The definitiRTEExampleEvaluate Jedz1/2OC istheupperhalf of theunit circle joining 1to-1.C isthelowerhalf of theunitcirclejoining 1to-1Solution.The function is analytic on the regionyCl(x+iy:x≤0,y=0j.Therefore(1)C:z=z(0)=eie,o≤≤元,wechoosetheprinciple branch of z in C is elei, henceleiieiedeLiede0d(i)=2(i-1)1FCV&IT13/53ofSci&TechNovember5,2019
The definition of complex integral Example Evaluate R C √ 1 z dz, 1 C is the upper half of the unit circle joining 1 to −1. 2 C is the lower half of the unit circle joining 1 to −1. O x y −1 1 Solution. The function √ 1 z is analytic on the region C\{x + iy : x ≤ 0, y = 0}. Therefore (1) C : z = z(θ) = e iθ , 0 ≤ θ ≤ π, we choose the principle branch of √ z in C is e 1 2 θi , hence Z C 1 √ z dz = Z π 0 e − 1 2 θi ieiθ dθ = i Z π 0 e − 1 2 iθ dθ = i 2 i Z π 0 e 1 2 iθ d( 1 2 iθ) = 2(i − 1) Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 13 / 53
Thedefinitintegra(2) C: z = z(0) = e-ie, ≤ ≤ 元, the principle branch in C of z ise-ie,henceeleiie-iededz=-liede"e-liea(-)i0) = 2(-i-1)haUni.ofSci&TechFCV&ITNovember5,201914/53ansWs
The definition of complex integral (2) C : z = z(θ) = e −iθ , 0 ≤ θ ≤ π, the principle branch in C of √ z is e − 1 2 iθ , hence Z C 1 √ z dz = − Z π 0 e 1 2 θi ie−iθ dθ = −i Z π 0 e − 1 2 iθ dθ = i 2 i Z π 0 e − 1 2 iθ d(− 1 2 iθ) = 2(−i − 1) Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 14 / 53
The definition of complex integralg3.2 Cauchy integral theoremgsha Uni..of Sci &Tech)FCV&ITNovember.5,201915/53angWan
The definition of complex integral §3.2 Cauchy integral theorem Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 15 / 53
auchvintraltheoremCauchyintegral theoremTheorem(CauchyTheorem）Supposethat f(z)is analytic,withf'continuous on and inside the simpleclosed curve C.Thenf f(2)dz = 0Proof.Settingf=u+iv,wehavef(z)dz=f(udx-vdy) +i [(udy+ vdx)thenapplyingGreentheoremapaQ[Pdx+ Qdy =dxdyaxayFCV&IT16/53shaUni.of Sci&Tech)November5.2019angWantChe
Cauchy integral theorem Cauchy integral theorem Theorem (Cauchy Theorem) Suppose that f(z) is analytic, with f 0 continuous on and inside the simple closed curve C. Then I C f(z)dz = 0 Proof. Setting f = u + iv, we have I C f(z)dz = I C (udx − vdy) + i Z C (udy + vdx) then applying Green theorem Z C Pdx + Qdy = Z Z D ∂Q ∂x − ∂P ∂y dxdy Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 16 / 53
Byapplying Green theoremto eachof theabove integral,weobtainaudodudud f(z)dz = /L (dxdy+dxdyaxaaxay/DNext we use the C-R equationsauaxayauauayaxit is easyto seethatz)dz=(Example(3.2.1)Let C be the unit squarw and let f(z) = sin(e). Thend sin(e")dz = 0CFCV&ITNovember.5,2019FangWang (ChangshaUni.of Sci&Tech)17/53
Cauchy integral theorem By applying Green theorem to each of the above integral, we obtain I C f(z)dz = ZZ D − ∂v ∂x − ∂u ∂y dxdy + ZZ D − ∂u ∂x − ∂v ∂y dxdy Next we use the C-R equations ∂u ∂x = ∂v ∂y ∂u ∂y = − ∂v ∂x it is easy to see that I C f(z)dz = 0 Example (3.2.1) Let C be the unit squarw and let f(z) = sin(e z 2 ). Then I C sin(e z 2 )dz = 0 Fang Wang (Changsha Uni. of Sci & Tech) FCV & IT November 5, 2019 17 / 53