===== Sine function =====
==== Function ====
| @#FFBB00: definiendum | @#FFBB00: $\mathrm{\sin}: \mathbb C\to\mathbb C$ |
| @#FFBB00: definiendum | @#FFBB00: $\sin(z) := \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)!}z^{2n+1} $ |

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$\theta\in\mathbb R$

^ $\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

$\sum_{n=a}^{b}\sin(2kn)=\dfrac{\sin (k (a-b-1)) \sin (k (a+b))} {\sin(k)}$

  * The following is kinda odd:
<code>
Integrate[Sin[a*x]*Sin[b*x]/x^2,{x,0,Infinity}]
</code>

<code>
Integrate[Sin[k*x]*Sin[(k+q)*x]/x^2,{x,0,Infinity}]
</code>

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=== Context ===
[[Infinite sum of complex numbers]], 
[[Factorial function]]
=== Related ===
[[Exponential function]]