(1/prod(x-cos(2*%pi*j/n+f),j,1,n) = 2^(n-1)/(cos(n*acos(x))-cos(f*n))) = 2^(n-1)*(sum(sin(2*%pi*k/n+f)/(x-cos(2*%pi*k/n+f)),k,1,n))/(n*sin(f*n)) n - 1 1 2 --------------------------- = ---------------- n T (x) - cos(f n) /===\ n | | 2 j pi | | (x - cos(------ + f)) | | n j = 1 n 2 k pi ==== sin(------ + f) n - 1 \ n 2 > ------------------- / 2 k pi ==== x - cos(------ + f) k = 1 n = -------------------------------- n sin(f n) where T (x) := cos(n acos(x)), the nth Tchebychev polynomial. So n 1/\t[n](x) = sum(sin(%pi*(4*k+1)/(2*n))/(x-cos(%pi*(4*k+1)/(2*n))),k,1,n)/n n (4 k + 1) pi ==== sin(------------) \ 2 n > --------------------- / (4 k + 1) pi ==== x - cos(------------) 1 k = 1 2 n ----- = ---------------------------, n>0. T (x) n n Dividing the first equation into itself with x=0, then x <- 1/x, then logderiv wrt x: sum(1/(x-sec(2*%pi*j/n+f)),j,1,n) = n/x-n*chebyshev_u[n-1](1/x)/((chebyshev_t[n](1/x)-cos(f*n))*x^2) n 1 ==== n U (-) \ 1 n n - 1 x > ------------------- = - - --------------------- / 2 j pi x 1 2 ==== x - sec(------ + f) (T (-) - cos(f n)) x j = 1 n n x where d 2 -- (T (x)) sqrt(1 - T (x)) sin(n acos(x)) dx n n U (x) = -------------- = ---------- = ---------------. n - 1 2 n 2 sqrt(1 - x ) sqrt(1 - x ) -d/dx squares the summand with considerable inelegance on the rhs: sum(1/(x-sec(2*%pi*j/n+f))^2,j,1,n) = n*(('chebyshev_u[n-1](1/x)*(2*x^2-1)/x+n*sin(f*n)^2/(cos(f*n)-'chebyshev_t[n](1/x))+n*cos(f*n))/((cos(f*n)-'chebyshev_t[n](1/x))*(x^2-1))+1)/x^2 n ==== \ 1 > ---------------------- / 2 j pi 2 ==== (x - sec(------ + f)) j = 1 n 1 2 U (-) (2 x - 1) 2 n - 1 x n sin (f n) -------------------- + ---------------- + n cos(f n) x 1 cos(f n) - T (-) n x n (---------------------------------------------------- + 1) 1 2 (cos(f n) - T (-)) (x - 1) n x = ------------------------------------------------------------. 2 x --rwg PAINT HORSES SENATORSHIP SPHERATIONS