Various definitions for a branch of logarithm

Question

I am confused by the definitions put up for a branch of logarithm. I have two questions in mind.

1. Why do some books define a branch on just an open subset whereas some books define it on an open connected subset?
2. Why some texts define a branch to be continuous but then some texts take the extra leap and define the branch to be a holomorphic function?

Here are some examples.

In Sarason's Complex Function Theory, it states the definition:

Let $G$ be an open connected subset of $\mathbb{C}-\{0\}$. A branch of $\log z$ in $G$ is a continuous function $\ell$ in $G$ such that for each $z \in G$, $\ell(z)$ is a logarithm of $z$.

In Ivan Wilde's Lecture Notes on Complex Analysis, we have:

A branch of the logarithm is a pair $(D, f)$, where $D$ is a domain (an open connected subset of $\mathbb{C}$) such that $0 \notin D$ and $f: D \to \mathbb{C}$ is continuous and satisfies $e^{f(z)} = z$ for all $z \in D$.

In Stein-Shakarchi's Complex Analysis, we have:

Suppose that $\Omega$ is simply connected with $1 \in \Omega$, and $0 \notin \Omega$. Then, in $\Omega$ there is a branch of logarithm $F(z) = \log_\Omega z$ so that $F$ is holomorphic in $\Omega$ and $e^{F(z)} = z$ for all $z \in \Omega$.

In lecture notes I found online:

Let $\Omega \subset \mathbb{C} - \{0 \}$ be open. Then, a branch of $\log z$ on $\Omega$ is a holomorphic function $L: \Omega \to \mathbb{C}$ such that $e^{L(z)} = z$ for every $z \in \Omega$.

A lot of the definitions assume that the logarithm is defined on a domain/region/open-connected subset of $\mathbb{C}$. Why is this so? Moreover, what do we lose if we define it on just an open, not necessarily connected subset of $\mathbb{C}$? Would this be the same case to as why we sometimes define holomorphic functions just on an open set rather than on a domain?

The only thing that I could think why assuming connectedness is important is to be more careful in the sense that the branch wouldn't take two different values on different subsets whose union is the subset where the branch is defined.

Similarly, what do we lose if we define the branch to be just a continuous function rather than a holomorphic one?

Answer 1

Usually, the domain of a holomorphic function is connected. Otherwise, what happens in a connected component is disconnected (pun intended) with what happens in the other ones. But no harm is done if we just assume that the domain is any non-empty open set.

And, in this context, being holomorphic and being continuous is the same thing. That's so because any branch of the logarithm has a left inverse (the exponential function) which is holomorphic and whose derivative is non-zero everywhere; every continuous map with such a left inverse is holomorphic. So, there is, in fact, no difference between these two definitions.

Answer 2

This is a old question for which I would like to add a few things. I will focus on the first two definitions outlined in the OP for they are not fully discussed in many courses in Complex Analysis where the main objective is the study of analytic (holomorphic) functions, and also because that are more general in nature.

Overview of basic concepts:

1. Recall that any $w\in\mathbb{C}\setminus\{0\}$ can be expressed as $$w=|w|e^{i\theta}$$ for some $\theta\in\mathbb{R}$, where $|w|=\sqrt{\operatorname{Re}^2(w) +\operatorname{Im}^2(w)}$. The set $$\operatorname{Arg}_w:=\{\theta\in\mathbb{R}: w=|w|e^{i\theta}\}$$ is the called the set of arguments of $w$. It is easy to check that if $\theta_1,\theta_2\in\operatorname{Arg}_w$, then $\theta_1-\theta_2\in 2\pi i\mathbb{Z}$. One may think of the argument as a relation $\operatorname{Arg}=\{(w,\theta)\in(\mathbb{C}\setminus\{0\})\times\mathbb{R}:w=|w|e^{i\theta}\}$.
2. Give $\theta_0$, for any $w\in\mathbb{C}\setminus\{0\}$, there is a unique $\theta_w\in[\theta_0,\theta_0+2\pi)\cap\operatorname{Arg}_w$. The map $\operatorname{arg}_{\theta_0}:\mathbb{C}\setminus\{0\}\rightarrow[\theta_0,\theta_0+2\pi)$ given by $w\mapsto\theta_w$ is called a branch of the argument.
3. The relation $\operatorname{Log}$ from $\mathbb{C}\setminus\{0\}$ to $\mathbb{C}$ such that $(w,z)\in \operatorname{Log}$ iff $e^z=w$ is called logarithm. For $w\in\mathbb{C}\setminus\{0\}$, the set $\operatorname{Log}_w=\{z\in\mathbb{C}: (w,z)\in\operatorname{Log}\}$ is called the set of logarithms of $w$. It is easy to check that $$\operatorname{Log}_w =\{ \ln|w|+i\theta:\theta\in\operatorname{Arg}_w\}$$ where $\ln:(0,\infty)\rightarrow\mathbb{R}$ is the natural logarithm function from Calculus or Real analysis.
4. Selecting a brach $\theta_0$ of $\operatorname{Arg}$ yields a function $\log_{\theta_0}:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}$, given as $w\mapsto \ln|w|+i\operatorname{arg}_{\theta_0}(w)$, where $\theta\in[\theta_0,\theta_0+2\pi)$. The map $\log_{\theta_0}$ is called a branch of logarithm.

