Saw a tweet today saying : "Given a regular n-gon inscribed in a unit circle, the product of the chords from one vertex of the polygon to all the others is equal to n." Which is pretty beautiful... so naturally I tried to work it out in Mathematica.. As a formal product its:
Product[Abs[1 - E^((2 Pi I theta)/n)], {theta, 1, n - 1}]
And Mathematica didn't solve it! (1)
There's a lot of stuff that could be causing that. Tried being very explicit about assumptions. And I was able to work it out using FullSimplify of course.
Assuming[{theta \[Element] Reals, n \[Element] Reals},
FullSimplify@ComplexExpand@Abs[1 - E^((2 Pi I theta)/n)]]
(3)
Did some investigating by converting it to a Sum with Logs.. and the real culprit is that this doesn't eval:
Sum[Sqrt[Sin[(Pi theta)/n]^2], {theta, 1, -1 + n}]
because Sum never feels comfortable making the jump from Sqrt[x^2] to Abs[x]. Branch Cuts are incredibly difficult for a computer algebra system to efficiently reason about. And it's why you have to work with them interactively.
(4).
Searke