A377302 Decimal expansion of the smallest positive real solution to Gamma(1+z) = Gamma(1-z).

2, 3, 6, 1, 1, 9, 1, 0, 8, 7, 1, 6, 3, 4, 1, 6, 6, 3, 4, 4, 9, 7, 3, 4, 1, 0, 3, 9, 6, 3, 2, 4, 0, 3, 7, 4, 3, 5, 4, 8, 5, 2, 8, 7, 1, 5, 7, 2, 5, 8, 1, 3, 5, 9, 6, 1, 0, 1, 9, 0, 4, 4, 3, 1, 6, 9, 2, 1, 3, 7, 4, 1, 0, 3, 7, 9, 1, 0, 9, 6, 4, 9, 3, 2, 0, 2, 2, 5, 2, 9, 0, 7, 4, 7, 5, 2, 8, 4, 6, 8, 7, 1, 1, 4, 9

Offset: 1

Author

Jwalin Bhatt, Oct 23 2024

Keywords

• nonn
• cons
• changed

Comments

This is the second smallest solution, the smallest one being purely imaginary A377297.

When expressed in terms of Gauss's Pi function, it is the smallest real solution to Pi(z) = Pi(-z).

Examples

			2.36119108716341663449734103963240374354852871572581359610190443169213741...
		

Crossrefs

Cf. A000796, A001620, A377297.

Programs

Maple

Digits:= 120:
fsolve(GAMMA(1+z)=GAMMA(1-z), z=1..3);  # Alois P. Heinz, Oct 25 2024

Mathematica

RealDigits[x /. FindRoot[Gamma[1 + x] == Gamma[1 - x], {x, 5/2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 23 2024 *)
RealDigits[x /. FindRoot[x*Sin[Pi*x]*Gamma[x]^2 == Pi, {x, 2}, WorkingPrecision -> 120]][[1]](* Vaclav Kotesovec, Oct 25 2024 *)

Python

from mpmath import mp, nstr, factorial, findroot
mp.dps = 120
root = findroot(lambda z: factorial(z)-factorial(-z), 2.4)
A377302 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']

Formula

Gamma(1+2.3611910871634166...) = Gamma(1-2.3611910871634166...) = 2.8607322727573070...

Smallest positive real root of the equation x*sin(Pi*x)*Gamma(x)^2 = Pi. - Vaclav Kotesovec, Oct 25 2024

Smallest positive real root of the equation Sum_{n>=1} x/n - arctanh(x/n) = x*euler_gamma - i*Pi. - Jwalin Bhatt, Aug 23 2025