Why are these \(\displaystyle\sum{\cos{}}\) and \(\displaystyle{\csc{}}\)

Adrien Wong

Adrien Wong

Answered question

2022-03-14

Why are these cos and csc equivalent?
Mathematica 'simplifies' this formula
k=1Rcos2kπxR
to this
12(cscπxRsin(2R+1)πxR1)

Answer & Explanation

i1a6ldihq

i1a6ldihq

Beginner2022-03-15Added 4 answers

Assuming you're a bit familiar with complex numbers:
eix=cos(x)+isin(x)
Let a=2πR then you want to find the sum
k=1Rcos(akx)
Notice that it is way easier to first compute (which we later can relate to
k=1Reakix=k=1R(eaix)k
which is a geometric series. This evaluates to
k=0R(eaix)k=eaix(R+1)1eaix1
But since your series starts at k=1 we have to subtract the first term, so that
k=1R(eaix)k=eaix(R+1)1eaix11
Now substitute eq. (1) back into (5) to get
k=1Rcos(akx)+ik=1Rsin(akx)=

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