Wednesday, 12 April 2017

complex analysis - Prove the taylor series of cos(2z)



First i turned cos(2z)=e2iz+e2iz2, then using the taylor series of ezI calculated the taylor series of both arguments.
e2iz2=n=0(i)n2n1znn!
e2iz2=n=0(i)n2n1znn!
From this stage i dont know how to continue.
Is my way to the solution is right?


Answer




Hint:
(i)n+(i)n=einπ2+einπ2=2cosnπ2.


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