∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 6 Theory

Complex Functions, Conformal Mappings & Contour Integration

Cauchy-Riemann equations, angle-preserving grid warps, and Cauchy's Residue Theorem

Morph a Cartesian grid z = x + iy under complex analytic mappings like w = z² and w = e^z, observing how orthogonal grid lines stay orthogonal at every intersection!

1. Conformal Mappings & The Cauchy-Riemann Equations

A complex function f(x + iy) = u(x,y) + i v(x,y) is complex-differentiable (holomorphic) if and only if its partials satisfy u_x = v_y and u_y = -v_x. Geometrically, its Jacobian matrix is a rotation-dilation, meaning it preserves angles between intersecting curves!

∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

Quick Reference & Cheat Sheet

Cauchy's Integral Formula & Residue Theorem
∮Cf(z)z−z0 dz=2πif(z0),∮Cg(z) dz=2πi∑Res(g,zk)\oint_C \frac{f(z)}{z - z_0}\,dz = 2\pi i f(z_0), \qquad \oint_C g(z)\,dz = 2\pi i \sum \text{Res}(g, z_k)
A closed contour integral depends solely on the singularities enclosed inside C.