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Homework Statement
Evaluate:
[tex] \int _{c} \dfrac{1- Log z}{z^{2}} dz [/tex]
where C is the curve:
[tex] C : z(t) = 2 + e^{it} ; - \pi / 2 \leq t \leq \pi / 2 [/tex]
Homework Equations
I know the independance of path in a domain where f(z) is analytical, but I tried the standard parametrization just to beging with someting.
The Attempt at a Solution
[tex] z^{2} = 4 + 4e^{it} + e^{2it} [/tex]
[tex] Log(2 + e^{it} ) = \frac{1}{2} \ln (5 + \cos t) +it [/tex]
[tex] dz = ie^{it} dt [/tex]
[tex] i \int _{- \pi / 2} ^{\pi / 2} \dfrac{1 -\frac{1}{2} \ln (5 + \cos t) -it }{4 + 4e^{it} + e^{2it}}e^{it} dt [/tex]
lol iam lost