A377297 Decimal expansion of the smallest positive imaginary solution to Gamma(1+z) = Gamma(1-z).

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

Offset: 1

Author

Jwalin Bhatt, Oct 23 2024

Keywords

• nonn
• cons
• changed

Comments

All solutions are either purely real or purely imaginary. The smallest solution (by absolute value) happens to be purely imaginary.

When expressed in terms of Gauss's Pi function, it is

- The smallest solution to Pi(z) = Pi(-z).

- The smallest `y` such that: Pi(i*y) is purely real or, equivalently, Gamma(i*y) is purely imaginary.

- Arg(Pi(i*y)) is given by Sum_{n>=1} y/n - arctan(y/n) - y*euler_gamma, so for Pi(i*y) to be purely real the argument must equal 2*Pi*k where k in some integer. - Jwalin Bhatt, Aug 23 2025

Examples

			1.8055470716051069198763666... .
		

Crossrefs

Cf. A000796, A001620, A212880, A377302.

Programs

Maple

Digits:= 120:
fsolve(GAMMA(1+z*I)=GAMMA(1-z*I), z=0..3);  # Alois P. Heinz, Oct 25 2024

Mathematica

RealDigits[x /. FindRoot[Gamma[1 + x*I] == Gamma[1 - x*I], {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 23 2024 *)
RealDigits[x /. FindRoot[Re[Gamma[I*x]] == 0, {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), 1.8j).imag
A377297 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']

Formula

Gamma(1+i*1.8055470716051069...) = Gamma(1-i*1.8055470716051069...) = 0.19754864094576264...

From Jwalin Bhatt, Aug 23 2025: (Start)

Smallest positive imaginary root of the equation x*sin(Pi*x)*Gamma(x)^2 = Pi.

Smallest positive real root of the equation Sum_{n>=1} x/n - arctan(x/n) = x * gamma where gamma = A001620. (End)