Cauchy-Riemann Equations — Mathematics Knowledge System

Overview

The Cauchy–Riemann equations are a pair of partial differential equations that characterize when a complex function is differentiable in the complex sense (holomorphic). Writing a complex function f(z) = f(x+iy) = u(x,y) + i v(x,y) in terms of its real part u and imaginary part v, f is complex-differentiable at a point exactly when u and v satisfy ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x there (together with u, v having continuous partial derivatives, which gives complex differentiability, not just satisfaction of the equations). These equations are far more restrictive than ordinary real differentiability: they force a two-variable real function pair (u, v) to behave, as a whole, like a single smooth complex function — the root reason complex analysis is so much more rigid than real analysis.

Intuition

In single-variable real calculus, a function is differentiable if it looks locally like a straight line. In complex analysis, being differentiable means f looks locally like multiplication by a single complex number f'(z) — which, geometrically, is always a rotation combined with a uniform scaling, never a shear or a direction-dependent stretch. If you approach z along the real direction versus the imaginary direction, both directional derivatives must agree with this same rotate-and-scale action. Writing that agreement out in terms of the real functions u(x,y) and v(x,y) that make up f produces exactly the Cauchy–Riemann equations: they are the algebraic signature of 'no shearing, only a uniform rotation and scaling' at every point.

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Formal Definition

Definition

Let f(z) = u(x,y) + i v(x,y) where z = x + iy and u, v are real-valued functions with continuous first partial derivatives in a neighborhood of a point. Then f is complex-differentiable (holomorphic) at that point if and only if:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

First Cauchy–Riemann equation

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

Second Cauchy–Riemann equation

f(z) = f(x+iy) = u(x,y) + i\,v(x,y)

Decomposition of a complex function into real and imaginary parts

Notation

Notation	Meaning
$u(x,y),\ v(x,y)$	Real and imaginary parts of f, each a real-valued function of two real variables
$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}$	The complex derivative, expressible via either pair of partials once the CR equations hold
$\text{holomorphic}$	Complex-differentiable in an open neighborhood of every point of a domain

Derivation

The complex derivative at z0 is defined as the limit of [f(z0+h) − f(z0)]/h as the complex number h → 0, and this limit must be the same regardless of the direction from which h approaches 0. Comparing the limit along the real axis (h real) with the limit along the imaginary axis (h = ik, k real) forces the Cauchy–Riemann equations.

f'(z_0) = \lim_{h\to 0} \frac{f(z_0+h)-f(z_0)}{h}

Definition of the complex derivative — the limit must be direction-independent

h \in \mathbb{R}:\quad f'(z_0) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}

Approaching along the real axis (h = Δx)

h = ik,\ k\in\mathbb{R}:\quad f'(z_0) = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}

Approaching along the imaginary axis (h = iΔy); dividing by i introduces the sign flip

\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}

Equating the two limits and matching real and imaginary parts yields the Cauchy–Riemann equations

Properties

Harmonicity

\text{If } f \text{ is holomorphic, both } u \text{ and } v \text{ satisfy Laplace's equation: } \nabla^2 u = 0,\ \nabla^2 v = 0

Example: u and v are called harmonic conjugates

Polar form

\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta},\qquad \frac{1}{r}\frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial r}

Condition: z = re^{i\theta} \neq 0

Necessary but not sufficient alone

\text{Satisfying the CR equations at a point does not guarantee differentiability unless } u,v \text{ have continuous partials there}

Applications

Because holomorphic functions' real and imaginary parts are harmonic, the Cauchy–Riemann equations underlie 2D potential theory: electrostatic potentials, steady-state heat flow, and incompressible irrotational fluid flow.

Worked Examples

Expand f(z) with z = x+iy.
$f(z) = (x+iy)^2 = x^2 - y^2 + 2ixy$
Read off the real and imaginary parts.
$u(x,y) = x^2-y^2,\qquad v(x,y) = 2xy$
Compute the four partial derivatives.
$\frac{\partial u}{\partial x}=2x,\quad \frac{\partial u}{\partial y}=-2y,\quad \frac{\partial v}{\partial x}=2y,\quad \frac{\partial v}{\partial y}=2x$
Check both equations: ∂u/∂x = ∂v/∂y gives 2x = 2x ✓; ∂u/∂y = −∂v/∂x gives −2y = −2y ✓.
$2x = 2x,\qquad -2y = -2y$

Answer: f(z) = z² satisfies the Cauchy–Riemann equations at every point, so it is entire (holomorphic on all of ℂ).

Practice Problems

Difficulty 6/10

For f(z) = e^z = e^x\cos y + i\,e^x \sin y, verify the Cauchy–Riemann equations hold.

Difficulty 7/10

Does f(z) = |z|² = x² + y² (so u = x²+y², v = 0) satisfy the Cauchy–Riemann equations anywhere, and if so where?

Common Mistakes

Common Mistake

Believing that satisfying the Cauchy–Riemann equations at a point is sufficient, by itself, to guarantee f is complex-differentiable there.

The CR equations are necessary for differentiability, but sufficiency additionally requires that the partial derivatives of u and v be continuous in a neighborhood of the point (this is the standard sufficient theorem). Without continuity, the equations can hold at isolated points without differentiability following.

Common Mistake

Mixing up the sign in the second equation, writing ∂u/∂y = ∂v/∂x instead of ∂u/∂y = −∂v/∂x.

The second Cauchy–Riemann equation has a minus sign: ∂u/∂y = −∂v/∂x. Forgetting it is the most common transcription error; re-derive it from f(z)=z² (u=x²−y², v=2xy) as a quick check: ∂u/∂y=−2y and −∂v/∂x=−2y match only with the minus sign.

Summary

f(z) = u(x,y) + iv(x,y) is holomorphic at a point exactly when ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x there, with u,v having continuous partials.
The equations arise from requiring the complex derivative's limit to agree whether h → 0 along the real or imaginary axis.
If f is holomorphic, both u and v are harmonic (satisfy Laplace's equation) — they are harmonic conjugates.
Functions like z̄ and |z|² generally fail the Cauchy–Riemann equations and are nowhere (or only pointwise) complex-differentiable.
The equations are the algebraic condition for f to act locally as a pure rotation-and-scaling map, the geometric hallmark of complex differentiability.

References

WebsiteWikipedia — Cauchy–Riemann equations