How constant shifts affect the zeros of certain rational harmonic functions [CL]

We study the effect of constant shifts on the zeros of rational harmomic functions $f(z) = r(z) – \conj{z}$. In particular, we characterize how shifting through the caustics of $f$ changes the number of zeros and their respective orientations. This also yields insight into the nature of the singular zeros of $f$. Our results have applications in gravitational lensing theory, where certain such functions $f$ represent gravitational point-mass lenses, and a constant shift can be interpreted as the position of the light source of the lens.

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J. Liesen and J. Zur
Mon, 27 Feb 17

Comments: 26 pages, 9 figures