Questions: Cauchy-Riemann Equations

Question 1

A function f(z) = u(x,y) + iv(x,y) has continuous partial derivatives at z₀ and satisfies ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x there. Which conclusion follows?

• A. f is real-differentiable at z₀ as a map from ℝ² to ℝ², but not necessarily complex-differentiable
• B. f is holomorphic (complex-differentiable) at z₀
• C. u and v are each harmonic on all of ℂ
• D. f can be extended to a real-analytic function on a neighborhood of z₀

Answer: (B)

The Cauchy-Riemann equations together with continuity of partial derivatives are necessary and sufficient for complex differentiability at a point. Option A is wrong: satisfying the CR equations with continuous partials implies complex differentiability, which is stronger than (and implies) real differentiability, not the other way around. Option C overstates scope — harmonicity holds locally at the point, not necessarily on all of ℂ.

Question 2

What is the geometric meaning of the Cauchy-Riemann equations, distinguishing holomorphic maps from arbitrary real-differentiable maps?

• A. They force u and v to be polynomials, restricting holomorphic functions to algebraic expressions
• B. They require f to map circles to circles, ruling out functions with varying scale
• C. They restrict the local linear approximation to rotation and uniform scaling, ruling out shear or differential stretching
• D. They require the Jacobian matrix of f to have positive determinant, ensuring orientation preservation

Answer: (C)

A real-differentiable map from ℝ² to ℝ² can have any 2×2 Jacobian. The CR equations restrict the Jacobian to matrices of the form [[a, −b],[b, a]], corresponding precisely to multiplication by the complex number a + ib — a rotation and uniform scaling. This eliminates shear, differential stretching, and orientation-reversing maps. Option D is weaker: positive determinant only ensures orientation preservation, whereas CR equations impose the much stronger condition of conformality.

Question 3

If f(z) = u(x,y) + iv(x,y) is holomorphic on a simply connected domain, the harmonic conjugate of u is determined uniquely.

• T. True
• F. False

Answer: False

Given a harmonic function u on a simply connected domain, the CR equations can be integrated to find v such that f = u + iv is holomorphic. But v is determined only up to an additive real constant: if v₀ is one harmonic conjugate, then v₀ + c for any real constant c also satisfies the CR equations with u. This mirrors how antiderivatives in single-variable calculus are defined only up to a constant.

Question 4

Complex differentiability is strictly stronger than real differentiability: there exist functions that are real-differentiable everywhere but nowhere holomorphic.

• T. True
• F. False

Answer: True

A real-differentiable function f: ℝ² → ℝ² only requires the linear approximation to exist; it can have any 2×2 Jacobian. Complex differentiability additionally requires that Jacobian to satisfy the CR equations. The function f(z) = z̄ (complex conjugate) is real-differentiable everywhere — its Jacobian is the constant matrix [[1, 0],[0, −1]] — but nowhere holomorphic because it reverses orientation and violates the CR equations (∂u/∂x = 1 but ∂v/∂y = −1).

Question 5

Derive the Cauchy-Riemann equations from the definition of complex differentiability by computing the limit of [f(z₀+Δz)−f(z₀)]/Δz along two different paths.

Model answer: Write f(z) = u(x,y) + iv(x,y). Along the real axis (Δz = Δx): the limit is ∂u/∂x + i∂v/∂x. Along the imaginary axis (Δz = iΔy): the limit is (1/i)∂u/∂y + ∂v/∂y = ∂v/∂y − i∂u/∂y. Setting the two limits equal — real parts equal and imaginary parts equal — gives ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x.

The derivation reveals why complex differentiability is so restrictive: the limit must be path-independent. Even requiring agreement along just two axis directions forces a coupling between all four partial derivatives. When partial derivatives are also continuous, this two-direction agreement is enough to guarantee the limit is the same from every direction — a non-trivial fact that makes holomorphic functions far more special than real-differentiable ones.