Complex Analysis/Complex Functions/Complex Derivatives

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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 function 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'(z_0) := \lim_{\Delta z\rightarrow 0} \frac {f(z_0+ \Delta z) - f(z_0)}{\Delta z}, where \Delta z is a complex number.

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

Here, \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'(\Delta z) independent of how \Delta z approaches zero. For this reason, the complex conjugation z \mapsto \bar z is nowhere differentiable. To see this consider the limit

\lim_{w \to 0} \frac{ \overline{z + w}- \overline{z}}{w}
= \lim_{w \to 0} \frac{ \overline{w}}{w}
= \lim_{x, y \to 0} \frac{{x -y i}}{x + y i}  \, .

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