Velocity Induced by a Plane Uniform Vortex Having the Schwarz Function of Its Boundary with Two Simple Poles

Table 3

Branches to be used for the square roots (𝐱−𝐱⁺_𝑖𝑜)^1/2 and (𝐱−𝐱⁻_𝑖𝑜)^1/2 in any 𝐱∈ℰ, expressed through the index 𝑘 (𝑘=2 equal, 𝑘=1 different). They are specified in terms of the kind of the vortex with respect to the first property (first column) and of the phase difference Δ𝜔^⋆ (second column). Here it has been assumed that sin𝜃^⋆>0, while if sin𝜃^⋆<0, the complementary value 3−𝑘 has to be considered.

Kind	Range of Δ𝜔^⋆	𝑘
1 or 3 with 𝒞̊𝑖(𝑡)⊃𝒞^⋆	(−2𝜋,−𝜋)∪(0,+𝜋)	1
	(−𝜋,0)∪(+𝜋,+2𝜋)	2
2 or 3 with 𝒞̊𝑖(𝑡)⊃𝒞	(−2𝜋,−𝜋)∪(0,+𝜋)	2
	(−𝜋,0)∪(+𝜋,+2𝜋)	1