Some analytical properties of solutions to a two-dimensional steadily translating inviscid bubble

We consider some analytical properties for a steadily translating two-dimensional bubble in an inviscid irrotational flow when surface tension effects are included but gravity neglected. For a general value of the Bernoulli constant, we show that a conformal mapping function from a unit circle to the exterior of the bubble has a very specific polar decomposition. The pole locations and corresponding residues are determined numerically, while, in specific limits, their values are determined asymptotically.