Is there study on polynomial-like functions of the following kind? $$f(z) = c_0 + a_1z+b_1\bar{z} + a_2z^2+b_2\bar{z}^2 + ...+ a_nz^n+b_n\bar{z}^n$$
My reason for studying it is polynomials are analytic so that they can't approximate non-analytic complex functions with arbitrary precision. While the above function is non-analytic so they might do the approximation.
Any pointer is appreciated. Thanks!
Yes, there are studies under the title harmonic maps of the plane. In general, a harmonic map is of the form $f+\overline{g}$ where $f,g$ are analytic. Most of this literature is about univalent harmonic maps (in a region) but there are some papers on the general case, including polynomials and entire functions in the whole plane. Here is a small sample:
P. Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, 156. Cambridge Univ. Press, Cambridge, 2004.
A. Lyzzaik, Local properties of light harmonic mappings, Canadian J. Math., 44 (1992) 135–153.
D. Khavinson and G. Swiatek, On the maximal number of zeros of cer- ´ tain harmonic polynomials, Proc. Amer. Math. Soc., 131 (2003) 409–414.
W. Bergweiler and A. Eremenko, On the number of solutions of a transcendental equation arising in the theory of gravitational lensing, CMFT, Comput. Methods Funct. Theory 10 (2010), No. 1, 303--324.
S. Nakane, D. Schleicher, On multicorns and unicorns. I. Antiholomorphic dynamics, hyperbolic components and real cubic polynomials. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 10, 2825–2844.
