Functions annihilated by a power of the Cauchy-Riemann operator rather than by the operator itself: a q-analytic (polyanalytic) function on a planar domain satisfies (d/d zbar)^q f = 0, and equivalently decomposes uniquely as a sum over j < q of a_j(z) times zbar^j with each coefficient a_j holomorphic. Alpha-analytic functions generalise this to several complex variables by annihilating a multi-index power of the conjugate derivatives. The theory studies which properties of holomorphic functions survive the weakening and which fail: zero sets become curves rather than isolated points, the identity theorem and the maximum principle require boundary or dimension hypotheses, and uniqueness must be recovered from data on hypersurfaces. Central topics include polyanalytic Bergman and Fock spaces with their reproducing kernels, boundary value problems of Riemann-Hilbert type, approximation by polyanalytic polynomials and rational functions, value distribution and growth of entire polyanalytic functions, and the relation to monogenic and Clifford-algebra-valued analysis.
A graduate-level monograph on polyanalytic (q-analytic) functions in complex analysis: the distribution solutions of the equation obtained by applying the q-th power of the Cauchy-Riemann operator d/…