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

1,2

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

Links

Table of n, a(n) for n=1..105.

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)

Example

1.8055470716051069198763666... .

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 *)

Prog

(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 != '.']

Crossrefs

Cf. A000796, A001620, A212880, A377302.

Sequence in context: A308739 A118253 A135001 * A146293 A019724 A195400

Adjacent sequences: A377294 A377295 A377296 * A377298 A377299 A377300

Keyword

nonn,cons

Author

Jwalin Bhatt, Oct 23 2024

Status

approved