- 105 -Chapter IVIntegralsIntegrals are extremely important in the study of functions of a complex variableThe theory of integration, to be developed in this chapter, is noted for itsmathematical elegance. The theorems are generally concise and powerful, and mostoftheproofsaresimple.S4.1.Derivatives ofComplex-ValuedFunctions ofOneRealVariableIn order to introduce integrals of f in a fairly simple way, we need to firstconsider derivative of a complex-valued function w of a real variable t.Wewritew(t) =u(t)+ iv(t),(4.1.1)where the functions u and y are real-valuedfunctions ofareal variable t.Definition 4.1.1. If the derivatives u'(t) and '(t) exists at t,then we saythat the function (4.1.1) is differentiable, or derivable, at t and its derivativew'(t), or d[w(t)]/dt,at 1 is defined asw'(t)=u(t)+iv'(t)(4.1.2)From definition (4.1.2), it follows that if a functionw(t)= u(t)+ iv() isdifferentiable at t, then for every complex constant zo =Xo +iyo, the functionzow is differentiable at t, andd.[zow(t)] =[(x + iy)(u() + iv(0))dt=[(xou(t)-yov(t)) +i(you(t)+Xov(t)=(xou(t)-yov(t))+i(you(t)+xov(t)=(xou(t)-yov(t)+i(you(t)+xov'(t)=(xo +iy)(u' +iv)= zow'(t).So, zow is differentiable at t and
- 105 - Chapter Ⅳ Integrals Integrals are extremely important in the study of functions of a complex variable. The theory of integration, to be developed in this chapter, is noted for its mathematical elegance. The theorems are generally concise and powerful, and most of the proofs are simple. §4.1. Derivatives of Complex-Valued Functions of One Real Variable In order to introduce integrals of in a fairly simple way, we need to first consider derivative of a complex-valued function of a real variable t . We write f w = + tivtutw )()()( , (4.1.1) where the functions u and v are real-valued functions of a real variable t . Definition 4.1.1. If the derivatives ′ tu )( and ′ tv )( exists at t ，then we say that the function (4.1.1) is differentiable, or derivable, at and its derivative , or , at is defined as t ′ tw )( /)]([ dttwd t ′ = ′ + ′ tvitutw )()()( . (4.1.2) From definition (4.1.2), it follows that if a function = + tivtutw )()()( is differentiable at t , then for every complex constant 000 = + iyxz , the function wz is differentiable at , and 0 t ).( ))(( ))()(())()(( ))()(())()(( ]))()(())()([( ]))()()([()]([ 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 twz viuiyx tvxtuyitvytux tvxtuyitvytux tvxtuyitvytux tivtuiyxtwz dt d = ′ += ′ + ′ = ′ − ′ + ′ + ′ −= ′ ++ ′ ++−= ′ ++= ′ . So, wz is differentiable at t and 0
- 106 -q[2ow(t)]= zow()(4.1.3)dtAnother expected rule that we shall often use isd-e0 = z0e(4.1.4)dtwhere zo = Xo + iyo. To verify this, we writee"o = eof eor = e"o cos yot + ie' sin yotand refer to definition (4.1.2) to see thatde* = (e' cos yot)'+i(e' sin yot).diFamiliar rules fromcalculus and some simplealgebra then lead us to the expressiond-e' = (xo + iyo)(e cos yot + ie"" sin yot).dtThis gives thatd"of = zoe"ofgo = Zoetodtand eqution (4.1.4)is established.Various other rules learned in calculus, such as the ones for differentiatingsums and products, apply just as they dofor real-valuedfunctions of t.As was thecase with property (4.1.3) and formula (4.1.4), verifications may be based oncorresponding rules in calculus. It should be pointed out, however, that not everyrule for derivatives in calculus carries over to functions of type (4.1.1).Thefollowingexampleillustrates thisExample.Supposethatthefunction wgivenby (4.1.1),is continuouson aninterval [a,b];that is, its component functions u and are continuous there.Even ifw'(t)existwhena<t<b,themeanvaluetheoremforderivativesnolonger applies. To be precise, it is not necessarily true that there is a number c inthe interval (a,b) such thatW(c) = w(b)-w(a)b-aTo see this, consider the function w(t)= ei on the interval [0,2元]. When thatfunction is used, we have [w'(t) |Hie" -l; and this means that the derivative w"is never zero, whilew(2元)-w(0)=0
