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 Tetration extension for bases between 1 and eta dantheman163 Junior Fellow Posts: 13 Threads: 3 Joined: Oct 2009 11/05/2009, 11:53 PM (This post was last modified: 11/06/2009, 12:24 AM by dantheman163.) I'm not sure how rigorous this is but here it is Proof: Assumption (1)The function $f(x)={}^x b$ is a smooth, monotonic concave down function Based on (1) we can establish (2)Any line can only pass through f(x) a maximum of 2 times (3)The intermediate value theorem holds for the entire domain of f(x) Take the region R bounded on the x axis by x=-1 and x=0 and bounded on the y axis by y=f(-1) and y=f(0) ( y=0 and y=1). Because of (3) every value, x, has a corresponding value, f(x), on the interval. If we take a point (x,y) in R and assume that it is on the curve f(x) we can then use the relation $f(x+1)=b^{f(x)}$ and obtain the new point $(x',y')$ which equals $(x+1,b^y)$. Applying this repeatedly we obtain the point $(x+k,{Exp}_b ^k (y))$. We can now establish (4)The point (x,y) in the region R is on the curve f(x) if $(x+k,{Exp}_b ^k (y))$ is not on the secant line that touches the curve at 2 other known points of f(x) for any value of k. Now we will find the equation of the secant line that touches f(x) at 2 consecutive known points. Using the point slope formula we find the equation to be $g(x)= ({}^k b-{}^{(k-1)} b)(x-k)+{}^k b$. We must also note that if a point (x,y) is above the curve in the region R then $(x+k,{Exp}_b ^k (y))$ is above the curve for any value of k. we shall now extend (4) to say (5)The point (x,y) in the region R is on or above the curve f(x) if $g(x+k) < {Exp}_b ^k (y)$for any value of k Finally if we take the limit as k approaches infinity we will find that the slope of the secant line approaches zero and therefore follows f(x) exactly because f(x) has an asymptote as x goes towards infinity. Therefore $g(x+k) = {Exp}_b ^k (y)$ for infinity large values of k Now solving for y and taking the limit as k approaches infinity we obtain the desired result: ${}^x b = \lim_{k\to \infty} (log_{b}^{ok}(x({}^k b- {}^{(k-1)} b)+{}^k b) )$ for $-1 \le x\le 0$ q.e.d « Next Oldest | Next Newest »

 Messages In This Thread Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 03:00 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/05/2009, 01:44 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/05/2009, 11:53 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 09:31 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 05:11 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/07/2009, 08:12 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 11/07/2009, 11:30 PM RE: Tetration extension for bases between 1 and eta - by bo198214 - 11/08/2009, 02:44 PM RE: Tetration extension for bases between 1 and eta - by mike3 - 11/12/2009, 07:11 PM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:01 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/15/2009, 01:40 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/15/2009, 01:48 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/17/2009, 02:40 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/17/2009, 10:59 AM RE: Tetration extension for bases between 1 and eta - by dantheman163 - 12/19/2009, 05:06 AM RE: Tetration extension for bases between 1 and eta - by bo198214 - 12/19/2009, 10:55 AM

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