Derivation.

Definition 3.2.1   Let $f$ be a function defined on a domain $R$ in $\mathbb{C}$ and let $z_0$ be an interior point of $R$ . The function $f$ is derivable (or differentiable) at $z_0$ if there exists a complex number $\alpha$ such that

$$\underset{z \rightarrow z_0}{\text{lim}} \frac {f(z)-f(z_0)}{z-z_0} = \alpha.$$

We denote this by $f'(z_0)=\alpha$ .

Remark 3.2.2   Another formulation of Def. 2.1 is as follows:

$$f'(z_0) = \underset{\Delta z \rightarrow 0}{\text{lim}} \frac {f(z_0 + \Delta z) - f(z_0)}{\Delta z}$$

Example 3.2.3   Let $f(z)=z^n$ , where $n \in \mathbb{N}$ . Then, for every $z_0 \in \mathbb{C}$ :

$$\frac {f(z)-f(z_0)}{z-z_0}=\frac {z^n-z_0^n}{z-z_0}=z^{n-1}+z^{n-2}z_0+ z^{n-3}z_0^2+ \dots + z_0^{n-1}$$

from what follows:

$$\underset{z \rightarrow z_o}{\lim}\frac {f(z)-f(z_0)}{z-z_0} = \u... ...rrow z_o}{\lim} (z^{n-1}+z^{n-2}z_0+ z^{n-3}z_0^2+ \dots + z_0^{n-1})=nz^{n-1}.$$

The function $f$ is differentiable at every point $z_0$ and $f'(z_0)=nz_0^{n-1}$ .

The algebra of derivable functions of a complex variable is (formally) identical to the algebra of derivable functions of a real variable seen in Calculus.

Proposition 3.2.4   Let $f$ and $g$ be two functions defined on a neighborhood of $z_0$ . We suppose that $f$ and $g$ are continuous at $z_0$ .

(i)

$f+g$ is differentiable at $z_0$ and $(f+g)'(z_0) = f'(z_0)+g'(z_0)$ .

(ii)

$fg$ is differentiable at $z_0$ and we have:

$$(fg)'(z_0)=f'(z_0)g(z_0) + f(z_0)g'(z_0).$$

(iii)

If $g(z_0) \neq 0$ , then $1/g$ is derivable at $z_0$ and we have $(1/g)'(z_0)= - g'(z_0)/(g(z_0)^2$ .

(iv)

If $g(z_0) \neq 0$ ,then $f/g$ is derivable at $z_0$ and we have:

$$\left( \frac {f}{g} \right)'(z_0)= \frac {f'(z_0)g(z_0)-f(z_0)g'(z_0)}{g(z_0)^2}$$

Example 3.2.5   If $f(z)=1/z$ , then $f$ is differentiable at every point of $\mathbb{C}^*$ and $f'(z)=-1/z^2$ .

Proposition 3.2.6   If $f$ is derivable at $z_0$ and if $g$ is derivable at $f(z_0)$ , then $gof$ is derivable at $z_0$ and we have:

$$(gof)'(z_0)=g'(f(z_0)) \cdot f'(z_0).$$

As easy consequences of Proposition 2.4, we have Corollary 2.7 and Corollary 2.8.

Corollary 3.2.7        A polynomial function is differentiable on the whole of $\mathbb{C}$ .

Corollary 3.2.8        A rational function is differentiable at every point of its domain of definition

Example 3.2.9   The function $f$ such that $f(z)= ((1+i)z-2i)/(z^2+1)$ is differentiable on $\mathbb{C} - \{ -i, i \}$ and we have:

$$\forall z \in \mathbb{C} - \{ -i, i \}, \; f'(z)= \frac {(1+i)(z^2+1)-((1+i)z-2i)2z}{(z^2+1)^2} = \frac {-(1+i)z^2+4iz+1+i}{(z^2+1)^2}.$$

Actually, we do not dispose of enough functions in order to display more interseting examples. For that purpose, we shall wait until next chapter (i.e. Chapter chapter analytic functions).