sin(nx)sin(x)=einxeinxeixeix=einxeixe2inx1e2ix1sin(θ)=eiθeiθ2i=einxeixk=0n1(e2ix)k=eix(n1)k=0n1e2ixk=k=0n1eix(n12k)k=0n1xk=xn1x1=eix(n1)+eix(n3)+...+eix(n3)+eix(n1)n term={eix(n1)+eix(n3)++e4ix+e2ix+e0k=n12+e2ix+e4ix+eix(n3)+eix(n1)n is oddeix(n1)+eix(n3)++e3ix+eix+eix+e3ix+eix(n3)+eix(n1)n is even={1+k=1n122e2kix+e2kix2n is oddk=1n22e(2k1)ix+e(2k1)ix2n is even={1+2k=1n12cos(2kx)n is odd2k=1n2cos((2k1)x)n is even\begin{aligned} \frac{\sin(nx)}{\sin(x)} &= \frac{e^{inx}-e^{-inx}}{e^{ix}-e^{-ix}} = \frac{e^{inx}}{e^{ix}} \cdot \frac{e^{-2inx}-1}{e^{-2ix}-1} & {\color{#D61F06} \sin(\theta) = \frac{e^{i\theta}-e^{-i\theta}}{2i} }\\ &= \frac{e^{inx}}{e^{ix}} \cdot \sum_{k=0}^{n-1} \left (e^{-2ix} \right )^k = e^{ix(n-1)} \cdot \sum_{k=0}^{n-1} e^{-2ixk}=\sum_{k=0}^{n-1} e^{ix(n-1-2k)} & {\color{#D61F06} \sum_{k=0}^{n-1} x^k = \frac{x^n - 1}{x-1}} \\ &= \underbrace{e^{ix(n-1)}+e^{ix(n-3)} +... {\color{#3D99F6} + e^{-ix(n-3)} + e^{-ix(n-1)}}}_{n \ term} \\ &= \left\{\begin{matrix} e^{ix(n-1)}+e^{ix(n-3)}+ \dots+e^{4ix}+e^{2ix}+\underbrace{e^0}_{k=\frac{n-1}{2}} + {\color{#3D99F6} e^{-2ix}+e^{-4ix} \dots + e^{-ix(n-3)} + e^{-ix(n-1)}} & n \ is \ odd\\ e^{ix(n-1)}+e^{ix(n-3)}+ \dots+e^{3ix}+e^{ix} + {\color{#3D99F6} e^{-ix}+e^{-3ix}\dots+ e^{-ix(n-3)} + e^{-ix(n-1)}} & n \ is \ even \end{matrix}\right.\\ &= \left\{\begin{matrix} 1+\sum_{k=1}^\frac{n-1}{2} {\color{#D61F06} 2} \cdot \frac{e^{2k ix}+e^{-2k ix}}{\color{#D61F06} 2} & n \ is \ odd\\ \sum_{k=1}^\frac{n}{2} {\color{#D61F06} 2} \cdot \frac{e^{(2k-1) ix}+e^{-(2k-1) ix}}{\color{#D61F06} 2} & n \ is \ even \end{matrix}\right.\\ &= \left\{\begin{matrix} 1+2 \sum_{k=1}^\frac{n-1}{2} \cos(2kx) & n \ is \ odd\\ 2 \sum_{k=1}^\frac{n}{2} \cos\left ( (2k-1)x \right ) & n \ is \ even \end{matrix}\right. \end{aligned}

Note by Hassan Abdulla
1 week, 3 days ago

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Relevant: Dirichlet kernel.

Pi Han Goh - 1 week, 3 days ago

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Can we generalise this to non integral values of nn ? What if we wanted to find 0πsin(xy)sinxdx\int_{0}^{\pi}\frac{\sin\left(xy\right)}{\sin x}dx when yy varies over the reals greater than 22 ? @Pi Han Goh @Hassan Abdulla

Aaghaz Mahajan - 1 week, 2 days ago

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If yy is not an integer, then the integrand (might) diverge at x=πx=\pi.

Pi Han Goh - 1 week, 2 days ago

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