- 106 - )()]([ 0 0 twztwz dt d = ′ . (4.1.3) Another expected rule that we shall often use is tz tz eze dt d 0 0 = 0 , (4.1.4) where . To verify this, we write 000 += iyxz tyietyeeee txtiytxtz tx 0 0 cos sin 0 00 0 0 == + and refer to definition (4.1.2) to see that )sin()cos( 0 0 0 0 0 = ′ + tyeityee ′ dt d txtz tx . Familiar rules from calculus and some simple algebra then lead us to the expression 00 cos)(( 0 0 )sin 0 0 0 tyietyeiyxe dt d tz tx tx += + . This gives that 0 00 0 0 0 tiytxtz tz ezeeze dt d = = . and eqution (4.1.4) is established. Various other rules learned in calculus, such as the ones for differentiating sums and products, apply just as they do for real-valued functions of . As was the case with property (4.1.3) and formula (4.1.4), verifications may be based on corresponding rules in calculus. It should be pointed out, however, that not every rule for derivatives in calculus carries over to functions of type (4.1.1). The following example illustrates this. t Example. Suppose that the function given by (4.1.1), is continuous on an interval ; that is, its component functions and v are continuous there. Even if exist when w ba ],[ u ′ tw )( < < bta , the mean value theorem for derivatives no longer applies. To be precise, it is not necessarily true that there is a number in the interval such that c ba ),( ab awbw cw − − ′ = )()( )( . To see this, consider the function on the interval it )( = etw π ]2,0[ . When that function is used, we have ; and this means that the derivative is never zero, while ′ == 1|||)(| it ietw w′ π − ww = 0)0()2(
107This shows that thedesired number c doed not exist for this function
107 This shows that the desired number c doed not exist for this function
$4.2.Definite Integrals of Functions wLet w bea complex-valued function ofa real variable t in [a,b], then it can be written asw(t)=u(t)+iv(t), Vte[a,b] ,(4.2.1)where u and y arereal-valued.Definition 4.2.1.Forafunction wasin(4.2.1),if u and y areRiemann integrable over[a,b], then we say that w is integrable on [a,b] and the definite integral of w over [a,b]is defined asJ"w(0)d = ['u(0)dt +if'v(n)dt.(4.2.2)ThusRe "w(t)dt = ['Re[w(t)]dt and Im [w(t)dt = ['Im[w(t)]dt. (4.2.3)Example 1. For an illustration of definition (4.1.2), we compute2(1+it)' dt = ['[1-t')+i2tldt = (1-t')dt+if’2tdt =-3Improperintegralsof woverunboundedintervalsaredefinedinasimilarway.The existenceof the integrals of u and v indefinition (4.2.2)is ensured if those functionsarepiecewise continuous on the interval [a,b].Such a function is continuous everywhere in thestated interval except possibly for a finite number of points where,although discontinuous, it hasone-sided limits. Of course, only the right-hand limit is required at a ; and only the left-hand limitis required at b.When both u and y are piecewise continuous, the function w is said to bepiecewise continuous.Thus,every piecewise continuous complex-valued function on the interval[a,b] is integrable over the interval.Some basic properties of the integrals defined here are listed in the following theoremTheorem 4.2.1. Suppose that the complex-valued functions w,w,w, are all integrableovertheinterval[a,bl,then(l)Thefunction w,+w,is integrableover [a,b] and[(w(0)+w,(t)dt=]w(0)d + Jw()dt;(4.2.4)(2) For every