The spectral method for an unbounded domain is developed and presented. For the expansion in the radial direction of polar coordinates \((r,\varphi)\) or \((r, \varphi, z)\) rational basis functions which are algebraically mapped Legendre functions are used. These functions satisfy the pole condition exactly at the coordinate singularity and their behavior as \(r\to\infty\) is suitable for expanding smooth functions which decay algebraically or exponentially as \(r\to\infty\). This method is not stiff when it is applied to an initial value problem despite the presence of coordinate singularity. Solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2. Examples which include the computation of vortex dynamics in two and three dimensions are also given.