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This research is about computing the Green's functions on unbounded doubly connected regions by using the method of boundary integral equation. The method depends on solving an exterior Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystr??m method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gaussian elimination method. Mathematica plots of Green's functions for several test regions are also presented. [ABSTRACT FROM AUTHOR]