Visualizing Complex Integration

An interactive demonstration of Tristan Needham's approach.

Domain (z-plane)

Range (w-plane) & Integral Sum

Controls

Approximation Result:

Visual Explanation

This visualizes the integral ∫γ f(z)dz as the sum: ∑ f(zⱼ) ⋅ Δzⱼ

  1. The Path γ (Domain): The light purple curve on the left is the path of integration.
  2. ■ Displacement Vectors (Δzⱼ): The path is broken into small red arrows. Each is a tiny step along the path.
  3. ● Sample Points (zⱼ): At the midpoint of each red arrow, we place a green dot. This is where we'll evaluate the function.
  4. Transformation (w-plane): Each red arrow Δzⱼ is multiplied by f(zⱼ). This multiplication rotates and scales the arrow, resulting in a new blue arrow in the right-hand canvas.
  5. ■ The Sum: These new blue arrows are chained together head-to-tail.
  6. ■ The Integral: The final purple arrow, from the start of the chain to the end, is the total sum—our approximation of the integral.