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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

Marvin Ray Burns, Paper on 20000 digits in a Mathematica notebook (Digits checked by different formula, computing more digits)

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-Fran├žois 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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Last modified March 30 18:30 EDT 2017. Contains 284302 sequences.