Continuous logarithm:

In many applications, one is interested in the possibility of selecting a single choice of $\operatorname{Arg}_z$ for each element $z$ on a domain $D\subset\mathbb{C}\setminus\{0\}$ continuously. This is not always possible, for example, there is not continuous choice of argument for $D=\mathbb{C}\setminus\{0\}$.

5. For $\theta_0\in\mathbb{R}$, define $L_{\theta_0}=\{re^{\theta_0 i}:r\geq0\}$. Using the properties of the exponential function along with the inverse function theorem from multivariate Calculus, it is easy to check that $\log_{\theta_0}:\mathbb{C}\setminus L_{\theta_0}\rightarrow(\mathbb{R}\times(\theta_0,\theta_0+2\pi)$, given by $w\mapsto\ln|w|+i\operatorname{arg}_{\theta_0}(w)$ is holomorphic and its inverse is the exponential function $\exp$. This fact can be used to construct continuous choices of arguments (and thus of logarithms).

Example 1: Suppose $w\in\mathbb{C}\setminus\{0\}$, and let $\alpha\in\operatorname{Arg}_w$. Then $B(w;|w|)\subset\mathbb{C}\setminus L_{\alpha+\pi}$. This means that the restriction of $\log_{\alpha+\pi}$ to $B(w;|w|)$ is continuous (in fact, holomorphic) and $\exp(\log_{\alpha+\pi}(z))=z$ for all $z\in B(w;|w|)$. Consequently, the map $\arg_{\alpha+\pi}(z)=\log_{\alpha+\pi}(z)-\ln(|z|)$ on $B(w;|w|)$ is a continuous function such that $\arg_{\alpha+\pi}(z)\in\operatorname{Arg}_z$ for all $z\in B(w;|w|)$.

6. The following is a slightly more general definition to definition in Ivan Wilde's Lecture Notes on Complex Analysis.

Definition Suppose $X$ is a path connected topological space, and $\phi:X\rightarrow\mathbb{C}\setminus\{0\}$ a continuous function. A continuous branch of $\operatorname{Log}_\phi$ is continuous function $f:X\rightarrow\mathbb{C}$ such that $\exp\circ f=\phi$. Similarly, a continuous brach of $\operatorname{Arg}_\phi$ is a function $\alpha:X\rightarrow\mathbb{R}$ such that $\alpha(x)\in\operatorname{Arg}_{\phi(x)}$ (equivalently $f(x)=\ln(|\phi(x)|)+i\alpha(x)\in\operatorname{Log}_{\phi(x)}$).

7. It is a well known and important result in Complex Analysis that when $g$ is a Holomorphic function on a simply connected domain $D$ and $g\geq0$, then there is a Holomorphic function $f$ on $D$ such the $\exp\circ f=g$. The function $f$ is of course a continuous choice of logarithm of $g$. Any other continuous choice $h$ of $\operatorname{Log}_g$ will differ from $f$ by a integer multiple of $2\pi$ and so $h$ will also be Holomorphic on $D$.

8. I will present two situations of continuous logarithms that are found in applications: logarithms of paths in $\mathbb{C}\setminus\{0\}$, and distinguished logarithms of continuous functions from $\mathbb{R}^n\rightarrow\mathbb{C}\setminus\{0\}$. I will only focused on the former for they form the foundation of the latter, and of many other results in complex analysis.
   The key to define logarithm branches of functions is the following result for paths:

Theorem 1: Let $\gamma:[a,b]\rightarrow\mathbb{C}\setminus\{0\}$ a path (continuous function). There exists a continuous function $\theta:[a,b]\rightarrow\mathbb{R}$ such that $\theta(t)\in\operatorname{Arg}_{\gamma(t)}$ for all $t\in[a,b]$. Furthermore, if $\psi:[a,b]\rightarrow\mathbb{R}$ is another continuous choice of argument of $\gamma$, then $$\theta(b)-\theta(a)=\psi(b)-\psi(a)$$ If in addition $\gamma$ is a closed path ($\gamma(a)=\gamma(b)$), then $\frac{\theta(b)-\theta(a)}{2\pi}$ is an integer that depends only on $\gamma$.

