The complex form of Vekua's characteristic factor: a derivation, and two sign corrections in §7 of Generalized Analytic Functions
Abstract
In 7 of Generalized Analytic Functions vekua, the reduction of a first-order elliptic system to canonical form proceeds through a factor of the characteristic equation, which Vekua selects in real form~(7.13) and then restates, without derivation, in complex (Beltrami) form~(7.14). We supply that conversion. With the standard Wirtinger convention used below, the complex form of~(7.13) is the negative of the coefficient printed in~(7.14) (p.~126, 1962 Pergamon edition), and we confirm the correct sign against Vekua's own factorization~(7.12) and his canonical coefficient~(7.17). A related sign defect appears in the second-order Beltrami coefficient~(7.23) (p.~127): a coordinate solving (7.23) as printed reduces the equation to the canonical form~(7.26) only in the special symmetric case a=c. In both instances the error is confined to the displayed coefficient and leaves the surrounding reduction, carried out independently of it, intact; we record the corrected coefficient in each case.
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