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 Tetration below 1 Daniel Fellow Posts: 66 Threads: 29 Joined: Aug 2007 08/31/2007, 12:47 AM jaydfox Wrote:While the sine wave idea makes sense from the standpoint of creating a simple and infinitely differentiable function, it doesn't seem to fit the idea of continuous iteration. If continuous iteration is to be considered, the sine wave idea doesn't fit. Given a value near the infinite limit, how would the function "know" which direction to go for a small fractional iteration? Is it on the upswing or the downswing? On the other hand, perhaps this is a situation where complex outputs are actually desirable? By having a spiral that hits the real plane only at the integer tetrations, we can embed information in the output that would "tell" the function where it is, and hence where to go next. Just an idea... For tetration $a$ converges to $h(a) = \;^{\infty}a = \frac{W(-\ln(a))}{-\ln(a)}$ $D^2 f^t(0) = f''(0) \sum_{k=0}^{t-1} f'(0)^{2t-k-2} \rightarrow D^2 \;^{n}a = 1/2 \sum_{k=0}^{n-1} {\ln(\;^{\infty}a )}^{2n-k} (1- \; \;^{\infty}a )^2$. The first few terms of the Taylor series for $\;^{n}a$ from it's fixed point are $\;^{n}a = \;^{\infty}a + \ln(\;^{\infty}a)^n \; (1- \; \;^{\infty}a) + 1/2 \sum_{k=0}^{n-1} {\ln(\;^{\infty}a )}^{2n-k} (1- \; \;^{\infty}a )^2 + \ldots$. This holds for any value of $a$ and and integer $n$. But this contains a geometrical progression that simplifies the equation based on the specific value of $a$, or more particularly the location and type of fixed point of $a^z$. As a result, $n$ will no longer be an index, but will be a complex number. « Next Oldest | Next Newest »

 Messages In This Thread Tetration below 1 - by bo198214 - 08/29/2007, 06:14 PM RE: Tetration below 1 - by Daniel - 08/30/2007, 09:32 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:08 PM RE: Tetration below 1 - by jaydfox - 08/30/2007, 11:20 PM RE: Tetration below 1 - by Daniel - 08/31/2007, 12:47 AM RE: Tetration below 1 - by GFR - 09/02/2007, 01:30 PM RE: Tetration below 1 - by bo198214 - 09/02/2007, 01:40 PM RE: Tetration below 1 - by GFR - 09/02/2007, 05:38 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 03:58 PM RE: Tetration below 1 - by bo198214 - 09/03/2007, 04:07 PM RE: Tetration below 1 - by jaydfox - 09/03/2007, 04:36 PM RE: Tetration below 1 - by jaydfox - 09/05/2007, 11:24 PM RE: Tetration below 1 - by GFR - 09/06/2007, 12:01 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 03:28 AM RE: Tetration below 1 - by jaydfox - 09/06/2007, 07:21 AM RE: Tetration below 1 - by bo198214 - 03/26/2008, 04:51 PM

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