Chapter 3:Complex Integralssha Uni. of Sci & Tech)FCV&ITNovember 5.20193/53angWan
Chapter 3: Complex Integrals Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 3 / 53
g3.1 The concept of complex integralshaUni.ofSci&TechFCV&ITNovember5,20194/53angWan
§3.1 The concept of complex integral Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 4 / 53
Thedefinitionofcomplexintegraly252xoangWatshaUni.ofSci&Tech)FCV&ITNovember5.20195/53
The definition of complex integral O x y z0 z1 z2 z3 z4 z5 z6 Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 5 / 53
The definLet C be a piecewise smoth curve,zo is its beginning point and Z is itsend point. Let f(z) be defined in C.Take the partition pointsZo,Z1,*+,Zn=ZalongthecurveC,wehaveSn==1f(k)zkwhere △zk = Zk-Zk-1, Vk E zk-1zk, let8 = max[|zk - zk-1l,k = 1,2, .-. ,n] ,if the limit of this sumlimSn=limf(x)Azk5→080k=1exists,f if called integrable along C.The limit is called the integral offalong C, denoted fcf(z)dz.FCV&ITUni.of Sci &Tech)November5,20196/53
The definition of complex integral Let C be a piecewise smoth curve, z0 is its beginning point and Z is its end point. Let f(z) be defined in C. Take the partition points z0, z1, · · · , zn = Z along the curve C, we have Sn = ∑ n k=1 f(ζk)∆zk , where ∆zk = zk − zk−1 , ∀ζk ∈ _ zk−1zk , let δ = max{|zk − zk−1 |, k = 1, 2, · · · , n} ,if the limit of this sum lim δ→0 Sn = lim δ→0 n ∑ k=1 f(ζk)∆zk exists, f if called integrable along C. The limit is called the integral of f along C, denoted R C f(z)dz. Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 6 / 53
The definition of conplexintegralThepropertiesoftheintegralsLetf,gbeintegrablealongC,(1) Jcαf +βg=α Jcf +βJc8(2) Jef= Je,f + Je,f, c=CiuC2(3) Jcf = - Jc(4)For continuous function f,if|f(z)/<M,the length of C isL,then/, f(2)dz/≤ /,1f(2)]ds ≤ MLSincef(R)Az≤EIf(3K)/Azk/≤If(3K)[Askand-|f()Ask≤M"-Ask=ML.Take→0bothsidesoftheaboveinequality,wehaveJ f(2)dz ≤ J/,1(2)ds ≤ MLNovember 5, 2019of Sci &Tech)FCV&IT7/53
The definition of complex integral The properties of the integrals Let f , g be integrable along C, (1) R C α f + βg = α R C f + β R C g (2) R C f = R C1 f + R C2 f , C = C1 ∪ C2 (3) R C f = − R C− f (4) For continuous function f , if | f(z)| ≤ M, the length of C is L, then Z C f(z)dz ≤ Z C | f(z)|ds ≤ ML Since n ∑ k=1 f(ζk)∆zk ≤ n ∑ k=1 | f(ζk)||∆zk | ≤ n ∑ k=1 | f(ζk)|∆sk and ∑ n k=1 | f(ζk)|∆sk ≤ M ∑ n k=1 |∆sk = ML. Take δ → 0 both sides of the above inequality, we have Z C f(z)dz ≤ Z C | f(z)|ds ≤ ML Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 7 / 53