Questions: Complex Differentiability

Question 1

The function f(z) = |z|² is smooth as a function of two real variables (x,y), yet it fails to be complex differentiable except at one point. Why?

• A. Because |z|² is not continuous in the complex plane
• B. Because the limit of the difference quotient depends on the direction h approaches 0
• C. Because |z|² cannot be expressed in the form u + iv
• D. Because the partial derivatives of u and v do not exist anywhere

Answer: (B)

In the complex sense, the difference quotient [|z₀+h|² − |z₀|²]/h must give the same result regardless of the direction h → 0. Computing along the real axis (h = a) versus the imaginary axis (h = bi) gives different limits, so the derivative doesn't exist. Real differentiability only requires that a linear approximation exists as a map ℝ² → ℝ² — it doesn't enforce path-independence of the complex ratio [f(z₀+h)−f(z₀)]/h.

Question 2

A function f(z) is complex differentiable at every point in an open region. Which of the following is an immediate consequence?

• A. f has exactly one complex derivative at each point, but not necessarily two
• B. f must be a polynomial
• C. f is automatically differentiable infinitely many times and equals its Taylor series on that region
• D. f must satisfy the triangle inequality at every point

Answer: (C)

This is the remarkable rigidity of holomorphic functions. In real analysis, being once differentiable says nothing about twice differentiable. In complex analysis, one 'free' derivative on an open set — holomorphicity — implies infinite differentiability and analyticity (equality with the Taylor series) for free. This has no real analogue and is one of the defining features of complex analysis.

Question 3

Complex differentiability is equivalent to real differentiability when f is viewed as a map from ℝ² to ℝ².

• T. True
• F. False

Answer: False

Real differentiability (as a map ℝ² → ℝ²) requires only that a linear approximation exists, placing no constraint on direction. Complex differentiability additionally requires that the ratio [f(z₀+h)−f(z₀)]/h gives the same complex number regardless of the direction h approaches zero — the Cauchy-Riemann condition. This rules out many real-differentiable functions, including f(z) = |z|², which is smooth on ℝ² but not complex differentiable.

Question 4

If a function satisfies the Cauchy-Riemann equations at a point, it is necessarily complex differentiable at that point.

• T. True
• F. False

Answer: False

Satisfying the Cauchy-Riemann equations is necessary but not sufficient. The partial derivatives of u and v must also be continuous at that point. A function can satisfy the Cauchy-Riemann equations at an isolated point while having discontinuous partials there, and in that case the complex derivative does not exist. The complete theorem requires: Cauchy-Riemann equations hold AND the partial derivatives are continuous.

Question 5

Explain why the requirement that the complex derivative be path-independent is so much more restrictive than real differentiability.

Model answer: In ℝ, h can only approach 0 from two directions (positive or negative). In ℂ, h is a complex number and can approach 0 from infinitely many directions — along the real axis, imaginary axis, any angle, any curved path. The derivative limit must give exactly the same complex number along all of them. This single constraint algebraically forces the Cauchy-Riemann equations, which rigidly link the partial derivatives of the real and imaginary parts. Real differentiability only requires a linear approximation to exist with no constraint coupling different directions.

Path-independence is the gateway to all of complex analysis. It forces harmonic conjugates, conformal mappings, and the remarkable rigidity that makes holomorphic functions infinitely differentiable from a single differentiability assumption.