complex number c, the function cw is integrable over [a,b] andJ'cw(t)dt =cf'w(t)dt ;(3) When a<c<b, w is integrable over [a,c] and [c,b],andI'w()dt =f' w(t)dt + f'w(t)dt ;(4)Thefunction w] is integrable over [a,b] and['w(t)dil " w(t)] dt(4.2.5)Proof. The proofs for (1) to (3) are easy to do by recalling corresponding results in calculus.Next, we give the proof for (4). By Definition 4.2.1, we know that the real and imaginary parts uand v of w are all Riemann integrable over [a,b], and so the functionI w(t) = /(u(t)* +(v(t)2is Riemann integrable over [a,b]. To show the inequality (4.2.4) is valid, we may assume thatthe integral on the left is a nonzero complex number. If ro is the modulus and 。 is anargument of the integral on the left, thenJ'w(t)dt = roe
§4.2. Definite Integrals of Functions w Let w be a complex-valued function of a real variable t in ba ],[ , then it can be written as = + tivtutw )()()( , ∀ ∈ bat ],[ ， (4.2.1) where u and v are real-valued. Definition 4.2.1. For a function as in (4.2.1), if and v are Riemann integrable over , then we say that is integrable on and the definite integral of over is defined as w u ba ],[ w ba ],[ w ba ],[ ∫ ∫ ∫ += b a b a b a )()()( dttvidttudttw . (4.2.2) Thus ∫ ∫ = b a b a )](Re[)(Re dttwdttw and . (4.2.3) ∫ ∫ = b a b a )](Im[)(Im dttwdttw Example 1. For an illustration of definition (4.1.2), we compute [ ] ∫ ∫ ∫ +=+−=+−=+ 1 0 1 0 1 0 2 1 0 2 2 3 2 2)1(2)1()1( itdtidttdttitdtit ∫ . Improper integrals of w over unbounded intervals are defined in a similar way. The existence of the integrals of and in definition (4.2.2) is ensured if those functions are piecewise continuous on the interval . Such a function is continuous everywhere in the stated interval except possibly for a finite number of points where, although discontinuous, it has one-sided limits. Of course, only the right-hand limit is required at ; and only the left-hand limit is required at . When both and are piecewise continuous, the function is said to be piecewise continuous. Thus, every piecewise continuous complex-valued function on the interval is integrable over the interval. u v ba ],[ a b u v w ba ],[ Some basic properties of the integrals defined here are listed in the following theorem. Theorem 4.2.1. Suppose that the complex-valued functions are all integrable over the interval , then 21, www ba ],[ (1) The function is integrable over and + ww 21 ba ],[ ∫ ∫ ∫ +=+ b a b a b a 1 2 1 2 )()())()(( dttwdttwdttwtw ; (4.2.4) (2) For every complex number c , the function cw is integrable over ba ],[ and ; ∫∫ = b a b a )()( dttwcdttcw (3) When << bca , w is integrable over ca ],[ and bc ],[ , and ∫ ∫ ∫ += c a b c b a )()()( dttwdttwdttw ; (4) The function w|| is integrable over ba ],[ and ∫∫ ≤ b a b a |)(|)( dttwdttw . (4.2.5) Proof. The proofs for (1) to (3) are easy to do by recalling corresponding results in calculus. Next, we give the proof for (4). By Definition 4.2.1, we know that the real and imaginary parts and of are all Riemann integrable over , and so the function u v w ba ],[ 2 2 += tvtutw ))(())((|)(| is Riemann integrable over . To show the inequality (4.2.4) is valid, we may assume that the integral on the left is a nonzero complex number. If is the modulus and ba ],[ 0r θ 0 is an argument of the integral on the left, then ∫ = b a i erdttw 0 0 )( θ
