For a non-degenerate planar harmonic mapping, the number of pre-images of any non-caustic point equals the total pole contribution plus twice the winding number of the caustics around that point.
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Number and location of pre-images under harmonic mappings in the plane
For a non-degenerate planar harmonic mapping, the number of pre-images of any non-caustic point equals the total pole contribution plus twice the winding number of the caustics around that point.