- 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