Sine 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 $\zeta_\theta-\overline{\zeta_\theta}=2i\sin(\theta)$ .

$\sum_{n=a}^{b}{\mathrm e}^{2kn}=\sum_{n=a}^{b}\left({\mathrm e}^{2k}\right)^n=\dots$

we get

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

Integrate[Sin[a*x]*Sin[b*x]/x^2,{x,0,Infinity}]

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