Sinc integrals and tiny numbers

@article{Basel2015SincIA,
  title={Sinc integrals and tiny numbers},
  author={Uwe Basel and Robert Baillie},
  journal={Elemente Der Mathematik},
  year={2015},
  volume={71},
  pages={7-20}
}
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 $\pi(1-t)/2$, 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… 
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References

SHOWING 1-10 OF 13 REFERENCES
Some Remarkable Properties of Sinc and Related Integrals
AbstractUsing Fourier transform techniques, we establish inequalities for integrals of the form $$\smallint _0^x \prod\limits_{k = 0}^n {\frac{{\sin (akx)_{} }}{{akx}}dx.} $$ We then give quite…
A remark concerning sinc integrals
We give a simple proof of Hanspeter Schmid's result that $K_n:=2\int_0^\infty\cos t\,\prod_{k=0}^n\mathrm{sinc}\left(\frac{t}{2k+1}\right)\mathrm{d}t=\pi/2$ if $n\in\{0,1,\ldots,55\}$, and…
On a sum involving powers of reciprocals of an arithmetical progression.
USTHB/ Faculte de Mathematiques, Algere-mail: hbelbachir@usthb.dz, hacenebelbachir@gmail.com, akhelladi@usthb.dz,akhelladi@wissal.dzSubmitted 25 September 2007; Accepted 9 December 2007AbstractOur…
A Remark on Stirling’s Formula
We shall prove Stirling’s formula by showing that for n=1, 2,… $$n! = \sqrt {2\pi } {n^{n + 1/2}}{e^{ - n}} \cdot {e^{{r_n}}}$$ (1) where rn satisfies the double inequality …
Surprising Sinc Sums and Integrals
We show that a variety of trigonometric sums have unexpected closed forms by relating them to cognate integrals.
Theorie und Anwendung der Unendlichen Reihen
The Integral 0 ∞ sin x x dx
Continuous Analogues of Series
Two curious integrals and a graphic proof
Advanced problem 6241
  • American Mathematical Monthly, vol. 87 (June-July,
  • 1980
...
...