) m ( ) a m ( ) ( ) a m ( ) ( )a m ( ) (.7) d ( ) m ( ) ( )a m ( ) ( ) ( )a m ( ) ( )( )a m ( ) (.8) Iteratig, d m m ( ) m ( )( ) ( m )a ( ) (m )!a ( )( ) ( m )a ( ) (.9) o ad the residue is d m ( m ) m (m )!a (m )!a d m a ( (m )! m ) m The residues of a holomorphic fuctio at its poles characterie a great deal of the structure of a fuctio, appearig for eample i the amaig residue theorem of cotour itegratio..7 EIDUE THEOEM Let there eist a aalytic fuctio whose Lauret series is give by a ( )

14 98 Fourier Trasform Methods i Fiace ad itegrate term by term usig a closed cotour ecirclig, a ( ) a ( ) a () ( ) a ( ) (.) The auchy itegral theorem requires that the first ad last terms vaish, so we have a () ( ) where a is the comple residue. Usig the cotour (t) e it gives so we have () ( ) (i e it dt) (e it ) i ia If the cotour ecloses multiple poles, the the theorem gives the geeral result i a A es ai where A is the set of poles cotaied iside the cotour. This amaig theorem therefore says that the value of a cotour itegral for ay cotour i the comple plae depeds oly o the properties of a few very special poits iside the cotour. Figure.4 shows a eample of the residue theorem applied to the illustrated cotour ad the fuctio 3 ( ) ( i) ( i) i ( 3 i) 5 ( i) Oly the poles at ad i are cotaied i the cotour, ad have residues of ad, respectively. The values of the cotour itegral is therefore give by i( ) 4 i Eample.7. osider agai the itegral ( ) d Now we are goig to solve it usig the residue approach. osider the comple-valued fuctio ( )

15 : omple Itegratio 99 The Lauret series of about i, the oly sigularity we eed to cosider, is 4( i) i 4( i) It is clear by ispectio that the residue is 3 6 i 8 ( i) 5 ( i) 64 i 4, so, by the residue theorem, we have ( ) i es i f i( i 4).8 JODAN LEMMA Jorda s lemma shows the value of the itegral alog the ifiite upper semicircle ad with a I f ()e ia d is for ice fuctios which satisfy f ( e i ) Thus, the itegral alog the real ais is just the sum of comple residues i the cotour. The lemma ca be established usig a cotour itegral I that satisfies I a () To derive the lemma, write e i (cos i si ) d ie i d ad defie the cotour itegral I f ( e i )e iacos asi ie i d The I f ( e i ) e iacos e asi i e i d f ( e i ) e asi d f ( e i ) e asi d (.) Now, if f ( e i ), choose a such that f ( e i ),so I e asi d But, for i [ ], si

16 Fourier Trasform Methods i Fiace so As log as the follows. I e a d e a a a ( e a ) (.), Jorda s lemma I a ()