e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... (/showthread.php?tid=580) e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - robo37 - 01/30/2011 ... what when put to the root of itself is equal to the cube root of 3? Can it be expressed in terms of e? I know this sounds like a bit of a random question but it's something I've always been curious in. RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - robo37 - 02/13/2011 I've calculated that the number is roughly 2.47805268, but I don't think I can calculate a way to express it without learning advanced mathematics (as you can probably tell I'm no mathematician). RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - bo198214 - 02/13/2011 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. RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - robo37 - 02/15/2011 Thank you very much for the reply. So, if you take the Lambert W function out of the equation, what are you left with? As I said I'm no mathematician I'm afraid so I don't fully understand the principle. RE: e is the global maximum of x root x, 2 root 2 = 4 root 4, so... - bo198214 - 02/15/2011 (02/15/2011, 03:20 PM)robo37 Wrote: So, if you take the Lambert W function out of the equation, what are you left with? As I said I'm no mathematician I'm afraid so I don't fully understand the principle. I dont think there might be a closed form solution without the Lambert W function. However if you are after calculating the number, that function really helps as it is available in most computer algebra packages.