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A197374
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Pi(3): fundamental real period of the Dixonian elliptic functions sm(z) and cm(z).
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1
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5, 2, 9, 9, 9, 1, 6, 2, 5, 0, 8, 5, 6, 3, 4, 9, 8, 7, 1, 9, 4, 1, 0, 6, 8, 4, 9, 8, 9, 4, 5, 3, 1, 6, 1, 0, 7, 7, 1, 5, 6, 0, 5, 6, 1, 4, 6, 0, 7, 6, 7, 2, 5, 9, 0, 3, 8, 0, 7, 1, 5, 7, 2, 5, 5, 0, 6, 3, 5, 9, 0, 0, 5, 1, 8, 4, 3, 2, 3, 7, 4, 0, 8, 1, 6, 4, 6, 0, 9, 8, 0, 0, 0, 0, 1, 5, 0, 7, 6, 1, 6, 5
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OFFSET
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1,1
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COMMENTS
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Pi(3) = 5.29991 62508 56349 87194 ... is the real period of the doubly-periodic Dixonian elliptic functions sm(z) (A104133) and cm(z) (A104134): sm(z+Pi(3)) = sm(z); cm(z+Pi(3)) = cm(z). The other period equals Pi(3)*w, where w = exp(2*I*Pi/3).
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REFERENCES
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A. C. Dixon, On the doubly periodic functions arising out of the curve x^3 + y^3 - 3 alpha xy = 1, Quarterly J. Pure Appl. Math. 24 (1890), 167-233.
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LINKS
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FORMULA
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Pi(3) = 3*int {0..1} 1/(1-t^3)^(2/3) dt = B(1/3,1/3) = Gamma(1/3)^2/Gamma(2/3) = sqrt(3)/(2*Pi)*Gamma(1/3)^3.
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EXAMPLE
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5.299916250856349...
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MATHEMATICA
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Sqrt[3]/(2*Pi)*Gamma[1/3]^3 // N[#, 103]& // RealDigits // First (* Jean-François Alcover, Jan 21 2013 *)
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PROG
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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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