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A086278
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Decimal expansion of Shapiro's cyclic sum constant mu.
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12
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9, 7, 8, 0, 1, 2, 4, 7, 8, 1, 8, 6, 6, 4, 6, 2, 2, 0, 2, 0, 1, 8, 2, 7, 9, 5, 9, 9, 7, 8, 6, 8, 2, 6, 8, 0, 9, 3, 2, 5, 3, 8, 6, 3, 5, 3, 4, 5, 9, 1, 4, 1, 8, 0, 9, 4, 9, 5, 3, 0, 4, 2, 0, 8, 3, 4, 5, 9, 9, 4, 4, 9, 2, 5, 8, 0, 7, 1, 0, 6, 9, 7, 5, 0, 0, 5, 5, 6, 6, 8, 9, 8, 5, 2, 0, 3, 9, 2, 6, 5, 9, 2, 4
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OFFSET
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0,1
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COMMENTS
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Based on the references, it seems that this constant was first defined by Elbert (1973). We have psi(0) = mu. Two auxiliary constants, b = -0.33060494... = -A335809 and c = 0.38755227... = A335810, are needed for the estimation of mu.
Here psi(x) is the convex hull of y = (1 + exp(x))/2 and y = (1 + exp(x))/(1 + exp(x/2)); i.e., psi(x) = (1 + exp(x))/2 for x <= b; psi(x) = (1 + exp(b))/2 + (((1 + exp(c))/(1 + exp(c/2)) - (1 + exp(b))/2)/(c - b)) * (x - b) for b <= x <= c; and psi(x) = (1 + exp(x))/(1 + exp(x/2)) for x >= c. (For b <= x <= c, we have the equation of the line segment tangent to both curves.)
It follows that mu = psi(0) = (1 + exp(b))/2 - b * (((1 + exp(c))/(1 + exp(c/2)) - (1 + exp(b))/2)/(c - b)) (where the y-axis crosses the line segment). Or by using the tangent line at x = b to the curve y = (1 + exp(x))/2, we find mu = psi(0) = (1 + exp(b))/2 - b * exp(b)/2. Or by using the tangent line at x = c to the curve y = (1 + exp(x))/(1 + exp(x/2)), we may get a third formula for mu = psi(0) in terms of c only.
Similar calculations were done by Drinfel'd (1971) for the Shapiro cyclic sum constant lambda = phi(0)/2 = A086277 = A245330/2. In this case, the corresponding curves are y = exp(-x) and y = 2/(exp(x) + exp(x/2)), while the corresponding x-coordinates at the tangent points are -A319568 = -0.20081... and A319569 = 0.15519... Here phi(x) is the convex hull of these two curves (and it becomes a line segment tangent to both curves for -A319568 <= x <= A319569).
Eric W. Weisstein, in the link below, has a summary of the above discussion (with contributions by Steven Finch). (End)
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 3.1, Shapiro-Drinfeld constant, p. 209.
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LINKS
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V. G. Drinfel'd, A cyclic inequality, Mathematical Notes of the Academy of Sciences of the USSR, 9 (1971), 68-71.
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FORMULA
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Solve the following system of equations to find the x-coordinates of the two points where the common tangent touches the two curves:
exp(b) = (-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2 and
(exp(b)*(c - b + 1) + 1)*(1 + exp(c/2)) = 2*(1 + exp(c)).
Then the constant equals (1 + exp(b)*(1 - b))/2. (End)
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EXAMPLE
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0.97801247818664622020182795997868268... = 1 - 0.02198752181335377979817204...
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MATHEMATICA
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eq = E^u + 2*E^(u + v/2) + E^(v/2) + E^(u + v) == 2*E^v + E^(3*v/2) && 2 + 2*E^(u + v/2) == 2*y + 2*E^v + E^u*(v - 2) && E^u*(u - v + 1) + 2*E^(u + v/2) + 1 == 2*E^v; mu = y /. FindRoot[eq , {{y, 1}, {u, -1/3}, {v, 1/3}}, WorkingPrecision -> 105]; RealDigits[mu, 10, 103] // First
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PROG
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(PARI)
default("realprecision", 200)
b(c) = log((-exp(c/2) + 2*exp(c) + exp(3*c/2))/(1 + exp(c/2))^2);
a = solve(c=-1, 1, (exp(b(c))*(c - b(c) + 1) + 1)*(1 + exp(c/2)) - 2*(1 + exp(c)));
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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