Cauchy-Riemann Equations

Introduction

The Cauchy-Riemann equations are the fundamental equations of complex analysis. They provide necessary and sufficient conditions (under mild regularity assumptions) for a complex function to be differentiable. These elegant partial differential equations reveal that complex differentiability is far more restrictive than real differentiability.

Named after Augustin-Louis Cauchy and Bernhard Riemann, these equations connect the real and imaginary parts of an analytic function in a precise way. They imply that the real and imaginary parts are harmonic functions, leading to deep connections with potential theory and physics.

Understanding the Cauchy-Riemann equations is essential for complex analysis. They explain why analytic functions have such remarkable properties: infinite differentiability, power series expansions, and the ability to be reconstructed from their boundary values.

Complex Functions and Differentiability

A complex function ƒ:C→C can be written in terms of its real and imaginary parts. If z=x+i*y and ƒ(z)=u(x,y)+i*v(x,y), then u and v are real-valued functions of two real variables.

ƒ(z)=ƒ(x+i*y)=u(x,y)+i*v(x,y)

A complex function f is differentiable at (z_0) if the limit exists:

ƒ′*((z_0))=(lim_h→0)((ƒ*((z_0)+h)−ƒ((z_0)))/h)

The crucial point is that h is complex, so it can approach 0 from any direction in the complex plane. This is much more restrictive than real differentiability.

Derivation of the Cauchy-Riemann Equations

Suppose ƒ is differentiable at z=x+i*y. Let us compute the derivative by approaching along two different paths.

Approach Along the Real Axis

Let h=t be real. Then:

ƒ′*(z)=(lim_t→0)((ƒ(x+t,y)−ƒ(x,y))/t)=∂(u)/∂(x)+i∂(v)/∂(x)

Approach Along the Imaginary Axis

Let h=i*t be purely imaginary. Then:

ƒ′*(z)=(lim_t→0)((ƒ(x,y+t)−ƒ(x,y))/(i*t))=1/i*(∂(u)/∂(y)+i∂(v)/∂(y))=∂(v)/∂(y)−i∂(u)/∂(y)

Equating the Two Expressions

Since ƒ is differentiable, the limit must be the same regardless of path. Equating real and imaginary parts:

∂(u)/∂(x)=∂(v)/∂(y)*and∂(v)/∂(x)=−∂(u)/∂(y)

These are the Cauchy-Riemann equations.

The Main Theorem

The Cauchy-Riemann equations provide both necessary and sufficient conditions for analyticity.

Necessary condition: If ƒ=u+i*v is differentiable at (z_0), then the Cauchy-Riemann equations hold at (z_0).

Sufficient condition: If u and v have continuous first partial derivatives in a neighborhood of (z_0) and the Cauchy-Riemann equations hold at (z_0), then f is differentiable at (z_0).

Moreover, when ƒ is differentiable, its derivative can be expressed as:

ƒ′*(z)=∂(u)/∂(x)+i∂(v)/∂(x)=∂(v)/∂(y)−i∂(u)/∂(y)

Examples

Example 1: f(z) = z^2

Let ƒ(z)=z2=(x+i*y)2=x2+i2*y2+2*i*x*y. So u=x2-y2 and v=2*x*y.

Check the Cauchy-Riemann equations:

(u_x)=2*x=(v_y)

(u_y)=−2*y=−(v_x)

The equations are satisfied everywhere, so ƒ(z)=z2 is analytic on all of C.

Example 2: f(z) = |z|^2

Let ƒ(z)=|z|2=x2+y2. here u=x2+y2 and v=0.

(u_x)=2*x,(v_y)=0

The equation (u_x)=(v_y) requires 2*x=0, so x=0. Similarly, (u_y)=-(v_x) requires 2*y=0. The Cauchy-Riemann equations hold only at the origin. So ƒ(z)=|z|2 is differentiable only at z=0.

Harmonic Functions

An important consequence of the Cauchy-Riemann equations is that both u and v satisfy Laplace's equation.

Differentiating the first Cauchy-Riemann equation with respect to x and the second with respect to y:

(u_x*x)=(v_y*x)*and*(v_x*y)=−(u_y*y)

Since mixed partials are equal (assuming continuity), (v_y*x)=(v_x*y), so:

(u_x*x)+(u_y*y)=0

This is Laplace's equation. A function satisfying Laplace's equation is called harmonic. Similarly, v is also harmonic.

The functions u and v are called harmonic conjugates. Given a harmonic function u, we can (under suitable conditions) find its harmonic conjugate v, constructing an analytic function ƒ=u+i*v.

Polar Form

In polar coordinates z=r*e(i*θ), with ƒ(z)=u(r,θ)+i*v(r,θ), the Cauchy–Riemann equations become:

∂(u)/∂(r)=1/r∂(v)/∂(θ),∂(v)/∂(r)=−1/r∂(u)/∂(θ)

These are useful when the function has circular symmetry.

Connection to Other Concepts

The Cauchy–Riemann equations have deep connections across mathematics and physics.

In vector calculus, they are equivalent to saying that the vector field (u,−v) is both irrotational (curl(=0) and incompressible (div(=0). This links directly to two-dimensional fluid flow.

In conformal mapping, analytic functions preserve angles. The Cauchy–Riemann equations ensure this angle-preserving property, which is why conformal maps are widely used in engineering and physics.

In potential theory, harmonic functions solve Laplace’s equation. Since the real and imaginary parts of analytic functions are harmonic, complex analysis becomes a powerful tool for solving potential problems.

In several complex variables, the Cauchy–Riemann equations generalize but become overdetermined, leading to a much richer and more constrained theory.

Summary

The Cauchy-Riemann equations (u_x)=(v_y) and (v_x)=−(u_y) are the fundamental conditions for complex differentiability. They relate the partial derivatives of the real and imaginary parts of a complex function.

If ƒ=u+i*v has continuous partial derivatives and satisfies the Cauchy-Riemann equations, then ƒ is analytic. This condition is both necessary and sufficient.

A key consequence is that u and v are harmonic functions, satisfying Laplace's equation. They are harmonic conjugates of each other.

The equations explain why complex differentiability is so restrictive: the derivative must be the same regardless of the direction of approach, forcing strong constraints on the component functions.

Applications include conformal mapping, potential theory, fluid dynamics, and electrostatics. The Cauchy-Riemann equations are central to all of complex analysis.

The equations can also be written compactly using the Wirtinger derivatives. Define the operators d/d(z) and d/d(z)*-bar. Then ƒ is analytic if and only if d(ƒ)/d(z)*-bar=0, which is equivalent to the Cauchy-Riemann equations.

Understanding the Cauchy-Riemann equations provides the foundation for all deeper results in complex analysis, from Cauchy's integral theorem to the residue theorem to the theory of conformal mappings.