The fact that simple functions such as \(\frac{\overline z}z\) do not have limits at certain points illustrates something special about complex numbers that has no parallel in the reals—we can express a function in a very compact way in one variable, yet it shows some peculiar behavior in the limit. We will repeatedly notice this kind of behavior; one reason is that when trying to compute a limit of a function \(f(z)\) as, say, \(z \to 0\text{,}\) we have to allow \(z\) to approach the point \(0\) in any way. On the real line there are only two directions to approach \(0\)—from the left or from the right (or some combination of those two). In the complex plane, we have an additional dimension to play with. This means that the statement A complex function has a limit … is in many senses stronger than the statement A real function has a limit …. This difference becomes apparent most baldly when studying derivatives.
If \(f\) is differentiable for all points in an open disk centered at \(z_0\) then \(f\) is called holomorphic 1
Some sources use the term analytic instead of holomorphic. As we will see in Chapter 8, in our context, these two terms are synonymous. Technically, though, these two terms have different definitions. Since we will be using the above definition, we will stick with using the term holomorphic instead of the term analytic.
at \(z_0\text{.}\) The function \(f\) is holomorphic on the open set \(E \subseteq G\) if it is differentiable (and hence holomorphic) at every point in \(E\text{.}\) Functions that are differentiable (and hence holomorphic) in the whole complex plane \(\C\) are called entire.
The difference quotient limit (2.1) can be rewritten as
\begin{equation*}
f'(z_0) \ = \ \lim_{ h \to 0 } \frac{ f(z_0+h) - f(z_0) }{ h } \,\text{.}
\end{equation*}
This equivalent definition is sometimes easier to handle. Note that \(h\) need not be a real number but can rather approach zero from anywhere in the complex plane.
The function \(f: \C \to \C\) given by \(f(z) = (\conj z)^2\) is differentiable at \(0\) and nowhere else; in particular, \(f\) is not holomorphic at \(0\text{.}\) To see why, let’s write \(z = z_0
+ r \, e^{ i \phi }\text{.}\) Then
\begin{align*}
\frac{ \conj{z}^2 - \conj{z_0}^2 }{ z - z_0 } \amp \ = \
\frac{ \left( \conj{ z_0 + r \, e^{ i \phi } } \right)^2 -
\conj{z_0}^2 }{ z_0 + r \, e^{ i \phi } - z_0 }\\
\amp \ = \
\frac{\left(\conj{ z_0 }+re^{-i\phi}\right)^2 -
\conj{z_0}^2}{re^{i\phi}}\\
\amp \ = \ \frac{ \conj{ z_0 }^2 + 2 \, \conj{z_0} \, r
\, e^{- i \phi } + r^2 e^{- 2 i \phi } - \conj{z_0}^2 }{ r
\, e^{ i \phi } }\\
\amp \ = \ \frac{ 2 \, \conj{z_0} \, r \, e^{ -i
\phi } + r^2 e^{ - 2 i \phi } }{ r \, e^{ i \phi } }\\
\amp \ = \ 2 \, \conj{z_0} \, e^{ -2i \phi } + r \, e^{
-3i \phi }\text{.}
\end{align*}
If \(z_0 \ne 0\) then taking the limit of \(f(z)\) as \(z \to z_0\) thus means taking the limit of \(2 \,
\conj{z_0} \, e^{ -2i \phi } + r \, e^{ -3i \phi }\) as \(r
\to 0\text{,}\) which gives \(2 \, \conj{z_0} \, e^{ -2i \phi }\text{,}\) a number that depends on \(\phi\text{,}\) i.e., on the direction that \(z\) approaches \(z_0\text{.}\) Hence this limit does not exist.
The basic properties for derivatives are similar to those we know from real calculus. In fact, the following rules follow mostly from properties of the limit.
A prominent application of the differentiation rules is the composition of a complex function \(f(z)\) with a path \(\gamma(t)\text{.}\) The proof of the following result gives a preview.
Suppose \(f\) is holomorphic at \(a \in \C\) with \(f'(a)\ne0\) and suppose \(\gamma_1\) and \(\gamma_2\) are two smooth paths that pass through \(a\text{,}\) making an angle of \(\phi\) with each other. Then \(f\) transforms \(\gamma_1\) and \(\gamma_2\) into smooth paths which meet at \(f(a)\text{,}\) and the transformed paths make an angle of \(\phi\) with each other.
Let \(\gamma_1(t)\) and \(\gamma_2(t)\) be parametrizations of the two paths such that \(\gamma_1(0) =
\gamma_2(0) = a\text{.}\) Then \(\gamma_1'(0)\) (considered as a vector) is the tangent vector of \(\gamma_1\) at the point \(a\text{,}\) and \(\gamma_2'(0)\) is the tangent vector of \(\gamma_2\) at \(a\text{.}\) Moving to the image of \(\gamma_1\) and \(\gamma_2\) under \(f\text{,}\) the tangent vector of \(f(\gamma_1)\) at the point \(f(a)\) is
and similarly, the tangent vector of \(f(\gamma_2)\) at the point \(f(a)\) is \(f'(a) \,
\gamma_2'(0)\text{.}\) This means that the action of \(f\) multiplies the two tangent vectors \(\gamma_1'(0)\) and \(\gamma_2'(0)\) by the same nonzero complex number \(f'(a)\text{,}\) and so the two tangent vectors got dilated by \(|f'(a)|\) (which does not affect their direction) and rotated by the same angle (an argument of \(f'(a)\)).
We end this section with yet another differentiation rule, that for inverse functions. As in the real case, this rule is only defined for functions that are bijections.
A function \(f: G \to H\) is one-to-one if, for every image \(w \in H\text{,}\) there is a unique \(z \in
G\) such that \(f(z)=w\text{.}\) The function is onto if every \(w \in H\) has a preimage \(z \in G\) (that is, there exists \(z \in G\) such that \(f(z) = w\)). A bijection is a function that is both one-to-one and onto. If \(f: G \to H\) is a bijection, then \(g: H \to G\) is the inverse of \(f\) if \(f(g(w)) = w\) for all \(w \in H\) and \(g(f(z)) = z\) for all \(z \in G\text{;}\) in other words, the composition \(f \circ g\) is the identity function on \(H\text{,}\) and \(g \circ f\) is the identity function on \(G\text{.}\)
Suppose \(G, H \subseteq \C\) are open sets, \(f: G \to H\) is a bijection, \(g: H \to G\) is the inverse function of \(f\text{,}\) and \(w_0 \in H\text{.}\) If \(f\) is differentiable at \(g(w_0)\) with \(f'(g(w_0)) \ne 0\) and \(g\) is continuous at \(w_0\text{,}\) then \(g\) is differentiable at \(w_0\) with
