Complex Limits, Continuity, Differentiability & Cauchy-Riemann Equations (Cartesian & Polar)

Explanation

Complex differentiability is vastly more restrictive than real multivariable differentiability. In ℝ², Δz = Δx + iΔy can approach 0 from infinitely many radial and spiral directions. Equating the horizontal limit (Δy = 0, Δx → 0) with the vertical limit (Δx = 0, Δy → 0) yields the C-R equations. When satisfied with continuous partial derivatives, conformal geometry, infinite differentiability, and analyticity follow immediately.

Key points

• Derivative: $f^{\prime }(z)={\lim }_{\Delta z\to 0}\frac{f(z+\Delta z)-f(z)}{\Delta z}$. Must be identical along every approaching path in the complex plane.
• Analytic Function: $f(z)$ is analytic at $z_0$ if it is differentiable at $z_0$ and throughout some open neighborhood around $z_0$.
• Cauchy-Riemann (C-R) Equations in Cartesian Coordinates: $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$.
• Necessary Condition: If $f(z)=u+iv$ is differentiable at $z$, C-R equations MUST hold.
• Sufficient Condition: If $u,v$ and all four first partial derivatives $u_x,u_y,v_x,v_y$ exist, are continuous, and satisfy C-R equations, then $f(z)$ is analytic.
• Cauchy-Riemann Equations in Polar Coordinates ($z=r{e}^{i\theta },f(z)=u(r,\theta )+iv(r,\theta )$): $\frac{\partial u}{\partial r}=\frac{1}{r}\frac{\partial v}{\partial \theta }$ and $\frac{\partial v}{\partial r}=-\frac{1}{r}\frac{\partial u}{\partial \theta }$.

Example

Code examples

// Rigorous Derivation of Cauchy-Riemann Equations in Cartesian Form:
// Let f(z) = u(x, y) + i v(x, y) be differentiable at z = x + iy.
// By definition: f'(z) = lim_{Δz -> 0} [ Δw / Δz ]
// where Δw = [u(x + Δx, y + Δy) - u(x, y)] + i [v(x + Δx, y + Δy) - v(x, y)]
// and Δz = Δx + i Δy.
//
// Path 1: Approach along the Real Axis (Horizontal: Δy = 0, Δz = Δx -> 0):
//   f'(z) = lim_{Δx -> 0} [ (u(x+Δx, y) - u(x, y))/Δx + i (v(x+Δx, y) - v(x, y))/Δx ]
//   f'(z) = ∂u/∂x + i (∂v/∂x)  ---------- (Equation 1)
//
// Path 2: Approach along the Imaginary Axis (Vertical: Δx = 0, Δz = i Δy -> 0):
//   f'(z) = lim_{Δy -> 0} [ (u(x, y+Δy) - u(x, y))/(i Δy) + i (v(x, y+Δy) - v(x, y))/(i Δy) ]
//   Since 1/i = -i:
//   f'(z) = -i (∂u/∂y) + ∂v/∂y
//   f'(z) = ∂v/∂y - i (∂u/∂y)  ---------- (Equation 2)
//
// Since f(z) is differentiable, the limits along Path 1 and Path 2 must be EQUAL:
//   ∂u/∂x + i (∂v/∂x) = ∂v/∂y - i (∂u/∂y)
//
// Equating Real and Imaginary parts:
//   ∂u/∂x = ∂v/∂y   and   ∂u/∂y = -∂v/∂x  (Q.E.D.!)

Exam questions

• [8 Marks] Derive the Cauchy-Riemann equations in polar coordinates: ∂u/∂r = (1/r)(∂v/∂θ) and ∂v/∂r = -(1/r)(∂u/∂θ).
• [8 Marks] Prove that f(z) = |z|² is differentiable only at the origin, but is nowhere analytic.
• [7 Marks] If f(z) = u + iv is an analytic function and u - v = (x - y)(x² + 4xy + y²), find f(z) in terms of z.