Let's say I want to evaluate the following integral using complex methods -
∫2π011+cosθdθ
So I assume this is not very hard to be solved using real analysis methods, but let's transform the problem for the real plane to the complex plane, and instead calculate -
∫2π011+cosθdθ⟹[z=eiθ,|z|=1]⟹∫|z|=111+z+1z2dziz
So now after few algebric fixed this is very easily solvable using the residue theorem.
My question is why can I just decide that I want to change the integration bounds for [0,2π] to |z|=1. If I wanted to change the integrating variable to z=eiθ aren't the integration bounds suppose to transform to [1,1] (because ei2πk=1)? I'm just having hard time figuring out why is this mathematicaly a right transform.
Thanks in advance!
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