In real analysis, the equation $e^y = x$ has a unique solution $y = \ln x$ for each $x > 0$: the real exponential is a bijection from $\mathbb{R}$ onto $(0, \infty)$, and its inverse is the natural logarithm. In complex analysis, the situation is fundamentally different. The complex exponential $\exp : \mathbb{C} \to \mathbb{C} \setminus \{0\}$ is surjective but not injective — every nonzero complex number has infinitely many preimages, differing by integer multiples of $2\pi i$. The "complex logarithm" is therefore not a function in the classical sense but a *multivalued* object, and the same problem afflicts square roots, arbitrary powers, and inverse trigonometric [functions](/page/Function).

The theory of branch cuts provides a systematic way to extract single-valued holomorphic functions from these multivalued objects. The idea is geometric: by removing a carefully chosen curve from the domain — the branch cut — one destroys the closed paths that would force the function to take multiple values, and the resulting restricted domain admits a well-defined holomorphic "branch." This interplay between topology (the fundamental group of the punctured domain) and analysis (holomorphicity of the selected branch) is one of the distinctive features of complex function theory.

## Motivation

[motivation] ### Inverting the Exponential The complex exponential is the entire function \begin{align*} \exp : \mathbb{C} &\to \mathbb{C} \setminus \{0\} \\ w &\mapsto e^w. \end{align*} It is periodic with period $2\pi i$: for every $w \in \mathbb{C}$ and every $k \in \mathbb{Z}$, $\exp(w + 2\pi i k) = \exp(w)$. In particular, $\exp$ is not injective — each value in $\mathbb{C} \setminus \{0\}$ has infinitely many preimages. If $z = r e^{i\theta}$ with $r > 0$ and $\theta \in \mathbb{R}$, then every solution of $e^w = z$ has the form \begin{align*} w = \ln r + i(\theta + 2\pi k), \qquad k \in \mathbb{Z}, \end{align*} where $\ln r$ denotes the ordinary real logarithm of the positive number $r$. The [set](/page/Set) of all "logarithms of $z$" is the infinite discrete set $\{\ln r + i(\theta + 2\pi k) : k \in \mathbb{Z}\}$ — a vertical arithmetic progression in the complex plane with common difference $2\pi i$. To define a "logarithm function," we must select exactly one of these infinitely many values for each $z \in \mathbb{C} \setminus \{0\}$. The question is whether this selection can be made continuously — and, ideally, holomorphically. ### The [Continuity](/page/Continuity) Obstruction Suppose, seeking a contradiction, that there exists a continuous function $L : \mathbb{C} \setminus \{0\} \to \mathbb{C}$ satisfying $\exp(L(z)) = z$ for all $z \neq 0$. Consider the unit circle parametrised by $\gamma(t) = e^{it}$ for $t \in [0, 2\pi]$. The composition $t \mapsto L(e^{it})$ is a continuous function with \begin{align*} \exp(L(e^{it})) = e^{it} \qquad \text{for all } t \in [0, 2\pi]. \end{align*} Since $L(e^{it})$ is a continuous logarithm of $e^{it}$, it must have the form $L(e^{it}) = it + 2\pi i k(t)$ for some function $k : [0, 2\pi] \to \mathbb{Z}$. But $L$ is continuous and $k(t)$ is integer-valued and continuous, so $k$ is constant: $k(t) = k_0$ for all $t$. Evaluating at the endpoints: \begin{align*} L(e^{i \cdot 0}) &= L(1) = 0 + 2\pi i k_0 = 2\pi i k_0, \\ L(e^{i \cdot 2\pi}) &= L(1) = 2\pi i + 2\pi i k_0. \end{align*} The first equation gives $L(1) = 2\pi i k_0$ and the second gives $L(1) = 2\pi i(k_0 + 1)$. These are contradictory. The [topological](/page/Topology) explanation is that $\mathbb{C} \setminus \{0\}$ has nontrivial fundamental group $\pi_1(\mathbb{C} \setminus \{0\}) \cong \mathbb{Z}$: there exist closed curves (such as the unit circle) that cannot be continuously contracted to a point. The logarithm "detects" these curves via the winding number: each trip around the origin shifts the imaginary part of $\log z$ by $2\pi$. ### The Resolution: Cutting the Domain The obstruction above relies on the existence of closed curves encircling the origin. If we remove a curve connecting the origin to infinity — say, the closed ray $(-\infty, 0]$ — the resulting domain $\mathbb{C} \setminus (-\infty, 0]$ is simply connected: every closed curve in this domain can be continuously deformed to a point without leaving the domain. In particular, no closed curve in $\mathbb{C} \setminus (-\infty, 0]$ winds around the origin, so the obstruction vanishes. On this simply connected domain, we can make a continuous — in fact holomorphic — selection of $\log z$ by restricting the argument to the interval $(-\pi, \pi)$. The removed ray $(-\infty, 0]$ is the *branch cut*, the origin is the *branch point*, and the resulting single-valued holomorphic function is a *branch* of the logarithm. [/motivation]

## Definition

Before giving formal definitions, we need to make precise the three concepts that appeared in the motivating discussion: the point around which values change (the branch point), the curve we remove (the branch cut), and the single-valued function that results (the branch). These are logically distinct notions — the branch point is a property of the multivalued function, the branch cut is a choice of surgery on the domain, and the branch is the holomorphic function that lives on the surgered domain.

