Sinc integrals and tiny numbers
This is an incremental mathematical result for specialists in oscillatory integrals and special functions.
The authors evaluate a sequence of highly-oscillatory integrals involving products of many sinc functions, obtaining values of the form π(1−t)/2 with extremely tiny t. For the tenth integral (product of 68,100,152 sinc functions), t is approximately 9.65×10^{-554,381,308}.
We apply a result of David and Jon Borwein to evaluate a sequence of highly-oscillatory integrals whose integrands are the products of a rapidly growing number of sinc functions. The value of each integral is given in the form $π(1-t)/2$, where the numbers $t$ quickly become very tiny. Using the Euler-Maclaurin summation formula, we calculate these numbers to high precision. For example, the integrand of the tenth integral in the sequence is the product of 68100152 sinc functions. The corresponding $t$ is approximately $9.6492736004286844634795531209398105309232 \cdot 10^{-554381308}$.