Solving for ro, we writew(t)dt(4.2.6)Now the leff-hand side of this equation is a real number, and so the right-hand side is real, too.Thus, using the fact that the real part of a real number is the number itself and referring to the firstof properties (4.2.3), we see that the right-hand side of equation (4.2.6) can be rewritten in thefollowing way:['e-0 w(1)dt = Ref'e- w(1)dt = f'Re(e% (0)dt.0=1(4.2.7)ButRe(e-10o w(t) ≤e-18ow(t) Hl e-0 Il w(0) H| w(t) ],Vt e [a,b];and so,accordingto equation (4.2.7),wehaver≤['1w(0)/dtThe proof is completed.The fundamental theorem of calculus, involving antiderivatives (i.e., primitive functions),can be extended so as to applyto integrals of thetype (4.2.2).Theorem 4.2.2. Suppose that the functionsw(t) =u(t)+ iv(t) and W(t) =U(t)+iV(t)are continuous on the interval [a,b] and W'(t)=w(t) when te[a,b], thenJ w(t)dt = W(b) -W(a) = W(t)l°Proof. Since W'(t)= w(t), we have U'(t)= u(t) and V'(t) = v(t). Hence, from thefundamental theorem of calculus and Definition 4.2.1, we obtain that'w()dt = J'u()dt+if'()dt=U(0) +iV(0)。=[U(b) + iV(b)] -[U(a)+iV(a)] Thus, we deduce that["'w(t)dt = W(b) -W(a) = W(t)。This completes the proof.Example 2. Since (e")'= ie" (See Sec. 4.1), we have e =(-ie")' and soJ"*ed -i']" -ie*+i+i=元112
Solving for , we write 0r ∫ − = b a i )( dttwer 0 0 θ . (4.2.6) Now the left-hand side of this equation is a real number, and so the right-hand side is real, too. Thus, using the fact that the real part of a real number is the number itself and referring to the first of properties (4.2.3), we see that the right-hand side of equation (4.2.6) can be rewritten in the following way: ∫ ∫∫ − − − = = = b a b a b a i i i ))(Re()(Re)( dttwedttwedttwer 0 0 0 0 θ θ θ . (4.2.7) But ],[|,)(||)(||||)(|))(Re( 0 0 0 battwtwetwetwe i i i ≤ = ∈∀= θ− θ− θ− ; and so, according to equation (4.2.7), we have ∫ ≤ b a |)(| dttwr0 . The proof is completed. The fundamental theorem of calculus, involving antiderivatives (i.e., primitive functions), can be extended so as to apply to integrals of the type (4.2.2). Theorem 4.2.2. Suppose that the functions = + tivtutw )()()( and = + tiVtUtW )()()( are continuous on the interval ba ],[ and ′ = twtW )()( when ∈ bat ],[ , then ∫ =−= b a b a tWaWbWdttw )()()()( . Proof. Since ′ = twtW )()( , we have ′ = tutU )()( and ′ = tvtV )()( . Hence, from the fundamental theorem of calculus and Definition 4.2.1, we obtain that ∫ ∫ ∫ += b a b a b a )()()( dttvidttudttw b a b a += tVitU )()( = + − + aiVaUbiVbU )]()([)]()([ . Thus, we deduce that ∫ =−= b a b a tWaWbWdttw )()()()( . This completes the proof. Example 2. Since (See Sec. 4.1), we have and so it it )( ′ = iee −= )( ′ it it iee . 2 1 1 2 1 22 1 4/ 0 4/ 4 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ −+=+⎠ ⎞ ⎜ ⎝ ⎛ +−= +−=−= ∫ ii i i iieiedteit it i π π π
$4.3.PathsIntegrals of complex-valued functions of a complex variable are defined on curves in the complexplane,rather than on just intervals of the real line.Classes of curves that are adequatefor the studyofsuchintegralsareintroducedinthissectionDefinition 4.3.1. A set C of points z = (x, y) in the complex plane is said to be an arc ifthereexistcontinuousfunctions x and y oftherealparameter t onaninterval [a,b] suchthatC = ((x(0), y(0) : t e[a,b))(4.3.1)See Fig.4-1 below.y(x(t),y(t)c0xFig. 4-1This definition establishes a continuous mapping from the interval [a,b] into the xy, orz-plane.If the image points are ordered according to increasing (resp.decreasing)values of tthen we call the arc to be positively (resp. negatively) oriented, shortly, oriented. It is convenienttodescribethepoints of Cbymeans oftheequationz=z(t)(a≤t≤b),(4.3.2)where(4.3.3)z(t)= x(t)+iy(t)Definition 4.3.2. The oriented arc C given by (4.3.2) is called to be a simple arc, or aJordan arc, ifit does not cross itself, that is, for a ≤t, ≤t, ≤ b, we havez()=z(t)=t =t2,or (t,5)=(a,b)When the oriented arc C is simplesuchthatyz(b)=z(a), we say that C is asimple closed1+i2+i1curve, or a closed Jordan curve.The geometric nature of a particulararcoftensuggestsdifferentnotationfortheparametertcasein thein equation (4.3.2). This is, in fact, theexamplesbelow.x0i2Example 1. The polygonal line(Sec.1.10)defined by means of the functionFig. 