The Cauchy-Riemann Equations

Theorem 3.1 (Cauchy-Riemann Equations). If $f(z) = u(x, y) + iv(x, y)$ is differentiable at $z = x + iy$ Then

Proof. Compute the limit along the real axis ($h \in \mathbb{R}$, $h \to 0$):

Compute along the imaginary axis ($h = ik$, $k \in \mathbb{R}$, $k \to 0$):

Equating real and imaginary parts: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial v}{\partial x} = -\frac{\partial u}{\partial y}$. $\blacksquare$

Theorem 3.2. If $u$ and $v$ have continuous first partial derivatives on an open set $U$ and Satisfy the Cauchy-Riemann equations on $U$ Then $f = u + iv$ is analytic on $U$.

Proof. Since $u_x, u_y, v_x, v_y$ are continuous on $U$, $u$ and $v$ are (real) differentiable. Let $\Delta z = \Delta x + i\Delta y$. By real differentiability:

$u(x + \Delta x, y + \Delta y) - u(x, y) = u_x\,\Delta x + u_y\,\Delta y + \varepsilon_1$ $v(x + \Delta x, y + \Delta y) - v(x, y) = v_x\,\Delta x + v_y\,\Delta y + \varepsilon_2$

Where $\varepsilon_1, \varepsilon_2 = o(|\Delta z|)$. Therefore

$\frac{f(z + \Delta z) - f(z)}{\Delta z} = \frac{(u_x + iv_x)\Delta x + (u_y + iv_y)\Delta y + \varepsilon_1 + i\varepsilon_2}{\Delta x + i\Delta y}$

By CR: $u_y + iv_y = -v_x + iu_x = i(u_x + iv_x)$. Substituting:

$= (u_x + iv_x)\frac{\Delta x + i\Delta y}{\Delta x + i\Delta y} + \frac{o(|\Delta z|)}{\Delta z} \to u_x + iv_x$

As $\Delta z \to 0$. $\blacksquare$

When the Cauchy-Riemann equations hold:

Definition. A real-valued function $\phi(x, y)$ is harmonic if $\phi_{xx} + \phi_{yy} = 0$ (Laplace’s equation).

Proposition 3.3. If $f = u + iv$ is analytic, then $u$ and $v$ are harmonic.

Proof. From the Cauchy-Riemann equations: $u_x = v_y$ and $u_y = -v_x$. Differentiating: $u_{xx} = v_{yx}$ and $u_{yy} = -v_{xy}$. By equality of mixed partials, $u_{xx} + u_{yy} = v_{yx} - v_{xy} = 0$. Similarly for $v$. $\blacksquare$

Definition. If $u$ and $v$ are harmonic on $U$ and satisfy the Cauchy-Riemann equations, then $v$ is the harmonic conjugate of $u$.

Proposition 3.4. If $U$ is a connected domain and $u$ is harmonic on $U$ Then $u$ has A harmonic conjugate on $U$ Unique up to an additive constant.

Proof. Define $v(x, y) = \int_{(x_0, y_0)}^{(x, y)} (-u_y\, dx + u_x\, dy)$. The integrand is closed (since $(-u_y)_y = -u_{yy} = u_{xx} = (u_x)_x$) and since $U$ is Connected, $v$ is well-defined (path-independent) by Green’s theorem. Then $v_x = -u_y$ and $v_y = u_x$ Which are the CR equations. $\blacksquare$

Mistake 1: Assuming that the Cauchy-Riemann equations are sufficient for differentiability. The Cauchy-Riemann equations are necessary but not sufficient for complex differentiability. Even if $u$ and $v$ satisfy the Cauchy-Riemann equations at a point, $f$ may not be differentiable there if the partial derivatives are not continuous. The sufficiency condition requires continuous partial derivatives in a neighborhood.

Mistake 2: Forgetting the negative sign in the second Cauchy-Riemann equation. The Cauchy-Riemann equations are $u_x = v_y$ and $u_y = -v_x$. The second equation has a negative sign. Forgetting this sign leads to incorrect conclusions about analyticity. Always check both equations carefully.

Mistake 3: Confusing the formula for the derivative. When the Cauchy-Riemann equations hold, the derivative is $f'(z) = u_x + iv_x = v_y - iu_y$. Do not use $f'(z) = u_x + iu_y$ or other incorrect combinations. The derivative must be expressible in terms of either $u_x$ and $v_x$ or $v_y$ and $u_y$.

Mistake 4: Assuming that a function satisfying the Cauchy-Riemann equations at a single point is analytic. Analyticity requires differentiability in a neighborhood, not just at a point. A function can satisfy the Cauchy-Riemann equations at a single point without being analytic there. For example, $f(z) = |z|^2$ satisfies the Cauchy-Riemann equations only at $z = 0$, but it is not analytic anywhere.

Mistake 5: Forgetting to check that partial derivatives are continuous. The sufficiency condition for the Cauchy-Riemann equations requires that the partial derivatives $u_x, u_y, v_x, v_y$ be continuous in a neighborhood. If the partial derivatives are not continuous, the function may not be differentiable even if the Cauchy-Riemann equations hold. Always verify continuity of partial derivatives.

The Cauchy-Riemann equations are the complex analogue of requiring a function to have the same derivative regardless of direction. In the real world, partial derivatives in x and y can be chosen independently, but complex differentiability forces them to be coupled. This coupling means the real part u and imaginary part v are harmonic conjugates, each satisfying Laplace’s equation. Think of a fluid flow: u might represent pressure and v the velocity potential, and the Cauchy-Riemann equations ensure the flow is irrotational and incompressible simultaneously.

• Complex Functions and Analyticity: Analytic functions satisfy the Cauchy-Riemann equations and are infinitely differentiable.

• Complex Integration: Cauchy’s integral theorem relies on the Cauchy-Riemann equations for its proof.

• Taylor and Laurent Series: Analytic functions can be represented as power series, which follows from the Cauchy-Riemann equations.

• Classical Mechanics

• Electromagnetism

• Analytic Continuation
• Applications of Contour Integration
• Argument Principle and Rouché’s Theorem
• Cauchy’s Integral Formula