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Sine function
Function
| definiendum | $\mathrm{\sin}: \mathbb C\to\mathbb C$ |
| definiendum | $\sin(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)!}z^{2n+1} $ |
$\theta\in\mathbb R$
| $\sin(\theta) = \frac{1}{2i}(\mathrm e^{i\theta}-\mathrm e^{-i\theta}) $ |
|---|
i.e. if $\zeta:=\mathrm e^{i\theta}$, then $\zeta_\theta-\overline{\zeta_\theta}=2i\sin(\theta)$.
Theorem
From $\sum_{n=a}^{b}{\mathrm e}^{2kn}=\sum_{n=a}^{b}\left({\mathrm e}^{2k}\right)^n=\dots$ we get
$\sum_{n=a}^{b}\sin(2kn)=\dfrac{\sin (k (a-b-1)) \sin (k (a+b))} {\sin(k)}$
Context
Related
