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open problems survey
#3
Conjecture (Part 1):
The exponential factorial (EF) is uniquely determined by the assumptions:


  1. EF(x) is invertible for
  2. EF(x) is real analytic for
  3. for
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 , the Euler-Macheroni constant.

Discussion:
Starting with the definition of the exponential factorial , and differentiating we get and evaluating at one, we get . 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 .

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:





Below I have attached of some graphs made with these approximations. Notice that values of the function for are required to be complex.


Attached Files
.pdf   expfac2.pdf (Size: 13.25 KB / Downloads: 635)
.pdf   expfac4.pdf (Size: 45.15 KB / Downloads: 557)
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Messages In This Thread
open problems survey - by bo198214 - 05/17/2008, 10:03 AM
Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM
Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM
A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM
Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM
RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM
Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM
The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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