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A093341
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Decimal expansion of "lemniscate case".
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15
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1, 8, 5, 4, 0, 7, 4, 6, 7, 7, 3, 0, 1, 3, 7, 1, 9, 1, 8, 4, 3, 3, 8, 5, 0, 3, 4, 7, 1, 9, 5, 2, 6, 0, 0, 4, 6, 2, 1, 7, 5, 9, 8, 8, 2, 3, 5, 2, 1, 7, 6, 6, 9, 0, 5, 5, 8, 5, 9, 2, 8, 0, 4, 5, 0, 5, 6, 0, 2, 1, 7, 7, 6, 8, 3, 8, 1, 1, 9, 9, 7, 8, 3, 5, 7, 2, 7, 1, 8, 6, 1, 6, 5, 0, 3, 7, 1, 8, 9, 7, 2, 7, 7, 7, 7
(list;
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OFFSET
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1,2
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, Section 18.14.7, p. 658.
Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software, 1990, p. iii.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.1 Gauss' Lemniscate Constant, p. 421.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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GAMMA(1/4)^2/(4*(Pi)^(1/2)). - Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
Also equals ellipticK(1/sqrt(2)) = Pi/2*hypergeom([1/2,1/2],[1],1/2),
or also the smallest positive root of cs(x/sqrt(2)|-1), where cs is the Jacobi elliptic function, or also the real half-period of the Weierstrass Pe function (Cf. Finch p. 422). - Jean-François Alcover, Apr 30 2013, updated Aug 01 2014
Equals Integral_{x = 0..oo} 1/sqrt(1 + x^4) dx = 2 * Integral_{x = 0..1} 1/sqrt(1 + x^4) dx = sqrt(2) * Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
Equals 2 * Sum {n >= 0} (-1/4)^n * binomial(2*n,n) * 1/(4*n + 1). (End)
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EXAMPLE
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1.854074677301371918433850347195260046217598823521766905585928045056021...
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MAPLE
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MATHEMATICA
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PROG
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(PARI) { allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/(4*(Pi)^(1/2)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b093341.txt", n, " ", d)); } \\ Harry J. Smith, Jun 19 2009
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CROSSREFS
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KEYWORD
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AUTHOR
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Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 26 2004
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EXTENSIONS
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More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
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STATUS
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approved
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