• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 e is the global maximum of x root x, 2 root 2 = 4 root 4, so... bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 02/13/2011, 10:27 AM Oh, now with the picture I see what you mean. To get a formula we can use the Lambert W function. The Lambert W function is the inverse of the function M(x)=x*e^x and we can express the self root x^(1/x) with help of M: y = x^(1/x) = exp(-M(-ln(x))) You can verify this with a bit of calculation. Then we can obtain the inverse: exp(-W(-ln(y))) = x More exactly x can be two values, left and right from e, which correspond to the two branches of W: $x_L = \exp(-W_0(-\ln(y)))$ $x_R = \exp(-W_{-1}(-\ln(y)))$ So when you want to get the left value - as in your case - you choose x_L and get: $x_L = \exp(-W_0(-\ln(3^{1/3})))\approx 2.47805268028830$ $x_R$ would be simply 3 again. « Next Oldest | Next Newest »

 Messages In This Thread e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by robo37 - 01/30/2011, 12:05 PM RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by robo37 - 02/13/2011, 12:30 AM RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by bo198214 - 02/13/2011, 10:27 AM RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by robo37 - 02/15/2011, 03:20 PM RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - by bo198214 - 02/15/2011, 05:08 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Can we get the holomorphic super-root and super-logarithm function? Ember Edison 10 4,716 06/10/2019, 04:29 AM Last Post: Ember Edison The super 0th root and a new rule of tetration? Xorter 4 4,548 11/29/2017, 11:53 AM Last Post: Xorter Super-root 3 andydude 10 11,913 01/19/2016, 03:14 AM Last Post: andydude Self-root function and reciprocal self-power function have same integrals Ztolk 5 9,260 04/11/2010, 12:52 AM Last Post: Ztolk Approximation method for super square root Ztolk 2 9,325 03/23/2010, 02:33 PM Last Post: Ztolk Lambert W function and the Super Square Root James Knight 3 9,536 10/29/2009, 06:30 AM Last Post: andydude Iterative square root like square root recursion limit bo198214 1 4,109 02/23/2009, 03:01 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)