4-2x+ix, when 0≤x≤](4.3.4)-when1≤x≤2x+iconsisting ofa line segmentfromthepoint O to the point 1+i followed by one from the point1+i to the point 2 +i (Fig. 4-1), is a simple arc.Example 2.The unit circlez=ei(0≤≤2元)(4.3.5)about the origin is a simple closed curve, oriented in the counterclochwise direction. So is thecircleZ= Zo +R-ei0(0≤0≤2元),(4.3.6)
§4.3. Paths Integrals of complex-valued functions of a complex variable are defined on curves in the complex plane, rather than on just intervals of the real line. Classes of curves that are adequate for the study of such integrals are introduced in this section. Definition 4.3.1. A set C of points = yxz ),( in the complex plane is said to be an arc if there exist continuous functions x and of the real parameter on an interval such that y t ba ],[ = ∈ battytxC ]},[:))(),({( . (4.3.1) See Fig. 4-1 below. x y • (x(t), y(t)) C O Fig. 4-1 This definition establishes a continuous mapping from the interval [a,b] into the xy , or - plane. If the image points are ordered according to increasing (resp. decreasing) values of , then we call the arc to be positively (resp. negatively) oriented, shortly, oriented. It is convenient to describe the points of by means of the equation z t C z = z(t)(a ≤ t ≤ b), (4.3.2) where z(t) = x(t) + iy(t). (4.3.3) Definition 4.3.2. The oriented arc given by (4.3.2) is called to be a simple arc, or a Jordan arc, if it does not cross itself; that is, for C a ≤ t ≤ t ≤ b 1 2 , we have ( ) ( ) , or { , } { , } 1 2 1 2 1 2 z t = z t ⇒ t = t t t = a b . When the oriented arc is simple such that , we say that is a simple closed curve, or a closed Jordan curve. C z(b) = z(a) C The geometric nature of a particular arc often suggests different notation for the parameter in equation (4.3.2). This is, in fact, the case in the examples below. t Example 1. The polygonal line (Sec.1.10) defined by means of the function ⎩ ⎨ ⎧ + ≤ ≤ + ≤ ≤ = , when 1 2, , when 0 1; x i x x ix x z (4.3.4) Fig. 4-2 consisting of a line segment from the point 0 to the point 1+ i followed by one from the point 1+ i to the point 2 + i (Fig. 4-1) , is a simple arc. Example 2. The unit circle (0 θ 2π ) θ = ≤ ≤ i z e (4.3.5) about the origin is a simple closed curve, oriented in the counterclockwise direction. So is the circle (0 2 ) = 0 + ⋅ ≤ θ ≤ π iθ z z R e , (4.3.6)
centered at the point =。 and with radius R (See Sec. 1.6, Fig. 4-3)山R20OxFig. 4-3The same set of points can make updifferent orientedarcs.Example3.Theoriented arcz=e-10(0≤0≤2元）(4.3.7)oriented in the clockvise direction, is not the same as the oriented arc described by equation(4.3.5):Thesetisthesame,butnowthecircleistraversedintheclockwisedirectionExample4.Theset ofpointsonthearc≥=e2i0(0≤0≤2元)(4.3.8)is the same as the set making up the arcs (4.3.5) and (4.3.7). The arc here differs, however, fromeachof those arcs sincethecircle is traversed wice in thecounterclockwisedirection.Suppose nowthatthecomponents x'(t)andy'()ofthederivative(4.3.9)='(t)= x'(t)+iy'(t)of thefunction(4.3.3),used to represent C,arecontinuouson the entire interval[a,b].Such anarc C is often called a differentiable arc,and thereal-valuedfunction1=()= /[x(0P +[y(0)is integrable over the interval [a,b]. Moreover, according to the definition of arc-length incalculus,thelengthof