Can we directly calculate some function involves complex arguments by the result of real arguments?
For example, the beta function $$\rm{Beta}(n+1,2) = \Gamma(n+1)/\Gamma(n+3) = 1/((n+1)(n+2))$$ $$|\rm{Beta}(jn+1,2)| = |\Gamma(jn+1)/\Gamma(jn+3)| = 1/\sqrt{(n^2+1)(n^2+4)}$$ and by Wolfram Mathematica, it seems like $\Gamma(jn+1)/\Gamma(jn+3) = 1/((jn+1)(jn+2))$.
Can we confirm $\rm{Beta}(jn+1,2) = 1/((jn+1)(jn+2))$? Can we apply the same rule to other functions?
Thanks in advance!