A377297 - OEIS

%I #27 Nov 22 2024 11:14:21

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

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

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

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

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

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

%e 1.8055470716051069198763666... .

%p Digits:= 120:

%p fsolve(GAMMA(1+z*I)=GAMMA(1-z*I), z=0..3);  # _Alois P. Heinz_, Oct 25 2024

%t RealDigits[x /. FindRoot[Gamma[1 + x*I] == Gamma[1 - x*I], {x, 2}, WorkingPrecision -> 120]][[1]] (* _Amiram Eldar_, Oct 23 2024 *)

%t RealDigits[x /. FindRoot[Re[Gamma[I*x]] == 0, {x, 2}, WorkingPrecision -> 120]][[1]] (* _Vaclav Kotesovec_, Oct 25 2024 *)

%o (Python)

%o from mpmath import mp, nstr, factorial, findroot

%o mp.dps = 120

%o root = findroot(lambda z: factorial(z)-factorial(-z), 1.8j).imag

%o A377297 = [int(d) for d in nstr(root, n=mp.dps)[:-1] if d != '.']

%Y Cf. A377302.

%K nonn,cons

%O 1,2

%A _Jwalin Bhatt_, Oct 23 2024