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

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, 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, 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

Offset: 0

Author

David Ulgenes, Sep 19 2023

Keywords

• nonn
• cons

Comments

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.

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

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

Examples

			0.4428773964847274374520325165206056717103645380663664...
		

Links

• K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 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.
• Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Crossrefs

Cf. A030169, A030171.

Programs

Maple

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

Mathematica

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

PARI

solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Formula

Equals ((((1/2)!/2)!/2)!/2)!/2...

Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.

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