Polyanalytic functions with equal modulus

Abstract

In [1], M. B. Balk obtains a representation for polyanalytic functions which have constant modulus on some region. Balk's result suggests the more difficult problem of obtaining necessary and sufficient conditions that two polyanalytic functions have equal modulus on some region. This problem is answered in Theorem 1 and as an immediate corollary to Theorem 1, we obtain the result of Balk cited in [1]. In Theorem 2 and subsequent corollaries, the question of establishing criteria that two polyanalytic functions have equal amplitudes or equal real parts or equal imaginary parts is considered. In what follows, G will denote a region of the finite complex plane r and G* will denote the region G*= {zEr|EeG}. DEFINITION. A function f: G-,r is said to be polyanalytic on G or n-analytic on G if and only if there exist n ?1 functions fo, fl, fr-i which are analytic on G such thatf _1 is not identically zero on G whenever n > 2 and such that f(z) = EZkfk (z), for all z E G, where k=0, 1, * * * , n-I and where z denotes the complex conjugate of z. We now need the following preparatory lemma: