Paper 2, Section II, 23G
Part II, 2011
Let be a lattice in generated by 1 and , where is a fixed complex number with non-zero imaginary part. Suppose that is a meromorphic function on for which the poles of are precisely the points in , and for which as . Assume moreover that determines a doubly periodic function with respect to with for all . Prove that:
(i) for all .
(ii) is doubly periodic with respect to .
(iii) If it exists, is uniquely determined by the above properties.
(iv) For some complex number satisfies the differential equation .