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 A157852 Decimal expansion of the absolute value of limit_{N -> infinity} Integral_{x=1..2*N} e^(i Pi x)*x^(1/x). 5
 6, 8, 7, 6, 5, 2, 3, 6, 8, 9, 2, 7, 6, 9, 4, 3, 6, 9, 8, 0, 9, 3, 1, 2, 4, 0, 9, 3, 6, 5, 4, 4, 0, 1, 6, 4, 9, 3, 9, 6, 3, 7, 3, 8, 4, 9, 0, 3, 6, 2, 2, 5, 4, 1, 7, 9, 5, 0, 7, 1, 0, 1, 0, 1, 0, 7, 4, 3, 3, 6, 6, 2, 5, 3, 4, 7, 8, 4, 9, 3, 7, 0, 6, 8, 6, 2, 7, 2, 9, 8, 2, 4, 0, 4, 9, 8, 4, 6, 8, 1, 8, 8, 7, 3, 1, 9, 2, 9, 3, 3, 4, 3, 3, 5, 4, 6, 6, 1, 2, 3, 2, 8, 6 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS This constant is the integral analog of the constant described in A037077 since e^(i Pi x) =(-1)^x. While A037077 has was called the MRB constant by Simon Pluoffe, Marvin Ray Burns named this constant MKB after his wife at the time. This constant is hard to integrate and very slow to converge, so it takes a combination of modern methods to calculate many digits! This constant could be written as a special value, for omega=Pi, of the function f(omega)=Limit_{N->infinity} Integral_{x=Pi/omega,2N(Pi/omega)}(exp(i*omega*x)*x^(1/x)), a kind of discretely sampled Fourier transform of x^(1/x). This stresses the fact that it is a complex entity. People who desire to underline the similarity of this integral to the MRB alternating series (A037077) often write the factor exp(i*Pi*x) as (-1)^x, which can be a bit confusing because it hides the imaginary unit. - Stanislav Sykora, Apr 08 2016 LINKS Marvin Ray Burns, Table of n, a(n) for n = 0..19999 Marvin Ray Burns, Author's public inquiry 1 Marvin Ray Burns, Author's public inquiry 2 Marvin Ray Burns, Some known record calculations of this constant R. J. Mathar, Numerical evaluation of the oscillatory integral over exp(i*pi*x)x^(1/x) between 1 and infinity, arxiv:0912.3844 [math.CA], 2009-2010. FORMULA Equals sqrt(A255727^2 + A255728^2). - Joerg Arndt, Apr 05 2016 EXAMPLE After integrating from 1 to 15 million the absolute value of the integral is approximately 0.687652_7, after integrating from 1 to 20 million approximately 0.687652_6. MATHEMATICA a = NIntegrate[ x^(1/x)*Cos[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; b = NIntegrate[ x^(1/x)*Sin[Pi*x], {x, 1, 10^20}, WorkingPrecision -> 30, MaxRecursion -> 70]; RealDigits[ Sqrt[a^2 + b^2], 10, 18] // First (* Jean-François Alcover, Feb 14 2013 *) (* Other program: For large calculations. Tested for 1000-35000 digits-- see post at http://community.wolfram.com/groups/-/m/t/366628?p_p_auth=KA7y1gD4 and search for "analog" to find pertinent replies. Designed to include 40000 digits. A157852 is saved as c, the real part as a and the imaginary part as b. *) Block[{\$MaxExtraPrecision = 200}, prec = 40000(* Replace 40000 with number of desired digits. 40000 digits should take two weeks on a 3.5 GH Pentium processor. *); f[x_] = x^(1/x); ClearAll[a, b, h]; Print[DateString[]]; Print[T0 = SessionTime[]]; If[prec > 35000, d = Ceiling[0.002 prec],   d = Ceiling[0.264086 + 0.00143657 prec]]; h[n_] :=   Sum[StirlingS1[n, k]*     Sum[(-j)^(k - j)*Binomial[k, j], {j, 0, k}], {k, 1, n}]; h[0] = 1; g = 2 I/Pi - Sum[-I^(n + 1) h[n]/Pi^(n + 1), {n, 1, d}]; sinplus1 :=   NIntegrate[    Simplify[Sin[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity},    WorkingPrecision -> prec*(105/100),    PrecisionGoal -> prec*(105/100)]; cosplus1 :=   NIntegrate[    Simplify[Cos[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity},    WorkingPrecision -> prec*(105/100),    PrecisionGoal -> prec*(105/100)]; middle := Print[SessionTime[] - T0, " seconds"]; end := Module[{}, Print[SessionTime[] - T0, " seconds"];    Print[c = Abs[a + b]]; Print[DateString[]]]; If[Mod[d, 4] == 0,   Print[N[a = -Re[g] - (1/Pi)^(d + 1)*sinplus1, prec]];   middle;   Print[N[b = -I (Im[g] - (1/Pi)^(d + 1)*cosplus1), prec]];   end]; If[Mod[d, 4] == 1,   Print[N[a = -Re[g] - (1/Pi)^(d + 1)*cosplus1, prec]];   middle;   Print[N[b = -I (Im[g] + (1/Pi)^(d + 1)*sinplus1), prec]]; end]; If[Mod[d, 4] == 2,   Print[N[a = -Re[g] + (1/Pi)^(d + 1)*sinplus1, prec]];   middle;   Print[N[b = -I (Im[g] + (1/Pi)^(d + 1)*cosplus1), prec]];   end]; If[Mod[d, 4] == 3,   Print[N[a = -Re[g] + (1/Pi)^(d + 1)*cosplus1, prec]];   middle;   Print[N[b = -I (Im[g] - (1/Pi)^(d + 1)*sinplus1), prec]];   end]; ] (* Marvin Ray Burns, Aug 06 2015 *) CROSSREFS Integrating A037077 instead of summing. Cf. A037077, A255727 (real part), A255728 (imaginary part). Sequence in context: A097668 A133748 A225037 * A088608 A176104 A011481 Adjacent sequences:  A157849 A157850 A157851 * A157853 A157854 A157855 KEYWORD nonn,cons AUTHOR Marvin Ray Burns, Mar 07 2009 EXTENSIONS Edited by N. J. A. Sloane, Mar 13 2009 Corrected and edited by Marvin Ray Burns, Apr 03 2009 8 more digits from R. J. Mathar, Nov 30 2009, 3 more Jan 03 2011, 3 more on Feb 25 2013 15 more digits added by Marvin Ray Burns, Feb 26 2013 Many more digits added by Marvin Ray Burns, May 11 2015 Edited by Marvin Ray Burns, Aug 06 2015 STATUS approved

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