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 $\ds \lim_{h \mathop \to 0} \ \frac {\map f {z_0 + h} - \map f {z_0} } h$ 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 $\map {f'} z$:

$\ds \forall z \in D : \map {f'} z := \lim_{h \mathop \to 0} \frac {\map f {z + h} - \map f z} h$

Also known as

Some sources refer to a derivative as a differential coefficient, and abbreviate it D.C.

Some sources call it a derived function.

Notation

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

$\dfrac {\d f} {\d x}$

$\map {\dfrac \d {\d x} } f$

$\dfrac {\d y} {\d x}$ when $y = \map f x$

$\map {f'} x$

$\map {D f} x$

$\map {D_x f} x$

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

$\map {f'} {x_0}$

$\map {D f} {x_0}$

$\map {D_x f} {x_0}$

$\map {\dfrac {\d f} {\d x} } {x_0}$

$\valueat {\dfrac {\d f} {\d x} } {x \mathop = x_0}$

and so on.

Leibniz Notation

Leibniz's notation for the derivative of a function $y = \map f x$ with respect to the independent variable $x$ is:

$\dfrac {\d y} {\d x}$

Newton Notation

Newton's notation for the derivative of a function $y = \map f t$ with respect to the independent variable $t$ is:

$\map {\dot f} t$

or:

$\dot y$

which many consider to be less convenient than the Leibniz notation.

This notation is usually reserved for the case where the independent variable is time.