Definition:Derivative/Complex Function

Definition

Let $f \left({z}\right): \C \to \C$ be a single-valued continuous complex function in a domain $D \subseteq \C$.

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 as a finite number and is independent of how the complex increment $h$ tends to $0$.

Then this limit is called the derivative of $f$ at the point $z_0$ and is variously denoted:

• $f^\prime \left({z_0}\right)$

• $D f \left({z_0}\right)$

• $D_z f \left({z_0}\right)$

• $\dfrac{\mathrm d f}{\mathrm d z} \left({z_0}\right)$

Further, let $f$ be complex-differentiable at all points in $D$.

Then $f^\prime: D \to \C$ is defined as the complex function whose value at each point $z \in D$ is $f^\prime \left({z}\right)$.

It can be variously denoted as:

• $\dfrac {\mathrm d f}{\mathrm d z}$

• $\dfrac {\mathrm d}{\mathrm d z} \left({f}\right)$

• $f^\prime \left({z}\right)$

• $D f \left({z}\right)$

• $D_z f \left({z}\right)$