Wednesday, 23 November 2016

Evaluating a real integral using complex methods

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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