Lecture17

# Lecture17 - By Green’s theorem integraldisplay C u dx −...

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Unformatted text preview: By Green’s theorem integraldisplay C ( u dx − v dy ) = integraldisplay U ( − v x − u y ) dx dy, (721) integraldisplay C ( v dx + u dy ) = integraldisplay U ( u x − v y ) dx dy. (722) From the Cauchy-Riemann equations (Thm. 2) u x = v y , u y = − v x , we conclude that (721), (722) vanish. square Examples 1. If f is an entire function, the integral over any closed curve C vanishes: contintegraldisplay C e z dz = 0 , contintegraldisplay C cos z dz = 0 , contintegraldisplay C z n dz = 0 , n ∈ N . (723) 2. Unit circle C = { z ∈ C || z | = 1 } . Functions which are not entire but holomorphic on and inside of C : tan z is holomorphic except at (2 k +1) π/ 2, k ∈ Z , ( z 2 +4) − 1 is holomorphic except at z = ± 2 i . These points lie outside of C , and therefore contintegraldisplay C tan z dz = 0 , contintegraldisplay C 1 z 2 + 4 dz = 0 . (724) 3. γ ( t ) = e it , f ( z ) = z is not holomorphic, and therefore Thm. 8 is not applicable. We compute, with Thm. 7: contintegraldisplay C z dz = 2 π integraldisplay e − it ie it dt = 2 πi negationslash = 0 . (725) 4. Domains which are not simply connected: f ( z ) := z − 1 is holomorphic in C \{ } , but this domain is not simply connected. We have (Example 1 after Thm. 7, Section 14.1), for γ ( t ) = e it : contintegraldisplay C 1 z dz = 2 πi negationslash = 0 . (726) 121 Independence of Path Theorem 9 If the complex function f is holomorphic in a simply connected domain U ⊆ C , then the integral of f is independent of path in U , i. e. its value depends only on the two endpoints of C . Proof: Let z 1 , z 2 be any points in U . Consider two curves C 1 , C 2 from z 1 to z 2 which do not intersect each other (so z 1 , z 2 are the only common points of C 1 and C 2 ). The path from z 1 to z 2 on C 1 and back from z 2 to z 1 on − C 2 is closed, and from Thm. 8 we know that integraldisplay C 1 f dz + integraldisplay − C 2 f dz = 0 ⇒ integraldisplay C 1 f dz = − integraldisplay − C 2 f dz = integraldisplay C 2 f dz. (727) For paths with a finite number of common points, we may integrate over loops between these points. square Principle of Deformation of Path From the path independence (Thm. 9), we conclude that the integral of f over any continuous transformation of a curve C is the same as the integral of f over C , as long as the deforming path contains only points at which f is holomorphic....
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Lecture17 - By Green’s theorem integraldisplay C u dx −...

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