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 $g(z) \ne 0$

Analytic Function

A function $f$ is analytic at $z_0$ iff it is differentiable at each point in some neighborhood of $z_0$. $f$ is analytic in an open set iff it is differentiable at every point in that open set.

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

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,

Let $u$ and $v$ be real-valued functions with continuous first-order partial derivatives on an open set $U \subseteq \mathbb{C}$, and define $f(z) = u(x,y)+iv(x,y)$, $z = x+iy$. Then $f$ is analytic in $U$ iff $u$ and $v$ satisfy the Cauchy-Riemann equations throughout $U$.

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,