Mind you, I did notice the $\displaystyle \frac{i}{t-i}$ thing and thought it is a little odd the second assumed period is pure imaginary, but of course I didn't check it (as I am not checking it, either

) .

Well, then the two periods are a $\displaystyle \mathbb{R}-$basis for the complex, so then the only reason why that function wouldn't be an elliptic one I can think of right now is that the function isn't meromorphic on a fundamental parallelogram. Now, either the function is there holomorphic and thus bounded and thus constant (but STILL would be consider elliptic, imo), or else it has a singularity that it is not a pole....oh, I think I see now! Zero is there...hmmm.

Anyway, it's late here so I shall check this closer tomorrow, perhaps.