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 An alternate power series representation for ln(x) bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/07/2011, 10:45 PM (05/07/2011, 08:41 PM)JmsNxn Wrote: This proof involves the use of a new operator: $x \bigtriangleup y = ln(e^x + e^y)$ and it's inverse: $x \bigtriangledown y = ln(e^x - e^y)$ and the little differential operator: $\bigtriangleup \frac{d}{dx} f(x) = \lim_{h\to\ -\infty} [f(x \bigtriangleup h) \bigtriangledown f(x)] - h$ (The notation is more unambiguous than in your previous thread ) But your operator can be expressed with the classical differentiation, see: $\begin{eqnarray} \bigtriangleup \frac{d}{dx} f(x) &=& \lim_{h\to\ -\infty} [f(x \bigtriangleup h) \bigtriangledown f(x)] - h\\ &=& \lim_{h\to\ -\infty} \quad\ln[\exp(f(x \bigtriangleup h)) - \exp(f(x))] - h\\ &=& \ln\quad\lim_{d\to 0} \frac{\exp(f(x \bigtriangleup \log(d))) - \exp(f(x))}{d}\\ &=& \ln\quad\lim_{d\to 0} \frac{\exp(f(\ln(e^x + d)) - \exp(f(x))}{d}\\ &=& \ln\quad\lim_{d\to 0} \frac{\exp(f(\ln(e^x + d)) - \exp(f(\ln(\exp(x)))))}{d}\\ & =& \ln((\exp\circ f\circ \ln)'(\exp(x))) \end{eqnarray}$ Or purely functional with the composition operation $\circ$: $\bigtriangleup \frac{d}{dx} f = \ln\circ(\exp\circ f\circ \ln)'\circ\exp$ PS: when you write ln with backslash in front: Code:$$\ln(x)$$ you get a better ln-typesetting. « Next Oldest | Next Newest »

 Messages In This Thread An alternate power series representation for ln(x) - by JmsNxn - 05/07/2011, 08:41 PM RE: An alternate power series representation for ln(x) - by JmsNxn - 05/07/2011, 09:43 PM RE: An alternate power series representation for ln(x) - by bo198214 - 05/07/2011, 10:45 PM RE: An alternate power series representation for ln(x) - by JmsNxn - 05/07/2011, 11:20 PM RE: An alternate power series representation for ln(x) - by bo198214 - 05/08/2011, 01:38 PM RE: An alternate power series representation for ln(x) - by JmsNxn - 05/08/2011, 07:54 PM RE: An alternate power series representation for ln(x) - by bo198214 - 05/08/2011, 08:28 PM RE: An alternate power series representation for ln(x) - by JmsNxn - 05/09/2011, 01:02 AM

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