Simplify or FullSimplify doesn't work on Gamma function

Question

I have this expression:

Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I]

which returns real values for all $n$. However, neither Simplify nor FullSimplify can reduce it to a form without involving $i$. Is there a way to force these commands to do a better job?

Answer 1

Check to see that the imaginary part is so small that you are "sure" it is zero. If so return just the real part:

\[Epsilon] = 10^-10;
g[n_] := Gamma[n - 0.691 + 2.35 I] Gamma[n - 0.691 - 2.35 I]
h[n_] := If[Im[g[n]] < \[Epsilon], Re[g[n]], g[n]]

g[55.5] returns 1.04586*10^142 + 0. I

h[55.5] returns 1.04586*10^142

Answer 2

Using the fact that $\overline{\Gamma(x)}=\Gamma(\overline{x})$, you can rewrite your expression as

Abs[Gamma[n - 0.691 + 2.35 I]]^2

which is manifestly real (even though it still contains an I). Honestly, I would be very surprised if you found a useful expression without any I, so this might be the best you can do.