Infinite Cosine Product Integral

At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey 2003, p. 26; Bailey et al. 2006)

			(1)
			(2)

(OEIS A091473). Although this is within  of ,

(3)

(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .

Interestingly, the integral

			(4)
			(5)

(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form  or .

The identity

(6)

can be expanded to yield

(7)

In fact,

(8)

where  is a Borwein integral.

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Kapoor, V.; and Weisstein, E. W. "Ten Problems in Experimental Mathematics." Amer. Math. Monthly 113, 481-509, 2006b.

Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.

Sloane, N. J. A. Sequences A091473 and A091494 in "The On-Line Encyclopedia of Integer Sequences."

Trott, M. "The Mathematica Guidebooks Additional Material: Infinite Cosine Product Integral." http://www.mathematicaguidebooks.org/additions.shtml#N_2_01.