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Infinite Cosine Product Integral

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

C_2=int_0^inftycos(2x)product_(n=1)^(infty)cos(x/n)dx
(1)
=0.39269908169...
(2)

(OEIS A091473). Although this is within 10^(-42) of pi/8,

1/8pi-C_2=7.407346566316950557...×10^(-43)
(3)

(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for C_2.

Interestingly, the integral

C_0=int_0^inftyproduct_(n=1)^(infty)cos(x/n)dx
(4)
=0.78538...
(5)

(Borwein et al. 2004, pp. 101-102) has a value fairly close to pi/4=0.78539..., but no other similar relationships seem to hold for other multipliers of the form cos(nx) or sin(nx).

The identity

sinc(x)=product_(k=1)^inftycos(x/(2^l))
(6)

can be expanded to yield

product_(k=0)^inftysinc((2x)/(2k+1))=product_(n=1)^inftycos(x/n).
(7)

In fact,

C_0=pi/2lim_(n->infty)(2n+1)!!I_(2n+1),
(8)

where I_(2n+1) is a Borwein integral.

See also

Borwein Integrals

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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.

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Infinite Cosine Product Integral

Cite this as:

Weisstein, Eric W. "Infinite Cosine Product Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InfiniteCosineProductIntegral.html

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