MaplePrimes - comments on Post, MRB Constant J

MaplePrimes - comments on Post, MRB Constant J http://www.mapleprimes.com/posts/80837-MRB-Constant-J en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Fri, 12 Jun 2026 12:29:41 GMT Fri, 12 Jun 2026 12:29:41 GMT The latest comments added to the Post, MRB Constant J http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Post, MRB Constant J http://www.mapleprimes.com/posts/80837-MRB-Constant-J n=E Gives a local maximum http://www.mapleprimes.com/posts/80837-MRB-Constant-J?ref=Feed:MaplePrimes:MRB Constant J:Comments#comment88272 Concerning the unlikely approximation to the MRB constant, MKB(e^-1,e) = Re(int((-1)^n*n^(1/n), n = e^-1..e)) = 0.1877790313..., one note to myself and anyone else working on this. n=E Gives a maximum for f in the function f(n)=Abs((-1)^n*n^(1/n)) in the range (0, oo). That critical point is (e, e^(1/e)). Another note: f(n)=abs((-1)^n*n^(1/n)). n = 1/e,  f = (1/e)^(e). The latest comments added to the Post, MRB Constant J 88272 Sun, 09 May 2010 05:59:37 Z Marvin Ray Burns Marvin Ray Burns I don't see where to go from here http://www.mapleprimes.com/posts/80837-MRB-Constant-J?ref=Feed:MaplePrimes:MRB Constant J:Comments#comment88270 I asked, "Is there a connection between the MRB constant and MKB(E^-1,E) ?" One person wrote me back and suggested that, "The two constants are different without doubt. Generally speaking there may be some approximate mapping with the Euler-Maclaurin formula to convert the sum and the integral with favorite cancellation because the derivative involves the logarithm, [ d/dn of n^(1/n) is n^(1/n)*(1-log n)/n^2 ] but beyond this approximation this is just coincidence." On the other hand, I was trying to work on a connection by means of the following: At n=1/2 and n=5/2  the real parts of (-1)^n*n^(1/n) are 0. eval((-1)^n*n^(1/n), n = 5/2) = 0+I*(5/2)^(2/5). eval((-1)^n*n^(1/n), n = 1/2) = 0+I*(1/4). Since (-1)^n*n^(1/n) is continuous from 1/e to e, int((-1)^n*n^(1/n), n = exp(-1) .. exp(1)) = int((-1)^n*n^(1/n), n = exp(-1) .. 1/2) + int((-1)^n*n^(1/n), n = 1/2 .. 5/2) + int((-1)^n*n^(1/n), n = 5/2 .. exp(1)). So, Re(int((-1)^n*n^(1/n), n = exp(-1) .. exp(1))) = Re(int((-1)^n*n^(1/n), n = exp(-1) .. 1/2)) + Re(int((-1)^n*n^(1/n), n = 1/2 .. 5/2)) + Re(int((-1)^n*n^(1/n), n = 5/2 .. exp(1))). Or approximately, 0.18777903132304277043 = 0.0032282119429439541474 + 0.28848088137392251193 + (-.10393006199382369565) But I don't see anywhere to go from there.     The latest comments added to the Post, MRB Constant J 88270 Wed, 12 May 2010 21:29:23 Z Marvin Ray Burns Marvin Ray Burns Real part of the nth deriviatives http://www.mapleprimes.com/posts/80837-MRB-Constant-J?ref=Feed:MaplePrimes:MRB Constant J:Comments#comment88273 For f(x)=(-1)^x*x^(1/x) I said there was a critical point at x=e. At x=e:   n        real part of the nth deriviative   1        -Pi^1 * e^(1/e) * sin(e*Pi) within 0   2        -Pi^2 * e^(1/e) *cos(e*Pi) within a remainder of 0.0455474...   3        Pi^3*e^(1/e)*sin(e*Pi) within a remainder of 0.440864...   4        Pi^4*e^(1/e)*cos(e*Pi) within a remainder of -3.830448090...   5       -Pi^5*e^(1/e)*sin(e*Pi) within a remainder of -6.35268...   6       -Pi^6*e^(1/e)*cos(e*Pi) within a remainder of  100.9196638... , where Pi^7>3020.   7      Pi^7*e^(1/e)*sin(e*Pi) within a remainder of -51.137422... , where Pi^8>9488     The latest comments added to the Post, MRB Constant J 88273 Thu, 13 May 2010 03:06:21 Z Marvin Ray Burns Marvin Ray Burns Is based mostly on chance http://www.mapleprimes.com/posts/80837-MRB-Constant-J?ref=Feed:MaplePrimes:MRB Constant J:Comments#comment88274 Concerning Re(int((-1)^n*n^(1/n), n = exp(-1) .. exp(1))) or the integral over cos(pi*x)x^(1/x) between 1/E and E,  According to the Euler–Maclaurin formula, the odd nth derivatives mentioned in my last post, which are rich with cancelation and approximation at the greater endpoint, e, do simplify the integral of concern. However our definite integral has a lesser endpoint of 1/e that possess very little cancelation within its higher derivatives. Thus I conclude that the nearness of the value of this integral to the value of the MRB constant is just coincidence. However the wealth of cancelation I demonstrated in my previous post will probably help me in writing a future blog. The value of the integral of concern is found in the online version of integer sequences under the A-number <a href="http://oeis.org/A177218">A177218</a>. For more on the Euler–Maclaurin formula see <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula">http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula</a>.    <a href="http://marvinrayburns.com">marvinrayburns.com </a> The latest comments added to the Post, MRB Constant J 88274 Tue, 18 May 2010 05:39:32 Z Marvin Ray Burns Marvin Ray Burns