Mathematica Solutions to the ISSAC '97 Systems
Challenge
Wolfram Research, Inc.
Problem 4
What is the coefficient of in the expansion of the
polynomial to 13 significant digits?
Result
Method 1: Calculate the residue numerically.
The coefficient of in the polynomial
can be obtained as the residue ofat .
(This old and widely used technique is used to calculate large
coefficients in enumeration theory [3].)
The default precision goal of NIntegrate is only 6, so we cannot be
sure of the last digits.
We use bignum arithmetic to make sure to get 13 correct digits. We
also use the trapezoidal rule because the integrand is periodic.
Method 2: Calculate the residue symbolically.
Method 3: Calculate the series up to order 3000.
Method 4: Expand the polynomial.
Since the polynomial is a square, we find the appropriate coefficient
in the square of the expansion of the square root.
