Thursday, 19 December 2013

Complex path Integral

Evaluate the following integral:



γ1ez1dz where γ:[0,1]C is a parameterization of the unit circle oriented counter clockwise.



Attempt at the solution:




Use the parameterization γ(t)=e2πit



γ1ez1dz=γez1ezdz=102πie2πitee2πit1ee2πitdt



let u(t)=e2πit so that du=2πie2πitdt



e2πi1eu1eudu



let v(t)=1eu so that dv=eudu




1e2πi1e11vdv=log(1e2πi)log(1e1)



I think there is something fishy going on here. Since the log function is not defined the same way as in the case of the real numbers. Can we treat the upper and lower limits in the integral in the same way as with real numbers? Are the change of variables techniques still applicable?

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