This paper presents the exact solution of the 2D modified Helmholtz equation which governs the discharge potential for a semi-confined rectangular aquifer with leakage due to line elements and imposing no-flow conditions on the boundaries. An application example is given for a constant straight line-sink at the centre of the aquifer and a rapidly convergent series representation of the solution is obtained involving only elementary functions. A precise comparison is performed with the Bessel function-based solutions usually derived assuming the aquifer of infinite extent.