2.2Holomorphic Functions

Updated 24 Jun 2026Chapter (PDF)

[2.2.1]Definition(Holomorphic Function)#

We say a function $f: A \to \bC$ , where $A \subseteq \bC$ , is differentiable at some $z \in \Lim(A)$ if

f'(z) = \lim_{h \to 0} \frac{f(z + h) - f(z)}{h}

is defined. If $f'$ is defined at all interior points of $A$ , we say $f$ is holomorphic.

[2.2.2]Corollary(Holomorphicity implies Continuity)#

If $f: A \to \bC$ , where $A \subseteq \bC$ , is complex differentiable at some $z \in A$ , then $f$ is continuous at $z$ .

Proof.

Note that $\displaystyle \lim_{h \to 0} f(z + h) - f(z) = \lim_{h \to 0} h \times \frac{f(z + h) - f(z)}{h} = 0 \times f'(z) = 0$ from standard multivariate limit theory.

[2.2.3]Remark(Derivation of Cauchy-Riemann Equations)#

Recall that for a limit to exist in $\bC \cong \bR^2$ , limits along all paths must exist and be equal. Treating $f: A \to \bC$ , where $A \subseteq \bC$ , from this multivariate lens, we write $f(z) = u(z) + iv(z)$ , as a vector function. For simplicity, let's say that $\Re(z) = x$ and $\Im(z) = y$ .

Leaving $y$ fixed and allowing $x$ to vary (traveling along $\bR$ ), we have that

f'(z) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} = \frac{\partial f}{\partial x} = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}.

Leaving $x$ fixed and allowing $y$ to vary (traveling along $i\bR$ ), and letting our step $h = ik$ for $k \in \bR$ , we have that

f'(z) = \lim_{k \to 0} \frac{f(z + ik) - f(z)}{ik} = -i\left( \frac{\partial f}{\partial y} \right) = -i \frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.

Letting these equal eachother, we arrive at the Cauchy-Riemann equations:

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \qquad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}.

The Cauchy-Riemann equations have major implications for holomorphic functions. For starters, $f'(z)$ can be expressed in four distinct ways in terms of the partials of $u, v$ . Additionally, we can write $|f'(z)|$ in a number of ways, with a very striking expression being

|f'(z)|^2 = \frac{\partial u}{\partial x} \frac{\partial v}{\partial y} - \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} = \frac{\partial (u, v)}{\partial (x, y)}.

Thus the squared norm of the derivative of a holomorphic function is exactly its Jacobian when written out as a vector function. This has many geometric implications, leading to the interpretation of holomorphic functions as conformal maps, which we will visit later.

Eventually, we will show that being once complex-differentiable is the same as being infinitely complex-differentiable, and this will imply Clairaut's Theorem (that mixed partials are equal under nice conditions). From this and the Cauchy-Riemann equations, we obtain

\Delta u = \frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0,

\Delta v = \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} = 0.

A function $u$ that satisfies Laplace's equation $\Delta u = 0$ is said to be harmonic. In the context of Cauchy-Riemann equations, the function $v$ is said to be conjugate harmonic to $u$ .

[2.2.4]Theorem(Criterion for Holomorphicity)#

A function $f(z) = u(z) + iv(z)$ is holomorphic if and only if $u, v$ have continuous, first-order partial derivatives that satisfy the Cauchy-Riemann equations.

Proof.

$(\Longrightarrow):$ This is implied from taking the limit of the difference quotient along $\bR$ and $i\bR$ .

$(\Longleftarrow):$ Suppose $u, v$ have continuous first-order partial derivatives that satisfy the Cauchy-Riemann equations. From multivariate theory, note that

\begin{align*} u(x + h, y + k) - u(x, y) &= u_x h + u_y k + o_1(h + ik),\\ v(x + h, y + k) - v ( x, y) &= v_x h + v_y k + o_2(h + ik). \end{align*}

Then note

f(z + [h + ik]) - f(z) = (u_x + iv_x)(h + ik) + [o_1(h + ik) + i o_2(h + ik)],

meaning

f'(z) = \lim_{h + ik \to 0} \frac{f(z + [h + ik]) - f(z)}{h + ik} = u_x + i v_x.