DanA> Some other values might be checked. E.g., since Gamma(z) Gamma(1-z) == pi/sin(pi z), setting z = 1/2 + i gives |Gamma(1/2 + i)|^2 = 2pi /(e^pi + e^-pi). --Dan So again, the big mystery is the arg. Plot[Arg[Gamma[1/2 + t*I]], {t, -4, 4}] shows a nice smooth curve vaguely resembling x^3-7x. What's the slope at the mystery point t=1? In[260]:= D[Arg[Gamma[1/2 + t*I]], t] Out[260]= I*Gamma[1/2 + I*t]*PolyGamma[0, 1/2 + I*t]* Derivative[1][Arg][Gamma[1/2 + I*t]] In[261]:= N[% /. t -> 1] Out[261]= -0.772374 + 1.01843 I A perfectly smooth, perfectly real function has a non-real derivative. Is this Jan 1 or April 1? Macsyma does this easily: (c111) RATSIMP(NEGFACTORIAL(SUBST(1,T,DIFF(%,T)))); 2 %i + 1 2 %i - 1 psi (--------) + psi (- --------) 0 2 0 2 (d111) --------------------------------- 2 (c112) DFLOAT(%); (d112) - 0.05176165099441d0 --rwg In a different reply I said, (42) and (43) raise the interesting question of evaluating Product[((1/(12*n)) + 1)*((n - 1/2)*(E^(1/n) - 1))^n,{n, 1, Infinity}] (i.e., prod(((n-1/2)*(%e^(1/n)-1))^n*(1+1/12/n),n,1,inf) This is similar to the age-old question of Product[Cos[x/n], {n, 1, \[Infinity]}] and will probably require at least one new special function.