Cisthenumber=(t)|dt：(4.3.10)If equation (4.3.2) represents a differentiable arc and if z'(t) + 0 anywhere in the intervala<t<b,then theunitangentvector≥(0)T1=(0) 1is well defined for all t in that open interval, with angle of inclination argz'(t).Also, when Tturns, it does so continuously as the parameter t varies over the entire interval (a,b). Thisexpression for T is the one learned in calculus when z(t) is interpreted as a radius vector.Definition4.3.3.Anorientedarcz=z(t)(a≤t≤b)is called smooth,if thederivative z'(t)is continuous on theclosed interval [a,bland nonzeroon the open interval (a,b)Definition 4.3.4. A path, or piecewise smooth arc, is an oriented arc consisting of a finitenumberofsmootharcs joinedendtoendHence, if equation (4.3.2) represents a path, then z is continuous, whereas its derivativeZis piecewise continuous. The polygonal line (4.3.4) is, for example, a path.Definition 4.3.5. When only the initial and final values of z in a path C are the same
centered at the point and with radius z0 R (See Sec. 1.6, Fig. 4-3). The same set of points can make up different oriented arcs. Fig. 4-3 Example 3. The oriented arc π≤θ≤= )20( iθ− ez , (4.3.7) oriented in the clockwise direction, is not the same as the oriented arc described by equation (4.3.5) . The set is the same, but now the circle is traversed in the clockwise direction. Example 4. The set of points on the arc )20( 2 πθ θ ≤≤= i ez (4.3.8) is the same as the set making up the arcs (4.3.5) and (4.3.7). The arc here differs, however, from each of those arcs since the circle is traversed twice in the counterclockwise direction. Suppose now that the components ′ tx )( and ′ ty )( of the derivative ′ = ′ + ′ tyitxtz )()()( (4.3.9) of the function (4.3.3), used to represent , are continuous on the entire interval . Such an arc is often called a differentiable arc, and the real-valued function C ba ],[ C 2 2 ′ = ′ + ′ tytxtz )]([)]([|)(| is integrable over the interval α b],[ . Moreover, according to the definition of arc-length in calculus, the length of C is the number ∫ = ′ b a |)(| dttzL . (4.3.10) If equation (4.3.2) represents a differentiable arc and if ′ tz ≠ 0)( anywhere in the interval << bta , then the unit tangent vector |)(| )( tz tz ′ ′ Τ = is well defined for all t in that open interval, with angle of inclination ′ tz )(arg . Also, when turns, it does so continuously as the parameter varies over the entire interval . This expression for is the one learned in calculus when is interpreted as a radius vector. Τ t ba ),( Τ tz )( Definition 4.3.3. An oriented arc = ≤ ≤ btatzz ))(( is called smooth, if the derivative is continuous on the closed interval and nonzero on the open interval . ′ tz )( ba ],[ ba ),( Definition 4.3.4. A path, or piecewise smooth arc, is an oriented arc consisting of a finite number of smooth arcs joined end to end. Hence, if equation (4.3.2) represents a path, then z is continuous, whereas its derivative z′ is piecewise continuous. The polygonal line (4.3.4) is, for example, a path. Definition 4.3.5. When only the initial and final values of z in a path C are the same
thenwe saythatthepath Cisasimpleclosedpath.Examples of simple closed arcs are the circles (4.3.5) and (4.3.6), as well as the boundary ofa triangle or a rectangle taken in a specific direction.The length of a path or a simple closed pathis the sum of the lengths of the smooth arcsthat make upthe pathTheorem 4.3.1(Jordan).Any simple closed pathCcuts the complex plane into two distinctdomains, one of which is the inside of C that is bounded and denoted by ins(C),the other isthe outside of C that is unbounded and denoted by out(C).So,thecomplexplanecanberepresentedasC= ins(C)UCUout(C) (See Fig. 4-4),where Cdenotestheset(=(t):a≤t≤b)iftheequationof Cis z=z(t)(a≤t≤b)It will be convenient to accept this statement, as geometrically evident, the proof is not easyFig. 4-4
