Conjecture (Part 1):
The exponential factorial (EF) is uniquely determined by the assumptions:
Conjecture (Part 2):
This is contingent on part 1, and if it is uniquely determined by these conditions, then = \gamma)
, the Euler-Macheroni constant.
Discussion:
Starting with the definition of the exponential factorial=x^{EF(x-1)})
, and differentiating we get =x^{EF(x-1)}(EF(x-1)/x+\ln(x)EF'(x-1)))
and evaluating at one, we get =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  \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.
These coefficients correspond to the functions:
_2 = 1 + 0.575571(x-1) + 0.151142(x-1)^2)
_3 = 1 + 0.575571(x-1) + 0.229718(x-1)^2 + 0.0785761(x-1)^3)
_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
are required to be complex.
The exponential factorial (EF) is uniquely determined by the assumptions:
- EF(x) is invertible for
- EF(x) is real analytic for
for
Conjecture (Part 2):
This is contingent on part 1, and if it is uniquely determined by these conditions, then
Discussion:
Starting with the definition of the exponential factorial
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