A157852 Decimal expansion of the absolute value of lim_{N -> infinity} Integral_{x=1..2*N} e^(i*Pi*x)*x^(1/x).

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

Offset: 0

Author

Marvin Ray Burns, Mar 07 2009

Keywords

• nonn
• cons

Comments

This constant is the integral analog of the constant described in A037077 since e^(i*Pi*x) =(-1)^x. While A037077 was named the MRB constant by Simon Plouffe, 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) = lim_{N->infinity} Integral_{x = Pi/omega .. 2N8(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

Examples

			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.
		

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, How I calculated the digits of the MKB constant
• Marvin Ray Burns, Paper on 20000 digits in a Mathematica notebook (Digits checked by different formula, computing more digits)
• Marvin Ray Burns, How 40,000 digits of A157852 was successfully computed and checked to 35,000 digits
• Marvin Ray Burns, Mathematica Notebook of 54,386 digits of A157852 computed with increasing MaxRecursion, with proof at bottom.
• Marvin Ray Burns, How_I_found_A157852_sum.
• Marvin Ray Burns, A157852 sum PDFK
• 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.

Crossrefs

Integrating A037077 instead of summing.

Cf. A037077, A255727 (real part), A255728 (imaginary part).

Programs

Maple

# After Marvin Ray Burns's "Program 3".
f := (n, x) -> seq(x, 0..n):
m := n -> (Pi/I)^n * MeijerG([[], [f(n, 1)]], [[1-n, f(n, 0)], []], -I*Pi):
s := n -> abs(add(m(k), k = 1..n) - 2)/Pi:
# s(n) approximates the constant for n -> oo and suitable chosen precision.
seq(evalf(s(n), 22), n = 1..3); # Peter Luschny, Nov 16 2021

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 *)
(* Program 2: to compute and verify 1000s of digits through a different formula. *)
g[x_] = x^(1/x); t = (Timing[
    MKB = -(I NIntegrate[(g[(1 + t I)]) ( Exp[-Pi t]), {t, 0,
           Infinity}, WorkingPrecision -> 2410,
          Method -> "Trapezoidal", MaxRecursion -> 10] + I/Pi)])[[
  1]]; Print["Timing for calculation=", t]; t = (Timing[
    MKB1 = (1/Pi  NIntegrate[
         g'[1 + I t] Exp[-Pi t], {t, 0, Infinity},
         WorkingPrecision -> 2410, Method -> "Trapezoidal",
         MaxRecursion -> 10] - 2 I/Pi)])[[
  1]]; Print["Timing for verification=", t]; err =
test - test2; Print["Error=", N[err, 20]];Abs[MKB] (* MaxRecursion -> 13 works for 10,000 digits. Marvin Ray Burns, Apr 18 2021 *)
(* Program 3: An infinite sum involving the Meijer G function. Compare the discussion near the end of "How I calculated the digits of the MKB constant" and all of Cloud Notebook "How_I_found_A157852_sum" in the link section. *)
f[n_] := MeijerG[{{},Table[1, {n+1}]}, {Prepend[Table[0, n+1], -n + 1], {}}, -I Pi];
Abs[Sum[(I/Pi)^(1 - n) N[f[n], 22], {n, 1, 15}] - 2 I/Pi] (* Marvin Ray Burns, Nov 15 2021 *)

Formula

Equals sqrt(A255727^2 + A255728^2). - Joerg Arndt, Apr 05 2016

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

Edited by Marvin Ray Burns, Jun 18 2017