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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.