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 open problems survey andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/26/2008, 03:24 PM (This post was last modified: 05/26/2008, 06:47 PM by bo198214.) Conjecture (Part 1): The exponential factorial (EF) is uniquely determined by the assumptions: $EF(1) = 1$ $EF(x) = x^{EF(x-1)}$ EF(x) is invertible for $x \in S$ EF(x) is real analytic for $x \in S$ $\frac{d^n}{dx^n}EF(x) > 0$ for $x \in S$ where S is an open interval on the real line that contains 1 but not 0. For a complex analytic extension, being bounded may be sufficient. Conjecture (Part 2): This is contingent on part 1, and if it is uniquely determined by these conditions, then $EF(0) = \gamma$, the Euler-Macheroni constant. Discussion: Starting with the definition of the exponential factorial $EF(x)=x^{EF(x-1)}$, and differentiating we get $EF'(x)=x^{EF(x-1)}(EF(x-1)/x+\ln(x)EF'(x-1))$ and evaluating at one, we get $EF'(1)=EF(0)$. So by finding EF(0) we are really finding the first coefficient of the power series expansion of EF (about x=1). A numerical approximation of the power series of EF using only these first principles will give a value of $EF(0) \approx 0.57$. Aside from the numerical approximations, if we assume that EF is invertible at one, then that means that EF'(1) is nonzero, which means that EF(0) is nonzero. Here is the first few real solutions. There seems to always be exactly 2 real solutions for every approximation, but the number of complex solutions increases with the approximation number. Code:M = Number of coefficients N = Number of solutions M N Solutions 2 2 {{0.575571, 0.151142}, {10.4244, 19.8489}} 3 2 {{0.575571, 0.229718, 0.0785761}, {10.4244, 39.0172, 19.1683}} 4 6 {{0.570807, 0.232292, 0.114153, 0.0234736}, {4.22694, 18.1158, -0.652778, -11.3147}, ...} These coefficients correspond to the functions: $EF(x)_2 = 1 + 0.575571(x-1) + 0.151142(x-1)^2$ $EF(x)_3 = 1 + 0.575571(x-1) + 0.229718(x-1)^2 + 0.0785761(x-1)^3$ $EF(x)_4 = 1 + 0.570807(x-1) + 0.232292(x-1)^2 + 0.114153(x-1)^3 + 0.0234736(x-1)^4$ Below I have attached of some graphs made with these approximations. Notice that values of the function for $x<-1$ are required to be complex. Attached Files   expfac2.pdf (Size: 13.25 KB / Downloads: 876)   expfac4.pdf (Size: 45.15 KB / Downloads: 791) « Next Oldest | Next Newest »

 Messages In This Thread open problems survey - by bo198214 - 05/17/2008, 10:03 AM eigenvalues of Carleman matrix for b^x, TPID 1 - by bo198214 - 05/17/2008, 10:23 AM Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM eigenvalues of Carleman matrix for (x+s)^p-s, TPID 3 - by bo198214 - 06/29/2008, 12:36 PM Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM sqrt(2) tetrational is completely discontinuous TPID 5 - by bo198214 - 05/01/2009, 09:20 AM Limit of self-super-roots is e^1/e. TPID 6 - by andydude - 10/07/2009, 12:03 AM A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM Elementary superfunction of polynomial without real fixed points TPID 8 - by bo198214 - 04/25/2010, 10:53 AM Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM Convergence of Eulers and Etas. TPID 10 - by dantheman163 - 10/31/2010, 07:13 PM RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM Tommy's conjecture about Eulers and Etas. TPID 11. - by tommy1729 - 12/01/2010, 03:56 PM convergence of self-tetra-root polynomial interpolation. TPID 12 - by bo198214 - 05/31/2011, 04:54 PM convergence of self-root polynomial interpolation. TPID 13 - by bo198214 - 05/31/2011, 07:02 PM Tommy's conjecture about andrew slog method. TPID 14 - by tommy1729 - 06/01/2011, 06:23 PM Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM Error terms for fake function theory TPID 17 - by tommy1729 - 03/28/2015, 10:48 PM The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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