I'm having trouble proving (\frac{\sin(\frac{n\theta}{2})}{\sin(\frac{\theta}{2})})^2=|\sum_{k=1}^{|n|}e^{ik\theta}|^2 where n\in\mathbb{Z} and \theta\in\mathbb{R}.

Taylor Haines

Taylor Haines

Answered question

2022-03-01

I'm having trouble proving
(sin(nθ2)sin(θ2))2=|k=1|n|eikθ|2
where nZ and θR.

Answer & Explanation

lucratifar1

lucratifar1

Beginner2022-03-02Added 5 answers

Using Euler's Formula , eix=cosx+isinx
So, eix=cos(x)+isin(x)=cosxisinx2isinx
=eixeix
If n>0,k=1|n|eikθ=k=1neikθ=eiθ(eθ1eiθ1)
=einθ2eiθ2(einθ2einθ2}{(eiθ2eiθ2)}
=(cos(n1)θ2+isin(n1)θ2)sinnθ2sinθ2
Taking modulus
k=1|n|eikθ
=(cos(n1)θ2+isin(n1)θ2)sinnθ2sinθ2
=|sinnθ2sinθ2|
Similiarly, for n<0

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