then we say that the path C is a simple closed path. Examples of simple closed arcs are the circles (4.3.5) and (4.3.6), as well as the boundary of a triangle or a rectangle taken in a specific direction. The length of a path or a simple closed path is the sum of the lengths of the smooth arcs that make up the path. Theorem 4.3.1(Jordan). Any simple closed path cuts the complex plane into two distinct domains, one of which is the inside of that is bounded and denoted by , the other is the outside of that is unbounded and denoted by . C C C)ins( C C)out( So, the complex plane can be represented as C = UU CCC )out()ins( (See Fig. 4-4), where C denotes the set ≤ ≤ btatz }:)({ if the equation of C is = ≤≤ btatzz ))(( . It will be convenient to accept this statement, as geometrically evident; the proof is not easy. Fig. 4-4
$4.4.PathIntegralsWe turn now to integrals of complex-valued functions f of the complex variable z. Such anintegral is defined in terms of the values f() along a given path C extending from a pointz=z, to a point z==, in the complex plane.It is, therefore, a line integral, and its valuedepends,ingeneral,onthepath Caswell asonthefunctionf,ItiswrittenJcf(2)dz or [f(z)dz,the latter notation often being used when the value of the integral is independent of the choice ofthe path taken between two fixed end points. While the integral may be defined directly as thelimit of a sum, we choose to define it in terms of a definite integral of the type introduced inSec.4.2Definition 4.4.1. Suppose that the equationz=z(t) (a≤t≤b)(4.4.1)represents a path C, extending from a point zi = z(a) to a point z2 = z(b). Let the functionf be piecewise continuous on C.We definethe path integral of f along C asfollows:J(z)dz=,[=(0)2(0)dt .(4.4.2)Note that, since C is a path, 2'(t) is also piecewise continuous on the interval [a,b];and so the existence of integral (4.4.2)is ensuredIt follows immediately from definition (4.4.2) and properties of integrals of complex-valuedfunctions w(t) mentioned in Sec.4.2 that,=0f(=)dz ==J.f(=)dz,(4.4.3)forany complex contents zo,andJ,[f(z) + g(=)]dz = J. J(z)dz +J,g(z)dz ,(4.4.4)provided the integrals on the right-hand sides exist.Associated with the path C used in integral (4.4.2) is the path -C, consisting of the sameset of points but with the order reversed so that the new path extends from the points z2 to thepoint z, (Fig. 4-5).The path -C has parametric representationz=z(-t) (-b≤t≤-a);and so, in view of Exercise l(a), Sec. 4.2,yC7oa可xFig.4-5c(2)dz= J [=(-1)]=(-1)dt =-[, F[2(-1)]2(-1)dt ,when =(-t) denotes the derivative of =(t) with respect to t, evaluated at -t, Making the
§4.4. Path Integrals We turn now to integrals of complex-valued functions f of the complex variable z . Such an integral is defined in terms of the values along a given path extending from a point to a point in the complex plane. It is, therefore, a line integral; and its value depends, in general, on the path as well as on the function . It is written zf )( C 1 = zz 2 = zz C f ∫C )( dzzf or , ∫ 2 1 )( z z dzzf the latter notation often being used when the value of the integral is independent of the choice of the path taken between two fixed end points. While the integral may be defined directly as the limit of a sum, we choose to define it in terms of a definite integral of the