Some Lemmas on Doubly Quasi-Periodic Analytic Functions in Multiplication

In this paper, some lemmas on doubly quasi-periodic analytic functions in multiplication are proved. Suchfunctions may not be identical to zero even if their real parts vanish on the boundary. Conditions for in which this case appears is also obtained. A concrete example is given to show that this case actually exists. Finally, the general solution of the considered Dirichlet problem of doubly quasi-periodic analytic functions with zero real parts on the boundary is obtained, provided the multipliers are not prescribed.