Definition:Derivative/Complex Function

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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.

Such a derivative is also known as an ordinary derivative.

This is to distinguish it from a partial derivative, which applies to functions of more than one independent variable.


In his initial investigations into differential calculus, Isaac Newton coined the term fluxion to mean derivative.


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.