Complex Analysis

Mathematical Principles & Theorems

Complex numbers \(z = x + iy = r e^{i\theta}\) in the Argand plane \(\mathbb{C}\). Holomorphic functions \(f(z) = u(x,y) + i v(x,y)\) satisfy the Cauchy-Riemann equations \(\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\) and \(\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\). Conformal maps preserve angles and local geometry. Roots of unity \(z^n = 1 \implies z_k = e^{i 2\pi k / n}\) form regular vertices of cyclic polygons in the unit circle.

Operating Instructions

• Select visualization mode: Argand Vector Arithmetic, Roots of Unity, or Conformal Mapping.
• Input complex numbers \(z_1, z_2\) or choose transformation function \(f(z)\).
• Drag vector heads in the complex plane to interactively observe modulus \(|z|\) and argument \(\arg(z)\).
• Inspect grid line deformation on the conformal stage to verify orthogonal curve preservation.