The Clairaut–Schwarz Theorem for Mixed Wirtinger Derivatives

In this note on the foundations of complex analysis, we present for Wirtinger derivatives a short proof of the analogue of the Clairaut–Schwarz theorem. It turns out that, via Fubini’s theorem for disks, it is a consequence of the complex version of the Gauss–Green formula relating planar integrals on disks to line integrals on the boundary circle. At the same time, we obtain a version of Pompeiu’s formula for partial aerolar derivatives (equivalent to Wirtinger derivatives) in several complex variables.