type introduced in Sec.4.2. Definition 4.4.1. Suppose that the equation = ≤ ≤ btatzz )()( (4.4.1) represents a path C , extending from a point )( 1 = azz to a point )( 2 = bzz . Let the function f be piecewise continuous on C . We define the path integral of f along C as follows: ∫ ∫ = ′ C b a )()]([)( dttztzfdzzf . (4.4.2) Note that, since C is a path, ′ tz )( is also piecewise continuous on the interval ; and so the existence of integral (4.4.2) is ensured. ba ],[ It follows immediately from definition (4.4.2) and properties of integrals of complex-valued functions tw )( mentioned in Sec. 4.2 that ∫ ∫ = C C )( )( dzzfzdzzfz0 0 , (4.4.3) for any complex contents , and 0 z ∫ ∫ ∫ =+ + C C C )()()]()([ dzzgdzzfdzzgzf , (4.4.4) provided the integrals on the right-hand sides exist. Associated with the path C used in integral (4.4.2) is the path − C , consisting of the same set of points but with the order reversed so that the new path extends from the points to the point (Fig. 4-5). 2 z 1z The path − C has parametric representation = − ()( − ≤ ≤ −atbtzz ) ; and so, in view of Exercise a)(1 , Sec. 4.2, Fig. 4-5 ∫ ∫ − ∫ − − − − −−=−−= ′ − C a b a b dttztzfdttz dt d tzfdzzf )()]([)( )()]([ , when ′ −tz )( denotes the derivative of tz )( with respect to t , evaluated at − t . Making the
substitution T =-t in this last integral and referring to Exercise l(a), Sec. 4.3, we obtain theexpressionJ- f(2)d =-f'[=(t)2(t)dt,which isthesameasJ-J(=)dz=-J.f(=)dz(4.4.5)Consider now a path Cwith representation (4.4.1) that consists of a path C, from z, to22 followed by a path C, from z2 to 23,the initial point of C, being the final point ofC,(Fig. 4-6). There is a value c of t with a<c<b such that z(c)= z2.Consequently, C, is represented by z=z(t)(a≤t≤c)and C, is represented byz=z(t) (c≤t≤b).yC2福23zoxFig. 4-6Also, by a rule for integrals of functions w(t) that was noted in Sec. 4.2z(0)](0)dt= [z()=()dt + J"[z()=()dtThus,[.f(2)= dz= J f(2)dz + Jc (2)dz .(4.4.6)Sometimes the path C is called the sum of its legs C, and C, and is denoted byC, +C,The sum of two paths C, and -C2 is well defined when C, and C have thesame final points, and it is written C, C, .Definite integrals in calculus can be interpreted as areas, and they have other interpretations aswell.,Except in special case, no corresponding helpful interpretation,geometric or physical,isavailableforintegralsinthecomplexplane
substitution τ −= t in this last integral and referring to Exercise , Sec. 4.3, we obtain the expression a)(1 ∫ ∫ − −= ′ C b a )()([)( dzzfdzzf τττ , which is the same as ∫ ∫ − −= C C )()( dzzfdzzf . (4.4.5) Consider now a path with representation (4.4.1) that consists of a path from to followed by a path from to , the initial point of being the final point of (Fig. 4-6). There is a value of t with C C1 1 z 2 z C2 2 z 3 z C2 C1 c < < bca such that 2 )( = zcz . Consequently, is represented by C1 = ≤ ≤ ctatzz )()( and is represented by . C2 ≤≤= btctzz )()( Also, by a rule for integrals of functions tw )( that was noted in Sec. 4.2, Fig. 4-6 ∫∫∫ ′ = ′ + ′ b a c a b c )()]([)()]([)()]([ dttztzfdttztzfdttztzf . Thus, ∫ ∫∫ == + C CC dzzfdzzfdzzf 1 2 )( )()( . (4.4.6) Sometimes the path is called the sum of its legs and and is denoted by . The sum of two paths and C C1 C2 + CC 21 C1 − C2 is well defined when and have the same final points, and it is written C1 C2 − CC 21 . Definite integrals in calculus can be interpreted as areas, and they have other interpretations as well. Except in special case, no corresponding helpful interpretation, geometric or physical, is available for integrals in the complex plane