1.5Holomorphic Functions

Updated 2 Aug 2026Chapter (PDF)

[1.5.1]Definition(Holomorphicity)#

Suppose $\Omega \subseteq \bC$ is an open set with $z_0 \in \Omega$ . Then a function $f$ is said to be holomorphic at $z_0$ if the value

f'(z_0) = \lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} \quad h \in \bC

exists.

A function $f$ is said to be holomorphic on $\Omega$ if it is holomorphic at every point in $\Omega$ . For a general, non-open set $C \subseteq \bC$ , we say $f$ is holomorphic on $C$ if it is holomorphic on some open set containing $C$ .

Note that every point in an open set is an interior point, and so the open disk $D_r(z_0) \subseteq \Omega$ for some small $r > 0$ (ensuring $z_0 + h \in \Omega$ for all complex $h$ with $|h| < r$ , which prevents vacuous edge-cases). This is the first definition that is truly complex in nature. While this resembles the real definition of differentiability, note that $h$ is not real-valued, but rather $h \in (\bC \cong \bR^2)$ . We have made complex differentiability far more strict: this limit must exist not just when approaching from the left and right as in $\bR$ , but along any path, from every available angle, as $h \to 0$ in the complex plane. Of course, also note that the limit that defines $f'(z_0)$ is referred to as the (complex) derivative of $f$ at $z_0$ .

[1.5.2]Theorem(Holomorphicity Implies Continuity)#

Suppose $f$ is holomorphic at some $z_0$ in $f$ 's domain. Then $f$ is continuous at $z_0$ .

Proof.

We show $f(z_0 + h) \to f(z_0)$ as $h \to 0$ as such:

\lim_{h \to 0} f(z_0 + h) - f(z_0) = \lim_{h \to 0} \left(h \times \frac{f(z_0 + h) - f(z_0)}{h}\right) = 0 \times f'(z_0) = 0.

Rearranging our definition of a complex derivative, we have that

\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h} = a \implies \underbrace{[f(z_0 + h) - f(z_0)]}_{\Delta f} - ah = o(h),

where $o(h)$ is some map such that $\lim_{h \to 0} o(h)/h = 0$ . This statement says that, for $h \in \bC$ of small magnitude, we can approximate the deviation of $f$ from $z_0$ (denoted $\Delta f = f(z_0 + h) - f(z_0)$ ) very well with the linear change $ah$ . This approximation is so good that the error goes to zero much quicker than $h$ can go to $0$ . Note that limits (and thus derivatives) are unique, and so $a$ is the unique constant that approximates $\Delta f$ by the linear function $ah$ . In multivariate theory, we say that the linear map $h \mapsto ah$ is the true derivative of $f$ , and this idea can be extended to all forms of finite-dimensional domains and codomains.

[1.5.3]Proposition(Basic Differentiation Rules for

\bC

Suppose $f, g$ are holomorphic functions on $\Omega$ .

$f + g$ is holomorphic in $\Omega$ , with $(f + g)' = f' + g'$ .
$fg$ is holomorphic in $\Omega$ , with $(fg)' = f'g + fg'$ .
$f/g$ is holomorphic in $\Omega$ at points for which $g \ne 0$ , with $(f/g)' = (f'g - fg')/g^2$ .
The chain rule holds. For the two-function case $f: \Omega \to U$ and $g: U \to \bC$ , where both are holomorphic, we have that
$(g \circ f)' = g'(f(z))\,f'(z) \quad \forall z \in \Omega.$

From this, we can prove basic results that polynomials, rational functions, and other nice classes of functions are holomorphic for suitable domains.