the Wirtinger derivatives
/ VEER-ting-er /
The Wirtinger derivatives are a clever bookkeeping device that treats z = x + i y and its conjugate z-bar = x - i y as if they were two independent variables, and differentiates with respect to each. They are defined by d/dz = (1/2)(d/dx - i d/dy) and d/dz-bar = (1/2)(d/dx + i d/dy), combinations of the ordinary partials in x and y.
The whole point is the slogan they make precise: a function is holomorphic exactly when df/dz-bar = 0 — that is, when f does not depend on z-bar at all. If you write f in terms of x and y, regroup into z and z-bar, and find no z-bar appears, the function is holomorphic; if z-bar survives (as in f = z-bar, or |z|^2 = z times z-bar), it is not. A direct expansion shows df/dz-bar = (1/2)[(u_x - v_y) + i(u_y + v_x)], which vanishes precisely when u_x = v_y and u_y = -v_x — the Cauchy-Riemann equations in disguise.
Used carefully the Wirtinger calculus is a real labour-saver: it lets you compute complex derivatives by formal manipulation, treating z-bar as constant when differentiating in z, and it generalizes cleanly to several complex variables. The honest caveat is that z and z-bar are NOT actually independent (each determines the other), so the notation is a formal convenience; the operators are genuine linear differential operators on smooth functions, and the slogan 'holomorphic = independent of z-bar' is shorthand for the precise statement df/dz-bar = 0.
For f = |z|^2 = z times z-bar: d/dz-bar (z z-bar) = z (treating z as constant in z-bar), which is nonzero off the origin. So |z|^2 is not holomorphic — matching the fact that its Cauchy-Riemann equations fail except at 0.
df/dz-bar = 0 is the Cauchy-Riemann equations repackaged into a single clean operator equation.
z and z-bar are not truly independent variables — the notation is a formal convenience. It works because the operators d/dz and d/dz-bar are well-defined real linear combinations of d/dx and d/dy.