2.1: The Derivative - Preliminaries

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In calculus we defined the derivative as a limit. In complex analysis we will do the same.

\[f'(z) = \lim_{\Delta z \to 0} \dfrac{\Delta f}{\Delta z} = \lim_{\Delta z \to 0} \dfrac{f(z + \Delta z) - f(z)}{\Delta z}. \nonumber \]

Before giving the derivative our full attention we are going to have to spend some time exploring and understanding limits. To motivate this we’ll first look at two simple examples – one positive and one negative.

Example \(\PageIndex{1}\)

Find the derivative of \(f(z) = z^2\).

Solution

We compute using the definition of the derivative as a limit.

\[\lim_{\Delta z \to 0} \dfrac{(z + \Delta z)^2 - z^2}{\Delta z} = \lim_{\Delta z \to 0} \dfrac{z^2 + 2z \Delta z + (\Delta z)^2 - z^2}{\Delta z} = \lim_{\Delta z \to 0} 2z + \Delta z = 2z. \nonumber \]

That was a positive example. Here’s a negative one which shows that we need a careful understanding of limits.

Example \(\PageIndex{2}\)

Let \(f(z) = \overline{z}\). Show that the limit for \(f'(0)\) does not converge.

Solution

Let's try to compute \(f'(0)\) using a limit:

\[f'(0) = \lim_{\Delta z \to 0} \dfrac{f(\Delta z) - f(0)}{\Delta z} = \lim_{\Delta z \to 0} \dfrac{\overline{\Delta z}}{\Delta z} = \dfrac{\Delta x - i \Delta y}{\Delta x + i \Delta y}. \nonumber \]

Here we used \(\Delta z = \Delta x + i \Delta y\).

Now, \(\Delta z \to 0\) means both \(\Delta x\) and \(\Delta y\) have to go to 0. There are lots of ways to do this. For example, if we let \(\Delta z\) go to 0 along the \(x\)-axis then, \(\Delta x\) goes to 0. In this case, we would have

\[f'(0) = \lim_{\Delta x \to 0} \dfrac{\Delta x}{\Delta x} = 1. \nonumber \]

On the other hand, if we let \(\Delta z\) go to 0 along the positive \(y\)-axis then

\[f'(0) = \lim_{\Delta y \to 0} \dfrac{-i \Delta y} {i \Delta y} = -1. \nonumber \]

The limits don’t agree! The problem is that the limit depends on how \(\Delta z\) approaches 0. If we came from other directions we’d get other values. There’s nothing to do, but agree that the limit does not exist.

Well, there is something we can do: explore and understand limits. Let’s do that now.