A branch point is characterised by what happens when one analytically continues a function element along a closed curve. If the function returns to a different value after a full circuit, the enclosed singularity is a branch point. The formal definition requires the notion of analytic continuation along a path, which we take as given.

[definition: Branch Point] Let $f$ be a multivalued analytic function on $\Omega \subseteq \mathbb{C}$ (that is, a collection of function elements connected by analytic continuation). A point $z_0 \in \mathbb{C}$ is a **branch point** of $f$ if there exists a closed curve $\gamma : [0,1] \to \Omega \setminus \{z_0\}$ with $\gamma(0) = \gamma(1)$ and $n(\gamma, z_0) \neq 0$ (where $n(\gamma, z_0)$ denotes the winding number of $\gamma$ around $z_0$) such that analytic continuation of some function element of $f$ along $\gamma$ returns a different function element at $\gamma(1) = \gamma(0)$. [/definition]

The winding number condition $n(\gamma, z_0) \neq 0$ ensures that $\gamma$ genuinely encircles $z_0$; curves that avoid the neighborhood of $z_0$ entirely will not detect the branching. For the complex logarithm, the origin is a branch point: continuing $\log z$ once around the origin shifts the value by $2\pi i$. For $z^{1/n}$, the origin is also a branch point, but the continuation is periodic — after $n$ circuits, the function returns to its original value. Such a branch point is called a *branch point of order $n - 1$* (or equivalently, the function has $n$ sheets). The logarithm, by contrast, has an *infinite-order* branch point (or *logarithmic branch point*), since no finite number of circuits returns the function to its starting value.

The next step is to remove enough of the domain to eliminate the problematic closed curves. A branch cut is a curve whose removal makes it impossible to wind around the branch point.

[definition: Branch Cut] Let $f$ be a multivalued analytic function with a branch point at $z_0 \in \mathbb{C}$. A **branch cut** for $f$ at $z_0$ is a simple arc $\Gamma \subset \mathbb{C}$ with one endpoint at $z_0$ and extending to another branch point or to infinity, such that the complement $\Omega \setminus \Gamma$ (where $\Omega$ is the domain of multivaluedness) is a domain on which a single-valued branch of $f$ can be defined. [/definition]

The branch cut is not unique — any arc from $z_0$ to another branch point or to infinity will do, provided the complement remains connected. For the logarithm, the standard choice is the ray $(-\infty, 0]$, but the positive real axis, a spiral, or even a wiggly curve from $0$ to $\infty$ would serve equally well. The choice is a matter of convention and computational convenience, not of mathematical necessity.

Finally, a branch is the holomorphic function that results from making the cut and selecting a value.

[definition: Branch] Let $f$ be a multivalued analytic function on $\Omega$ and let $\Gamma$ be a branch cut for $f$. A **branch** of $f$ on $\Omega \setminus \Gamma$ is a holomorphic function $F : \Omega \setminus \Gamma \to \mathbb{C}$ such that $F(z)$ is one of the values of $f$ at each point $z \in \Omega \setminus \Gamma$. [/definition]

Different branches on the same cut domain differ by the "sheet" they lie on. For the logarithm, the branches on $\mathbb{C} \setminus (-\infty, 0]$ are the functions $\log_k(z) = \ln|z| + i(\operatorname{Arg}(z) + 2\pi k)$ for $k \in \mathbb{Z}$, where $\operatorname{Arg}(z) \in (-\pi, \pi)$ is the principal argument. Each branch is holomorphic on the cut plane, and any two branches differ by the constant $2\pi i k$.

## The Complex Logarithm

### The Principal Branch

The most important multivalued function in complex analysis is the logarithm. To extract a holomorphic branch, we must choose a branch cut (removing a ray from the origin) and then select a consistent value of the argument on the resulting domain. The most common convention is to cut along the negative real axis $(-\infty, 0]$ and restrict the argument to the interval $(-\pi, \pi)$.

[definition: Principal Argument] The **principal argument** is the function \begin{align*} \operatorname{Arg} : \mathbb{C} \setminus (-\infty, 0] &\to (-\pi, \pi) \\ z = re^{i\theta} &\mapsto \theta, \qquad \text{where } \theta \in (-\pi, \pi) \text{ and } r > 0. \end{align*} [/definition]

The restriction to $(-\pi, \pi)$ rather than $(-\pi, \pi]$ ensures that $\operatorname{Arg}$ is continuous on $\mathbb{C} \setminus (-\infty, 0]$: approaching the negative real axis from above gives arguments near $\pi$, while approaching from below gives arguments near $-\pi$. Including the endpoint $\pi$ would create a discontinuity at the cut. With this convention in hand, the principal logarithm is defined by combining the real logarithm of the modulus with the principal argument.