Velocity induced by a plane uniform vortex having the Schwarz function of its boundary with two simple poles

The vortex boundary is described through a conformal map z --> x which transforms the unit circle C in that curve, the Schwarz function of which is assigned as a function of z having two simple poles. It is shown that the equation in zita: x(zita)=x(z) posseses solutions zita=z and zita=zitastar, zitastar being a rational function of z. For this reason, the new transformed plane zitastar has to be also considered, on which the image curve zitastar(C) results to be still a circle, called Cstar. A general classification of the vortices having Schwarz functions of the above form is proposed, based on the relative positions of the circles C and Cstar and on the global behaviour of the map z-->x. Once a vortex is classified, the inverse map x-->z can be easily built. The image of the physical plane through the inverse function results to be a certain neighbourhood of C, which in turn is transformed in the complementary set by the map z-->zitastar. The velocity field is analytically evaluated by using the map z-->x and a new integral form of the relation between Schwarz function and self-induced velocity. This relation is completely equivalent to a previous one, based on the splitting of the Schwarz function in the sum of two functions, one analytic inside and the other one analytic outside the vortex.Nevertheless, it appears to be of a few interest, being often of simpler application. The dependence of the velocity field on the vortex shape is investigated by comparing velocity and streamfunction with the ones of the equivalent Rankine vortex (which has the same vorticity, area and center of vorticity). By changing the parameters of the Schwarz function (poles and corresponding residues), rather complicated vortex shapes can be easily analyzed, some of them mimicing an incipient filamentation of the vortex boundary.