Simplify square of sinc functions

Question

I need to simplify if possible the following:

$$\left(i^n\cdot \operatorname{sinc}\big(\pi(x-\tfrac{n}{2})\big)+(-i)^n\cdot \operatorname{sinc}\big(\pi(x+\tfrac{n}{2})\big)\right)^2$$

with $n \in \mathbf{N}$ and $\operatorname{sinc}(x)=\sin(x)/x$.

Thanks

Answer 1

Note that $\;\pi\left(x-\frac n2\right)\;$ and $\;\pi\left(x+\frac n2\right)\;$ difference will be $n\pi$ so that they will be equal for $n$ even and of different sign for $n$ odd.

This allows to get :

• For $n$ even : $$\left(\frac{2\,x\sin(\pi x)}{\pi\left(x^2-(n/2)^2\right)}\right)^2$$
• while for $n$ odd : $$-\left(\frac{2\,x\cos(\pi x)}{\pi\left(x^2-(n/2)^2\right)}\right)^2$$