Complex Differentiability and Holomorphic Functions

A complex function is holomorphic (or analytic) if it is differentiable in the complex sense. Unlike real differentiability, complex differentiability is a very strong condition: it implies infinite differentiability and local power series expansions.

Complex Derivative

Definition2.1Complex derivative

Let $f: D \to \mathbb{C}$ be defined on a domain $D$ . The function $f$ is complex differentiable at $z_0 \in D$ if the limit

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

exists, where $h \in \mathbb{C}$ and $h \to 0$ means $|h| \to 0$ . The limit must be the same regardless of how $h$ approaches $0$ in the complex plane.

RemarkPath independence

The key difference from real differentiability: in $\mathbb{C}$ , the limit must exist as $h \to 0$ along any path in the complex plane. This is a much stronger condition than differentiability in $\mathbb{R}^2$ .

ExamplePolynomial derivative

For $f(z) = z^n$ , we have

$f'(z) = \lim_{h \to 0} \frac{(z+h)^n - z^n}{h} = \lim_{h \to 0} \frac{nz^{n-1}h + O(h^2)}{h} = nz^{n-1}.$

The usual power rule from real calculus holds.

ExampleConjugate is not differentiable

$f(z) = \bar{z}$ is not complex differentiable anywhere. To see this, compute the limit along two paths:

Along the real axis ( $h = t \in \mathbb{R}$ ): $\frac{\overline{z+t} - \bar{z}}{t} = \frac{\bar{z} + t - \bar{z}}{t} = 1$ .
Along the imaginary axis ( $h = it$ for $t \in \mathbb{R}$ ): $\frac{\overline{z+it} - \bar{z}}{it} = \frac{\bar{z} - it - \bar{z}}{it} = -1$ .

The limits differ, so $f'(z)$ does not exist.

Holomorphic Functions

Definition2.2Holomorphic function

A function $f: D \to \mathbb{C}$ is holomorphic on a domain $D$ if it is complex differentiable at every point of $D$ . Synonyms: analytic, regular, complex-analytic.

Definition2.3Entire function

A function $f: \mathbb{C} \to \mathbb{C}$ that is holomorphic on all of $\mathbb{C}$ is called entire.

ExampleExamples of entire functions

Polynomials: $p(z) = a_0 + a_1 z + \cdots + a_n z^n$
The exponential function: $e^z$
Trigonometric functions: $\sin z$ , $\cos z$
Any power series with infinite radius of convergence

ExampleHolomorphic but not entire

Rational functions: $f(z) = \frac{p(z)}{q(z)}$ are holomorphic on $\mathbb{C} \setminus \{\text{zeros of } q\}$ .
$f(z) = 1/z$ is holomorphic on $\mathbb{C} \setminus \{0\}$ .
$f(z) = \log z$ is holomorphic on $\mathbb{C} \setminus \mathbb{R}_{\leq 0}$ (with a branch cut).

Cauchy-Riemann Equations

Theorem2.1Cauchy-Riemann equations

Let $f(z) = u(x, y) + iv(x, y)$ where $z = x + iy$ . If $f$ is holomorphic at $z_0 = x_0 + iy_0$ , then the partial derivatives of $u$ and $v$ satisfy

$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

at $(x_0, y_0)$ . Conversely, if $u$ and $v$ are continuously differentiable and satisfy these equations, then $f$ is holomorphic.

Proof

Write $h = \Delta x + i \Delta y$ and compute

$\frac{f(z_0 + h) - f(z_0)}{h} = \frac{u(x_0 + \Delta x, y_0 + \Delta y) - u(x_0, y_0) + i(v(x_0 + \Delta x, y_0 + \Delta y) - v(x_0, y_0))}{\Delta x + i\Delta y}.$

Taking the limit as $h \to 0$ along the real axis ( $\Delta y = 0$ ) gives

$f'(z_0) = \frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}.$

Taking the limit along the imaginary axis ( $\Delta x = 0$ ) gives

$f'(z_0) = \frac{1}{i}\left(\frac{\partial u}{\partial y} + i \frac{\partial v}{\partial y}\right) = -i\frac{\partial u}{\partial y} + \frac{\partial v}{\partial y}.$

Equating real and imaginary parts yields the Cauchy-Riemann equations.

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ExampleExponential satisfies CR equations

For $f(z) = e^z = e^x \cos y + i e^x \sin y$ :

$u = e^x \cos y$ , $v = e^x \sin y$
$\frac{\partial u}{\partial x} = e^x \cos y = \frac{\partial v}{\partial y}$
$\frac{\partial u}{\partial y} = -e^x \sin y = -\frac{\partial v}{\partial x}$

The Cauchy-Riemann equations are satisfied, confirming that $e^z$ is entire.

Consequences of Holomorphicity

RemarkHolomorphic implies infinitely differentiable

A function that is holomorphic is automatically infinitely differentiable (has derivatives of all orders). This is not true in real analysis: a real function can be differentiable once but not twice.

Moreover, a holomorphic function is locally equal to its Taylor series. This is a consequence of Cauchy's integral formula.

RemarkHolomorphic implies conformal

A holomorphic function $f$ with $f'(z_0) \neq 0$ is conformal at $z_0$ : it preserves angles and orientation. This geometric property is fundamental to conformal mapping.

Singularities

Definition2.4Singular point

A point $z_0$ is a singular point (or singularity) of $f$ if $f$ is not holomorphic at $z_0$ but is holomorphic in some punctured neighborhood $0 < |z - z_0| < r$ .

ExampleTypes of singularities

$f(z) = 1/z$ has a pole at $z = 0$ (the Laurent series has finitely many negative powers).
$f(z) = e^{1/z}$ has an essential singularity at $z = 0$ (the Laurent series has infinitely many negative powers).
$f(z) = \sin(z)/z$ has a removable singularity at $z = 0$ (the limit $\lim_{z \to 0} f(z) = 1$ exists).

Singularities are classified in detail in residue theory.

Summary

RemarkKey properties of holomorphic functions

Complex differentiability is path-independent and much stronger than real differentiability.
Cauchy-Riemann equations provide a necessary and sufficient condition for holomorphicity.
Holomorphic functions are infinitely differentiable and locally analytic (equal to power series).
Holomorphic functions are conformal (angle-preserving) where the derivative is nonzero.
Singularities (poles, essential singularities, removable singularities) are central to the theory of meromorphic functions.

These properties make holomorphic functions the central objects of study in complex analysis.