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 e is the global maximum of x root x, 2 root 2 = 4 root 4, so... robo37 Junior Fellow Posts: 14 Threads: 5 Joined: Jun 2009 01/30/2011, 12:05 PM (This post was last modified: 02/13/2011, 12:35 AM by robo37.) ... 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. robo37 Junior Fellow Posts: 14 Threads: 5 Joined: Jun 2009 02/13/2011, 12:30 AM 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). bo198214 Administrator Posts: 1,386 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. robo37 Junior Fellow Posts: 14 Threads: 5 Joined: Jun 2009 02/15/2011, 03:20 PM (This post was last modified: 02/15/2011, 03:21 PM by robo37.) 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. bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 02/15/2011, 05:08 PM (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. « Next Oldest | Next Newest »

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