Cauchy-Riemann Equations

Definition

The Cauchy-Riemann (C-R) equations are a pair of partial differential equations that provide a necessary condition for a complex function $f(z) = u+iv$ to be differentiable: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

How to read: “The partial derivative of u with respect to x equals the partial derivative of v with respect to y, and the partial derivative of u with respect to y equals the negative partial derivative of v with respect to x.”
Meaning: The real part $u$ and imaginary part $v$ must satisfy these coupled derivative relations—otherwise $f(z)$ fails to be complex-differentiable (holomorphic).

Why It Matters

They provide the essential condition for complex functions to be “smooth,” enabling the powerful tools of complex analysis to be applied to problems in physics and engineering.

Core Concepts

Analytic Function: A function that is differentiable at every point in a region. Such functions are incredibly rigid; if you know an analytic function in a small patch, its behavior is determined everywhere (Analytic Continuation).
Differentiability vs. Analyticity: Satisfying C-R equations and having continuous partial derivatives is sufficient for analyticity.
Harmonic Functions: If $f=u+iv$ $f = u + i v$ is analytic, then $u$ $u$ and $v$ $v$ are harmonic conjugate functions, satisfying Laplace’s equation: $\nabla^2 u = 0$ $\nabla^{2} u = 0$ and $\nabla^2 v = 0$ $\nabla^{2} v = 0$ .
- How to read: “The value nabla squared u equals zero and nabla squared v equals zero.”
- Meaning: Real and imaginary parts are both harmonic (Laplacian vanishes)—no local maxima/minima in their interior; useful in physics (potential theory).
Orthogonality: The level curves $u(x, y) = c_1$ $u (x, y) = c_{1}$ and $v(x, y) = c_2$ $v (x, y) = c_{2}$ are always orthogonal at their points of intersection.
- How to read: “The function u of x comma y equals c one, and the function v of x comma y equals c two.”
- Meaning: Level curves of $u$ and $v$ are orthogonal wherever $f$ is analytic—they form a mutually perpendicular coordinate grid.

Cauchy-Riemann Equations

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes