Derivatives

Let $f$ be a function whose domain contains a neighborhood $|z-z_0| < \epsilon$ of $z_0$. The derivative of $f$ at $z_0$ is

provided this limit exists. $f$ is differentiable at $z_0$ when $f'(z_0)$ exists.

If $f$ is differentiable at $z_0$, then $f$ is continuous at $z_0$.

Differentiation Formulas

Let $c \in \mathbb{C}$, and let $f$, $g$ be functions differentiable at $z \in \mathbb{C}$. Then:

• $\dfrac{\text{d}}{\text{d}z}\,c = 0$
• $\dfrac{\text{d}}{\text{d}z}\,(cf(z)) = cf'(z)$
• $\dfrac{\text{d}}{\text{d}z}\,z^n = nz^{n-1}$, for positive integer $n$
• $\dfrac{\text{d}}{\text{d}z}\,(f(z)+g(z)) = f'(z)+g'(z)$
• $\dfrac{\text{d}}{\text{d}z}\,(f(z)g(z)) = f(z)g'(z)+f'(z)g(z)$
• $\dfrac{\text{d}}{\text{d}z}\,\dfrac{f(z)}{g(z)} = \dfrac{g(z)f'(z)-f(z)g'(z)}{(g(z))^2}$, provided $g(z) \ne 0$

Cauchy-Riemann Equations

Let $f(z) = u(x,y)+iv(x,y)$, and suppose $f'(z_0)$ exists at $z_0 = x_0+iy_0$. Then the first-order partial derivatives of $u$ and $v$ exist at $(x_0,y_0)$ and satisfy

at $(x_0,y_0)$, and

The equations follow from forcing the derivative limit to agree along the real and imaginary directions of approach.

The equations are necessary but not sufficient. They can hold at $z_0$ while $f'(z_0)$ does not exist. See Sufficient Conditions for Differentiability.

The Jacobian of $(x,y) \mapsto (u,v)$ is then

which acts as multiplication by $u_x + iv_x$. A complex-differentiable map is locally a rotation plus a uniform scaling.

Examples:

• $f(z) = z^2$: $u_x = v_y = 2x$, $u_y = -v_x = -2y$, and $f'(z) = 2x + i2y = 2z$.
• $f(z) = \bar{z}$: $u_x = 1$, $v_y = -1$, so the equations fail everywhere and $\bar{z}$ is nowhere differentiable.

Sufficient Conditions for Differentiability

Let $f(z) = u(x,y)+iv(x,y)$ be defined throughout a neighborhood $U$ of $z_0 = x_0+iy_0$. $f'(z_0)$ exists iff:

• The first-order partial derivatives of $u$ and $v$ with respect to $x$ and $y$ exist everywhere in $U$.
• Those partial derivatives are continuous at $(x_0,y_0)$ and satisfy the Cauchy-Riemann equations at $(x_0,y_0)$.

When these hold,

Polar Form of the Cauchy-Riemann Equations

Let $f(z) = u(r,\theta)+iv(r,\theta)$ be defined throughout a neighborhood of a non-zero point $z_0 = r_0e^{i\theta_0}$. $f'(z_0)$ exists iff:

• The first-order partial derivatives of $u$ and $v$ with respect to $r$ and $\theta$ exist everywhere in the corresponding polar-coordinate neighborhood.
• Those partial derivatives are continuous at $(r_0,\theta_0)$ and satisfy

at $(r_0,\theta_0)$.

When these hold,