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

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

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

$f"(z) = \lim_{h \to 0} \frac{u(x+h, y) - u(x, y)}{h} + i\lim_{h \to 0} \frac{v(x+h, y) - v(x, y)}{h} = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x}$

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

$f'(z) = \lim_{k \to 0} \frac{u(x, y+k) - u(x, y)}{ik} + i\lim_{k \to 0} \frac{v(x, y+k) - v(x, y)}{ik} = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}$

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:

$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}$

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. $\blacksquare$