Definition:Derivative/Complex Function

Definition

The definition for a complex function is similar to that for real functions.

At a Point

Let $D\subseteq \C$ be an open set.

Let $f : D \to \C$ be a complex function.

Let $z_0 \in D$ be a point in $D$.

Let $f$ be complex-differentiable at the point $z_0$.

That is, suppose the limit $\displaystyle \lim_{h \to 0} \ \frac {f \left({z_0 + h}\right) - f \left({z_0}\right)} h$ exists.

Then this limit is called the derivative of $f$ at the point $z_0$.

On an Open Set

Let $D\subseteq \C$ be an open set.

Let $f : D \to \C$ be a complex function.

Let $f$ be complex-differentiable in $D$.

Then the derivative of $f$ is the complex function $f': D \to \C$ whose value at each point $z \in D$ is the derivative $f' \left({z}\right)$:

$\displaystyle \forall z \in D : f' \left({z}\right) := \lim_{h \mathop \to 0} \frac {f \left({z + h}\right) - f \left({z}\right)} h$

Notation

There are various notations available to be used for the derivative of a function $f$ with respect to the independent variable $x$:

• $\dfrac {\d f} {\d x}$
• $\dfrac \d {\d x} \left({f}\right)$
• $\dfrac {\d y} {\d x}$ when $y = f \left({x}\right)$
• $f' \left({x}\right)$
• $D f \left({x}\right)$
• $D_x f \left({x}\right)$

When evaluated at the point $\left({x_0, y_0}\right)$, the derivative of $f$ at the point $x_0$ can be variously denoted:

• $f' \left({x_0}\right)$
• $D f \left({x_0}\right)$
• $D_x f \left({x_0}\right)$
• $\dfrac {\d f} {\d x} \left({x_0}\right)$

and so on.