Wrong:
(%i9) xxx : %e^-x*gamma(-x)*(x+1)^(x+1/2)*sin(%pi*x);
(%o9) %e^-x*gamma(-x)*(x+1)^(x+1/2)*sin(%pi*x)
(%i10) limit(xxx, x,inf);
(%o10) und
A correct value is
-%e * sqrt(%pi/2)
Negative integers are not in the natural domain of xxx, but xxx does analytically continue to the negative integers.