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A249542
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Decimal expansion of the average product of a side and an adjacent angle of a random Gaussian triangle in two dimensions.
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1
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1, 6, 3, 7, 7, 2, 9, 3, 2, 4, 8, 5, 6, 8, 6, 8, 0, 3, 2, 7, 8, 0, 1, 5, 6, 9, 5, 6, 7, 9, 8, 4, 7, 6, 4, 5, 5, 8, 2, 0, 3, 8, 1, 9, 8, 7, 0, 9, 0, 5, 9, 3, 4, 1, 7, 5, 4, 8, 7, 6, 5, 2, 2, 4, 7, 7, 1, 2, 0, 5, 6, 8, 9, 3, 3, 1, 1, 1, 6, 4, 9, 0, 2, 1, 5, 0, 7, 1, 1, 3, 4, 8, 3, 2, 2, 0, 7, 1, 2, 4, 6, 9, 9, 2, 8
(list;
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refs;
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OFFSET
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1,2
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COMMENTS
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Coordinates are independent normally distributed random variables with mean 0 and variance 1.
As of 2010, an exact expression of this constant was not known, according to Steven Finch.
This average product is noticeably smaller than the product of the averages sqrt(Pi)*Pi/3 = 1.8561..., the side length being negatively correlated with the adjacent angle value.
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LINKS
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Steven R. Finch, Random Triangles, January 21, 2010, p. 14. [Cached copy, with permission of the author]
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FORMULA
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Equals (1/(3*Pi)*Integral_{t=0..infinity} (Integral_{y=0..infinity} (Integral_{t=0..Pi} x^2*y*t*exp(-(1/3)*x^2-x*y*cos(t) + y^2) dt) dy) dx.
Equals (-sqrt(3)*log(3) + Pi^2 - 8*Li_2(2-sqrt(3)) + 2*Li_2(7-4*sqrt(3)))/(2*sqrt(Pi)), where Li_2 is the dilogarithm function.
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EXAMPLE
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1.6377293248568680327801569567984764558203819870905934...
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MATHEMATICA
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ex = (-Sqrt[3]*Log[3] + Pi^2 - 8*PolyLog[2, 2-Sqrt[3]] + 2*PolyLog[2, 7-4*Sqrt[3]])/(2*Sqrt[Pi]); RealDigits[ex, 10, 105] // First
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PROG
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(Python)
from mpmath import *
mp.dps=106
C = (-sqrt(3)*log(3) + pi**2 - 8*polylog(2, 2-sqrt(3)) + 2*polylog(2, 7 - 4*sqrt(3)))/(2*sqrt(pi))
print([int(n) for n in list(str(C).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 04 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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