Comments on: tetration https://blog.yhuang.org/?p=1084 here. Tue, 14 Oct 2025 11:10:14 +0000 hourly 1 http://wordpress.org/?v=3.1.1 By: Andrew https://blog.yhuang.org/?p=1084&cpage=1#comment-113887 Andrew Wed, 26 Jun 2013 16:37:54 +0000 http://allegro.mit.edu/~zong/wpress/?p=1084#comment-113887 This function is what I call the "exponential commutator" on the Tetration Forum. I think the fundamental problem with your proof is that you are dealing with inverse functions of functions that are not monotonic. When you try and find an inverse function of a nonmonotonic function, then you end up with a Riemann surface, which you might consider a multivalued function, and in order to think of it as a function, you must assign an index to each branch. Doing this, and going back to your proof, I would reword it as follows: Let c = \sqrt{2}. Let f_0(x) be the branch of the inverse function of x^{1/x} that includes the point (x=sqrt(2), f_0=2). Let f_1(x) be the branch of the inverse function of x^{1/x} that includes the point (x=sqrt(2), f_1=4). Then f_0(c) = 2. Similarly, f_1(c) = 4. But then 2 = f_0(c) != f_1(c) = 4. Your assumption that f_0 = f_1 is false, that is the where you went wrong. This function is what I call the “exponential commutator” on the Tetration Forum. I think the fundamental problem with your proof is that you are dealing with inverse functions of functions that are not monotonic. When you try and find an inverse function of a nonmonotonic function, then you end up with a Riemann surface, which you might consider a multivalued function, and in order to think of it as a function, you must assign an index to each branch. Doing this, and going back to your proof, I would reword it as follows:

Let c = \sqrt{2}. Let f_0(x) be the branch of the inverse function of x^{1/x} that includes the point (x=sqrt(2), f_0=2). Let f_1(x) be the branch of the inverse function of x^{1/x} that includes the point (x=sqrt(2), f_1=4). Then f_0(c) = 2. Similarly, f_1(c) = 4. But then 2 = f_0(c) != f_1(c) = 4.

Your assumption that f_0 = f_1 is false, that is the where you went wrong.

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