Holomorphic functions and the Cauchy–Riemann equations

Try

The functions z² and |z|² agree on the real axis, yet their behavior under imaginary increments differs sharply. At z=1, z² changes to first order by 2h in every complex direction. For |z|², a real increment has a nonzero first-order change while a purely imaginary increment has no first-order change.

A complex derivative is one limit over all nonzero complex increments approaching zero. It is more restrictive than the existence of two real partial derivatives, or agreement along a few selected lines.

Predict

Predict whether conjugation has a derivative. Along real increments its quotient is one; along imaginary increments it is minus one. A single complex derivative cannot equal both.

Now predict whether |z|² might still have a derivative at zero. A special point can satisfy the full limit even when no neighborhood does. The distinction between differentiability at a point and holomorphy on an open set will prevent several false conclusions later.

Experiment

Compare the real Jacobian with the matrix for complex multiplication.

Function	Real Jacobian	Complex derivative
z²	rows (2x,−2y), (2y,2x)	2z
conjugate(z)	rows (1,0), (0,−1)	Nowhere
∣z∣²	rows (2x,2y), (0,0)	Only at zero

In the final row, the vertical bars denote modulus, and the output is real. Use the matrix form to identify exactly which entries prevent complex linearity.

Understand

Write f=u+iv, where u and v are real-valued functions of x,y. If f is complex differentiable at a point, taking increments along the real direction gives f′=u_x+i v_x. Taking increments along the imaginary direction gives f′=v_y−i u_y. Equality yields the Cauchy–Riemann equations u_x=v_y and u_y=−v_x. They are necessary pointwise conditions.

Their sufficiency needs a genuine real differentiability hypothesis. Suppose f has a real-linear first-order approximation with remainder o(√(h²+k²)). The equations turn that linear approximation into multiplication by a=u_x+i v_x: Δf=a(h+ik)+o(|h+ik|). Dividing by the increment proves complex differentiability. Continuous first partial derivatives in a neighborhood are one convenient sufficient way to obtain real differentiability.

Existence of the partial derivatives and the equations at a single point, without a two-dimensional remainder estimate, does not justify this argument. Directional information can miss curved approaches.

A function is holomorphic on an open set when it is complex differentiable at every point there. A domain means a nonempty connected open set. Holomorphy is local, while later conclusions such as constancy or uniqueness often use connectedness to propagate local information across a domain.

Apply

Prove directly that |z|² has derivative zero at zero: its quotient by z is z̄, which tends to zero. Then use the necessary equations to rule out every other point. This gives a concrete example of isolated complex differentiability without holomorphy.

Repair this invalid proof: “The equations hold at the origin, so the function is analytic near the origin.” The claim changes both the set of points and the strength of the hypothesis. State what would need to be checked on an entire neighborhood.

Continue

Contour integration will reveal the surprising consequences of holomorphy: derivatives of all orders and convergent local power series. Those conclusions are theorems, not part of the derivative definition. Optional depth: interpret the real Jacobian of a nonzero complex derivative as an orientation-preserving similarity, connecting this lesson to conformal maps.