Discussion on a beautiful function, $\sin(nx) / \sin x$

Question

In this post, I may need help with your broad knowledge on a function

$$ \frac{\sin{Nx}}{\sin{x}} $$

where $N$ is an positive integer. The questions are as follows.

1. What is the name of this function

2. Where does it from in literature historically and mathematically

3. Any interesting properties we can use for math, physics, and engineering

Thanks a lot, and hope I can create a great discussion here on this function.

Answer 1

There is no reason to name this function, it's fairly easy to express with trigonometric functions, as you demonstrated.

It is fairly recognizable to be the difraction pattern of waves hitting N equally spaced slits - finite diffraction grating formula. In wave optics, you derive it as a sum of a finite geometric series:

$$\sum_{k=-N/2}^{N/2} e^{ikx}$$ which works for both odd and even $N$.

This is also a mathematical connection - it's a sum of equally spaced complex numbers on the unit circle, and can be viewed as a Fourier series.