Homework 6 Solutions - MATH 185 HOMEWORK 6 SOLUTIONS TO SELECTED PROBLEMS Exercise 1 For n N derive the formula 2 1 2 1 3 5(2n 1 2 4 6(2n cos2n(t dt = 0

# Homework 6 Solutions - MATH 185 HOMEWORK 6 SOLUTIONS TO...

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MATH 185, HOMEWORK 6: SOLUTIONS TO SELECTED PROBLEMSExercise 1.FornN, derive the formula12πZ2π0cos2n(t)dt=1·3·5· · ·(2n-1)2·4·6· · ·(2n)by integrating the function1zz+1z2naround the unit circle, parametrized by the curveγ(t) =eit, 0t2π.(Hint: Use the Binomial Theorem.)Solution.We prove the formula by computing an integral two different ways. Onthe one hand,Zγ1zz+1z2ndz=Z2π0e-it(eit+e-it)2ni eitdt= 22niZ2π0cos2nt dt.On the other hand,Zγ1zz+1z2ndz=2nXk=02nk¶ Zγz2(k-n)-1dz.Now we note thatZγz2(k-n)-1dz=(2πiifk=n0ifk6=n.HenceZγ1zz+1z2ndz=2nn2πi.Equating the two expressions, from computing the integral two different ways, we get12πZ2π0cos2nt dt= 2-2n2nn.From here it is fun and easy to get the final answer.I suggest writing out a fewspecial cases. But for practice, here is a proof by induction:1
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