Complex Analysis/Complex Functions/Complex Derivatives

With the concept of the limit of a complex function now established, we can now introduce the differentiability of a complex function.

2.3.1.: Derivatives

Definition: Let f be a complex valued funtion defined in a neighborhood of $z_0$. Then the derivative of f at $z_0$ is given by:

$\frac{df}{dz}(z_0)\equiv f^{\prime }(z_0):=\underset{\mathrm{\Delta }z\to 0}{\lim }\frac{f(z_0+\mathrm{\Delta }z)-f(z_0)}{\mathrm{\Delta }z}$, where $\mathrm{\Delta }z$ is a complex number.

Provided that this limit exists, f is said to be differentiable at $z_0$.

Here, $\mathrm{\Delta }z$ is any complex number, so it may approach zero in a number of different directions. However, for this limit to exist, it must reach a unique limit f'($\mathrm{\Delta }z$) independent of how $\mathrm{\Delta }z$ approaches zero. For this reason, the complex conjugation $z\mapsto \overline{z}$ is nowhere differentiable. To see this consider the limit

$\underset{w\to 0}{\lim }\frac{\overline{z+w}-\bar{z}}{w}=\underset{w\to 0}{\lim }\frac{\bar{w}}{w}=\underset{x,y\to 0}{\lim }\frac{x-yi}{x+yi}.$

For $x=0$ this limit is $-1$ and for $y=0$ this limit is $1$, so there does not exist a unique complex limit.

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