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 Generalized arithmetic operator tommy1729 Ultimate Fellow Posts: 1,358 Threads: 330 Joined: Feb 2009 03/22/2014, 12:13 AM (03/22/2014, 12:06 AM)hixidom Wrote: Quote:what lies between tetration and pentation ? Do we know what lies between addition and multiplication, or multiplication and exponentiation? I would be happy to know those first. I assume they would be simpler to find, but I can also imagine that they would be equally difficult to find. Quote:Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x)) Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ? I found an answer to part of your question. By that I mean I was able to find S^[1/2](exp(x)): By definition: $S^{1/2}(S^{1/2}(e^x))=e^x$ So we are trying to find some function $f$ such that $f(f(x))=e^x$ If we define $b$ such that $f(x)=e^{bx}$ then $f(f(x))=e^{be^{bx}}=e^x$ $\Rightarrow be^{bx}=x$ $\Rightarrow bx\cdot e^{bx}=x^2$ $\Rightarrow bx=W(x^2)$, where W is the Lambert W function $\Rightarrow b=W(x^2)/x$ $\Rightarrow f(x)\equiv S^{1/2}(e^x)=e^{bx}=e^{W(x^2)}$ There is your half-superfunction of exp(x). Sorry for not using tex before but By definition: $S^{1/2}(S^{1/2}(e^x))=S^{1/2+1/2}(e^x)=S(e^x)=sexp(x)$ that is sufficient to see your answer is wrong ... Sorry. regards tommy1729 « Next Oldest | Next Newest »

 Messages In This Thread Generalized arithmetic operator - by hixidom - 03/11/2014, 03:52 AM RE: Generalized arithmetic operator - by JmsNxn - 03/11/2014, 03:15 PM RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 06:24 PM RE: Generalized arithmetic operator - by MphLee - 03/11/2014, 10:49 PM RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 11:20 PM RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 11:18 AM RE: Generalized arithmetic operator - by JmsNxn - 03/12/2014, 02:59 AM RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 04:37 AM RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 06:19 PM RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 06:43 PM RE: Generalized arithmetic operator - by tommy1729 - 03/21/2014, 10:31 PM RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:06 AM RE: Generalized arithmetic operator - by tommy1729 - 03/22/2014, 12:13 AM RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:42 AM RE: Generalized arithmetic operator - by hixidom - 06/11/2014, 05:10 PM

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