cp's OEIS Frontend

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

%I A365797 #35 Dec 01 2024 11:37:42
%S A365797 4,4,2,8,7,7,3,9,6,4,8,4,7,2,7,4,3,7,4,5,2,0,3,2,5,1,6,5,2,0,6,0,5,6,
%T A365797 7,1,7,1,0,3,6,4,5,3,8,0,6,6,3,6,6,4,0,2,9,9,1,2,3,0,7,1,9,8,9,5,8,5,
%U A365797 2,4,8,2,2,8,4,1,7,4,0,8,0,4,0,7,7,0,0,9,3,7,7,2,9,8,4,4,8,2,2,1,0,8,3,6,3,4
%N A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.
%C A365797 Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.
%C A365797 Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - _David Ulgenes_, Feb 19 2024
%C A365797 x is the least positive real number where 1+Gamma(1+Gamma(1+Gamma...(x)...)) converges; it converges to 3. - _Colin Linzer_, Nov 25 2024
%H A365797 K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, <a href="https://doi.org/10.1109/SYNASC.2017.00020">Properties and Computation of the Functional Inverse of Gamma</a>, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.
%H A365797 Mitsuru Uchiyama, <a href="https://doi.org/10.1090/S0002-9939-2011-11023-2">The principal inverse of the gamma function</a>, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.
%F A365797 Equals ((((1/2)!/2)!/2)!/2)!/2...
%F A365797 Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.
%F A365797 Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - _David Ulgenes_, Feb 11 2024
%e A365797 0.4428773964847274374520325165206056717103645380663664...
%p A365797 Digits:= 140:
%p A365797 with(RootFinding):
%p A365797 NextZero(x -> (x - 1)! - 2, 0);
%t A365797 FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]
%o A365797 (PARI) solve(x=0.1, 1, gamma(x)-2) \\ _Michel Marcus_, Sep 19 2023
%Y A365797 Cf. A030169, A030171.
%K A365797 nonn,cons
%O A365797 0,1
%A A365797 _David Ulgenes_, Sep 19 2023