Company
Mathematica Solutions to the ISSAC '97 Systems Challenge

Wolfram Research, Inc.


Problem 4

What is the coefficient of [Graphics:ISSACChallengegr50.gif] in the expansion of the polynomial [Graphics:ISSACChallengegr51.gif] to 13 significant digits?

Result
[Graphics:ISSACChallengegr52.gif]


Method 1: Calculate the residue numerically.

The coefficient of [Graphics:ISSACChallengegr53.gif]in the polynomial [Graphics:ISSACChallengegr54.gif] can be obtained as the residue of[Graphics:ISSACChallengegr55.gif]at [Graphics:ISSACChallengegr56.gif]. (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.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr57.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr58.gif]

We use bignum arithmetic to make sure to get 13 correct digits. We also use the trapezoidal rule because the integrand is periodic.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr59.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr60.gif]


Method 2: Calculate the residue symbolically.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr61.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr62.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr63.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr64.gif]


Method 3: Calculate the series up to order 3000.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr65.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr66.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr67.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr68.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr69.gif]


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.

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr70.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr71.gif]

[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr72.gif]
[Graphics:ISSACChallengegr7.gif][Graphics:ISSACChallengegr73.gif]