Solution to Monthly Problem 11277 Page: 1 of 3
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Solution to Monthly Problem 11277
David H. Bailey* Jonathan M. Borweint
March 23, 2007
The Monthly problem #11277  asks to find
j /2 j/ 2 log(1 + sin 0 cos 0) sin 0 dOdo
1 + sin2 0 cos2 p
When the first of the present authors saw this problem, he quickly eval-
uated its numerical value, using the "QD" software package (which performs
arithmetic, together with 1-D, 2-D and 3-D numerical integration, to roughly
62-digit precision). The numerical value obtained was:
After pasting in this numerical value to the Inverse Symbolic Calculator tool,
available at http://oldweb.cecm.sfu. ca/projects/ISC/ISCmain.html he obtained
(using the "integer relation" option) the result that most likely
? r2 log 2
The first author then confirmed this detection by using Mathematica to evaluate
the right-hand side to over 100 digits, which agreed with a result subsequently
obtained for the integral using the "ARPREC" package (which performs arith-
metic, together with 1-D, 2-D and 3-D numerical integration, to hundreds or
thousands of digits as needed).
Separately, the second author, using Maple Version 10, obtained a numerical
value of roughly 20-digit accuracy (actually ten digits suffices), which was then
immediately recognized by Maple's "identify" function as the value obtained by
the first author's calculations.
In all but a technical sense, we were then done, because we had obtained an
analytic evaluation, confirmed numerically to exceedingly high precision. But
*Lawrence Berkeley National Laboratory, Berkeley, CA 94720, firstname.lastname@example.org. Sup-
ported in part by the Director, Office of Computational and Technology Research, Division of
Mathematical, Information, and Computational Sciences of the U.S. Department of Energy,
under contract number DE-AC02-05CH11231.
tFaculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada,
email@example.com. Supported in part by NSERC and the Canada Research Chair Pro-
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Bailey, David H. & Borwein, Jonathan M. Solution to Monthly Problem 11277, report, March 23, 2007; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc1015065/m1/1/: accessed September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.