I would like to obtain the conformal map from a uniform rectilinear fluid flowing in the x-direction, where the field is bounded below by the x-axis, to the flow in the w-plane. In the w-plane the flow is correspondingly bounded from below by a trochoid. (A trochoid is a continuous waveform shaped something a sine wave but with pointier tops.) With [itex]z=x+iy[/itex] [itex]w=u+iv[/itex], the trochoid boundary is given parametrically as [itex]u=a \theta – b sin \theta[/itex] [itex]v= -a + b cos \theta[/itex] where a is greater than b. But how do I map the x-axis to the trochoid? There seem to be an infinity of maps. How do I select the correct one? This problem seems to be isomorphic to finding the electric field and equipotentials of a charged trochoid shaped conductor with the return conductor at [itex]y=\inf[/itex].