

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
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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 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) = 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)
Marvin Ray Burns, Work done on this sequence and A037077
Marvin Ray Burns, 40,000 digit Mathematica notebook
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], 20092010.


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 (* JeanFrançois Alcover, Feb 14 2013 *)


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
Edited by Marvin Ray Burns, Jun 18 2017


STATUS

approved



