Differentiable Complex Functions

Differentiable Complex Functions

We will now touch upon one of the core concepts in complex analysis - differentiability of complex functions baring in mind that the concept of differentiability of a complex function is analogous to that of a real function.

Definition: Let $A \subseteq C$ be open, $z_0 \in A$, and let $f : A \to \mathbb{C}$. Then $f$ is said to be Complex Differentiable at $z_0$ if the limit $\displaystyle{f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}}$ exists. The limit, $f'(z_0)$ is called the Derivative of $f$ at $z_0$ provided that this limit exists.

For a simple example, consider the following function:

(1)
\begin{align} \quad f(z) = z \end{align}

We will show that this function is differentiable at every $z_0 \in \mathbb{C}$. We have that:

(2)
\begin{align} \quad f'(z_0) &= \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0} \\ &=\lim_{z \to z_0} \frac{z - z_0}{z - z_0} \\ &= \lim_{z \to z_0} 1 \\ &= 1 \end{align}

Therefore $f$ is differentiable at every point $z_0 \in \mathbb{C}$ and the derivative of $f$ at every $z_0 \in \mathbb{C}$ is $f'(z_0) = 1$.

For another example, consider the following function:

(3)
\begin{align} \quad f(z) = z^2 \end{align}

We will show that this function is differentiable at $0$. We have that:

(4)
\begin{align} \quad f'(0) &= \lim_{z \to 0} \frac{f(z) - 0}{z - 0} \\ &= \lim_{z \to 0} \frac{z^2 - 0}{z - 0} \\ &= \lim_{z \to 0} \frac{z^2}{z} \\ &= \lim_{z \to 0} z \\ &= 0 \end{align}
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