cp's OEIS Frontend

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

%I A377297 #45 Aug 27 2025 21:15:42
%S A377297 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,
%T A377297 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,
%U A377297 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
%N A377297 Decimal expansion of the smallest positive imaginary solution to Gamma(1+z) = Gamma(1-z).
%C A377297 All solutions are either purely real or purely imaginary. The smallest solution (by absolute value) happens to be purely imaginary.
%C A377297 When expressed in terms of Gauss's Pi function, it is
%C A377297 - The smallest solution to Pi(z) = Pi(-z).
%C A377297 - The smallest `y` such that: Pi(i*y) is purely real or, equivalently, Gamma(i*y) is purely imaginary.
%C A377297 - 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
%F A377297 Gamma(1+i*1.8055470716051069...) = Gamma(1-i*1.8055470716051069...) = 0.19754864094576264...
%F A377297 From _Jwalin Bhatt_, Aug 23 2025: (Start)
%F A377297 Smallest positive imaginary root of the equation x*sin(Pi*x)*Gamma(x)^2 = Pi.
%F A377297 Smallest positive real root of the equation Sum_{n>=1} x/n - arctan(x/n) = x * gamma where gamma = A001620. (End)
%e A377297 1.8055470716051069198763666... .
%p A377297 Digits:= 120:
%p A377297 fsolve(GAMMA(1+z*I)=GAMMA(1-z*I), z=0..3);  # _Alois P. Heinz_, Oct 25 2024
%t A377297 RealDigits[x /. FindRoot[Gamma[1 + x*I] == Gamma[1 - x*I], {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 23 2024 *)
%t A377297 RealDigits[x /. FindRoot[Re[Gamma[I*x]] == 0, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Vaclav Kotesovec_, Oct 25 2024 *)
%o A377297 (Python)
%o A377297 from mpmath import mp, nstr, factorial, findroot
%o A377297 mp.dps = 120
%o A377297 root = findroot(lambda z: factorial(z)-factorial(-z), 1.8j).imag
%o A377297 A377297 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']
%Y A377297 Cf. A000796, A001620, A212880, A377302.
%K A377297 nonn,cons,changed
%O A377297 1,2
%A A377297 _Jwalin Bhatt_, Oct 23 2024