From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Complex Trig Sum
Date: 17 Nov 1998 18:34:06 GMT
Patrick Jemmer wrote:
>Hello All!
>
>Arising from the theory of intermolecular interactions
>I have the following sum to evaluate:
>
>S(n)= Sum_{i=1}^{n} (Sin[Pi*(i-1)/n])^(-3)
^^^
You really want i=0 here, according to your tables.
>Since I (or Mathematica!) can calculate the Sin terms
>for n=2->6 (and 10), I can get the Sum for these values:
>
> S(n)
>n=2 1
>n=3 16*Sqrt[3]/9
>n=4 1+4*Sqrt[2]
>n=5 32*Sqrt[phi]/5^(3/4) phi=(1+Sqrt[5])/2
>n=6 17+16*Sqrt[3]/9
>
>Is it possible to find closed-form solutions _in general_
>for S(n); if so, what is a general expression for S(n) ?
Yes and no. Writing sin( pi (j/n) ) as (zeta^j-zeta^(2n-j))/2i where
zeta=exp(pi i/n) is a primitive 2n-th root of unity, it is possible to
rewrite your S(n) as a rational function in zeta. Multiplying
numerator N and denominator D by norm(D)/D, we can actually write
S(n)/i as a polynomial in zeta, indeed a polynomial with rational
coefficients, of degree at most phi(2n), the degree of the minimal
polynomial of zeta. As closed-forms go, I'd say that was pretty good.
But this is in general going to be an element "deep" into this
cyclotomic number field, that is, it's unlikely to be rational, or to
be of the form a+b sqrt(c) with a,b,c rational, etc.
For example, when n=7, I compute S(7) to be (take your pick):
32 sin(Pi/7) + (32/7) sin(2 Pi/7) + (96/7) sin(3 Pi/7)
(64/7)*sin(pi/7)*(3 + c - 14 c^2 - 8 c^3 + 32 c^4), c=cos(pi/7)
-(16 i)/7 * Z * (Z+1) * (7Z^4-6Z^3+9Z^2-6Z+7), Z=exp(pi i/7)
which is in the maximal real subfield of Q(zeta_14,i): it cannot be expressed
using only square roots. Indeed, it is a root of its minimal polynomial,
6 4 2
343 X - 740096 X + 515178496 X - 115578241024
whose roots can be found by extracting a square root of the roots of an
irreducible cubic. You can express the roots of the cubic using square
and _cube_ roots, but you will have to invoke cube roots of complex
numbers since the cubic has three real roots ("casus irreducibilis").
The preferred method of presenting the roots of the cubic involves
trigonometric functions...
dave