w = cos(40)+i*sin(40), 所以有，w^9 = 1; 而且，|w^k| = 1, k = 1,2,....

现在考察多项式 f(w) = w+2w^2+...+9w^9

= (w+w^2+...+w^9) + (w^2+...+w^9)+...+ (w^8+w^9) + w^9

= w*(1-w^9)/(1-x) + w^2*(1-w^8)/(1-x) + ... + w^8*(1-w^2)/(1-w) + w^9

= [(w-w^10) + (w^2-w^10) + ... + (w^8 - w^10) ]/(1-w) +w^9

= [(w+w^2+...+w^8)-8*w]/(1-w) + 1 (因为w^9 = 1, 所以w^10 = w)

= [(w+w^2+...+w^8) - 8w + (1-w)]/(1-w)

= [(1+w+w^2+...+w^8) - 9w]/(1-w)

因为1+w+...+w^8 = (1-w^9)/(1-w) = 0 (记住，w^9 = 1)

所以 f(w) = - 9w/(1-w), 而1/|f(w)| = |1-w|/|9w|

这里|9w| = 9*|w| = 9; 这是因为|w| = 1

而|1-w| = |1-cos40-i*sin40| = ((1-cos(40))^2 + sin(20)^2)^0.5

= [(1-2*cos(40)+cos(40)^2)+sin(40)^2]^0.5

= [2(1-cos(40))]^0.5

而1-cos(40) = 2*sin(20)^2

所以|1-w| = 2*sin(20)

1/|f(w)| = 2*sin(20)/9

花了好长时间，终于搞出个大概了。