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Next: Limits of the Search Up: Introduction Previous: Integer Relations

An Example

As an introductory example, let us consider the definite integral

 \begin{displaymath}
V=\int _{0}^{\infty }\!{\frac {\sqrt {x}\left (\ln x\right )^{5}}{
\left (1-x\right )^{5}}}{dx}.
\end{displaymath} (2)

Some computer algebra systems are able to evaluate V in a closed form while others are not. Most of these systems will nevertheless yield a numerical approximation to V to a few dozens of decimal digits very quickly; this value is

\begin{displaymath}V\approx -16.6994737192290704961872434007\cdots
\end{displaymath}

to the precision of 30 decimal digits.

Suppose that we knew in advance that the closed form evaluation of V was a rational linear combination of even powers of $\pi$. If moreover we knew that K is the maximal power of $\pi$that possibly can enter the expression for V, we could consider the integer relation problem (1) with a1=V, $a_2,a_3,\ldots,a_n=1,\pi^2,\ldots,\pi^{2(n-2)}$, where the value of nis determined by
2(n-2)=K.

In this ``toy'' example we can use any available implementation of the integer relation algorithms to obtain a solution. For example, using an LLL-based algorithm implemented in Maple, an initial trial for K=10(thus n=7) with all numerics performed at the precision of 30 decimal places gives within two seconds an answer

(c1,c2,c3,c4,c5,c6,c7)=(24, 0, 120, 140, -15, 0, 0)

or in other words

 \begin{displaymath}
V={5\over{24}}\pi^2(3\pi^4-28\pi^2-24).
\end{displaymath} (3)

We must have some reasonable idea about the form of the result sought if we hope to keep an integer relation search within reasonable limits (Section 1.3). In the case of the integral V, its closed form follows from the theory presented in [31], which contains a general symbolic algorithm for a family of definite integrals of the type (2). Indeed the example (4.8) of [31] is exactly our equation (3).

Similarly, multiplicative relations can be sought by taking logarithms and including logarithms of small primes, from which an eventual rational coefficient can be composed. Such ideas are discussed in detail by Bailey and Plouffe in [5].


next up previous
Next: Limits of the Search Up: Introduction Previous: Integer Relations
Agnes Szanto
2000-05-10