Calculation Pi^Pi^Pi in hexadecimal in High precision

Specific Domains Numerics
52

Seems that using higher precision Julia packages I’ve discovered that log(gamma( is related to PI ;
per using SpecialFunctions and using Nemo ?

using Julia SpecialFunctions and Nemo seems I’ve discovered that loggamma(-1 + 0i) = Infinity - i*pi (a Complex number) .
Also Wolframalpha answer is incomplete About: loggamma(-1 + 0i) or logΓ(-1 + 0 i)
Try this link loggamma(-1 + 0i) - Wolfram|Alpha
and you’ll see Wolframalpha shows loggamma(-1 + 0i) = Infinity

However it seems I’ve discovered/uncovered that loggamma(-1 + 0i) = Infinity - i*pi (a Complex number) per the following :

## show using SpecialFunctions 
## loggamma(-1 + 0i) = Infinity - i*pi (a Complex number)
julia> loggamma(-1+0im)
Inf - 3.141592653589793im

julia> imag(loggamma(-1+0im))/pi
-1.0

## Showing Effects of error propagation ?
julia> log(gamma(-1+0im))
Inf - 2.356194490192345im


## Reconfirmed suspicion via show using Nemo to 64 Bits / 18 Decimals
using Nemo # for arbitrary precision math calculations 
CC = ComplexField(64)

julia> using Nemo
julia> CC = ComplexField(64)
Complex Field with 64 bits of precision and error bounds

julia> RR = RealField(64)
Real Field with 64 bits of precision and error bounds

julia> Nemo.lgamma(RR("-1") + CC("0"))
nan + i*[-3.141592653589793238 +/- 5.14e-19]
julia> Nemo.const_pi(RR)
         [3.141592653589793239 +/- 5.96e-19]

–

Also here’s an analytical/symbolic equation question :
Is this the same or a different fact than the Euler Identity ?

Maybe start here Gamma, Beta, Erf > LogGamma[z]

Representations through more general functions >> Logarithm of the gamma function: Representations through more general functions (subsection 26/01)