Cauchy-Riemann Equations — Definition, Formula & Examples

The Cauchy-Riemann equations are a pair of partial differential equations that the real and imaginary parts of a complex function must satisfy for the function to be differentiable at a point in the complex plane.

Let $f(z) = u(x,y) + iv(x,y)$ be a complex-valued function where $z = x + iy$ . The function $f$ is complex-differentiable (holomorphic) at a point only if the partial derivatives of $u$ and $v$ exist there and satisfy $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ . When these equations hold and the partial derivatives are continuous, $f$ is analytic at that point.

Key Formula

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

Where:

$u(x,y)$ = Real part of the complex function f(z)
$v(x,y)$ = Imaginary part of the complex function f(z)
$x$ = Real component of z = x + iy
$y$ = Imaginary component of z = x + iy

How It Works

To check whether a complex function

f(z)

is analytic, first split it into its real part

u(x,y)

and imaginary part

v(x,y)

. Then compute all four first-order partial derivatives:

u_x

u_y

v_x

, and

v_y

. Verify whether

u_x = v_y

and

u_y = -v_x

at the point(s) of interest. If both conditions hold and the partials are continuous,

f

is holomorphic there. If either condition fails,

f

is not complex-differentiable at that point.

Worked Example

Problem: Verify that f(z) = z² satisfies the Cauchy-Riemann equations.

Expand f(z): Write z = x + iy, so z² = (x + iy)² = x² − y² + 2xyi. This gives u(x,y) = x² − y² and v(x,y) = 2xy.

f(z) = (x^2 - y^2) + i(2xy)

Compute partial derivatives: Find all four first-order partials of u and v.

u_x = 2x,\quad u_y = -2y,\quad v_x = 2y,\quad v_y = 2x

Check Cauchy-Riemann: Verify both conditions: u_x = v_y gives 2x = 2x, and u_y = −v_x gives −2y = −2y. Both hold everywhere.

u_x = v_y \;\checkmark, \qquad u_y = -v_x \;\checkmark

Answer: Both Cauchy-Riemann equations are satisfied for all (x, y), confirming that f(z) = z² is analytic on the entire complex plane.

Why It Matters

The Cauchy-Riemann equations are the gateway test for analyticity in complex analysis, which underpins contour integration, residue calculus, and conformal mappings. Engineers use these ideas in fluid dynamics and electrostatics, where analytic functions model irrotational, incompressible flows and electric potentials.

Common Mistakes

Mistake: Forgetting the negative sign in the second equation and writing u_y = v_x instead of u_y = −v_x.

Correction: The second Cauchy-Riemann equation has a minus sign: ∂u/∂y = −∂v/∂x. A sign error will incorrectly classify non-analytic functions as analytic or vice versa.