Before giving some details about the proof of this important result, I state one important consequence of Theorem 1. Denone by $\gamma^*=\gamma([a,b])$. Notice that for any $w\in\mathbb{C}$, $w\notin \gamma^*$ iff $0\notin (\gamma-w)^*$.

Definition: Let $\gamma:[a,b]\rightarrow\mathbb{C}$ a path and suppose $w\notin\gamma^*$. The index of $\gamma$ around $w$ is defined by $$\operatorname{Ind}_\gamma(w)=\frac{\theta(b)-\theta(a)}{2\pi}$$ where $\theta$ is any branch of $\operatorname{Arg}_{\gamma-w}$ on $[a,b]$. If $\gamma$ is closed, then $\operatorname{Ind}_\gamma(w)$ is an integer also known as the winding number of $\gamma$ around $w$.

Remark: when $\gamma$ in the definition above is continuous and piecewise differentiable, it can be seen that for any $w\notin\gamma^*$ $$ \operatorname{Ind}_\gamma(w)=\frac{1}{2\pi i}\int_{\gamma}\frac{1}{z-w}\,dz=\frac{1}{2\pi i}\int^b_a\frac{\gamma'(s)}{\gamma(s)-w}\,ds$$ and the for any $p\in\operatorname{Log}_{\gamma(a)-w}$, $f(t)=\int^t_a\frac{\gamma'(s)}{\gamma(s)-w}\,ds+p$ is a continuous branch of $\log_{\gamma-w}$.

9. Sketch of proof of Theorem 1: Since $\gamma$ is continuous on $[a,b]$ and $0\notin\gamma^*$, $$d(0,\gamma^*)=\inf_{a\leq t\leq b}|\gamma(t)|>0,$$ and $\gamma$ is uniformly continuous. Fix $0<\varepsilon<d(0,\gamma^*)$. There is $\delta>0$ such that $|t-s|<\delta$ implies $|\gamma(t)-\gamma(s)|<\varepsilon$. Let $a=t_0<t_1<\ldots<t_n=b$ be a partition of $[a,b]$ such that $|t_{j+1}-t_j|<\delta$ for $0\leq j<n$. Let $z_j=\gamma(t_j)$. The choice of the partition implies that $\gamma([t_j,t_{j+1}])\subset B(z_j;\varepsilon)$, and that $0\notin B(z_j;\varepsilon)$ for all $0\leq j <n$.
   As in Example 1, for each $0\leq j\leq n$ there is a continuous choice $\operatorname{arg}_j$ of argument on $B(z_j;\varepsilon)$. Hence $\theta_j=\arg_j\circ \gamma$ on $[t_j,t_{j+1}]$ is a continuous branch of $\operatorname{Arg}_\gamma$ on $[t_j,t_{j+1}]$.
   $\theta_0$ and $\theta_1$ may not coincide at $t_1$; however, since $\theta_0(t_1)$ and $\theta_1(t_1)$ are both arguments of $z_1$, $$\theta_0(t_1)-\theta_1(t_1)=2\pi n_1$$ for some $n_1\in\mathbb{Z}$. Define $$\theta(t)=\left\{\begin{matrix} \theta_0(t) & t_0\leq t\leq t_1\\ \theta_1(t)+2\pi n_1 & t_1\leq t\leq t_2\end{matrix}\right.$$ This defines a continuous choice of argument for $\gamma$ on $[t_0,t_2]$. We extend $\theta$ continuously as follows. Suppose we have extended $\theta$ continuously on $[t_0,t_k]$, $k<n$, so that $\theta$ on $[t_0,t_k]$ is a continuous choice of argument of $\gamma$ on $[t_0,t_k]$, Since $\theta(t_k)$ and $\theta_k(t_k)$ are both arguments of $z_k$, $$\theta(t_k)-\theta_k(t_k)=2\pi n_k$$ for some $n_k\in\mathbb{Z}$. Then, defining $\theta(t)=\theta_k(t)+2\pi n_k$ for $t\in[t_k,t_{k+1}]$ yields a continuous choice of argument for $\gamma$ on $[t_0,t_{k+1}]$. Proceeding this way, we obtain a continuous choice of $\operatorname{Arg}_\gamma$ $\Box$.

A nice treatment of Complex Analysis that exploits the concept of arguments is Beardon, A. F., Complex Analysis: The Argument Principle in Analysis and Topology, John Wiley & Sons. New York, 1979.

10. Here I only state a more general result about distinguished logarithms.

Theorem 2: Suppose $g:\mathbb{R}^n\rightarrow\mathbb{C}\setminus\{0\}$ is a continuous function an that $g(\boldsymbol{0})=1$. There exists a unique continuous function $f:\mathbb{R}^d\rightarrow\mathbb{C}$ such that $f(\boldsymbol{0})=0$ and $\exp\circ f= g$. The function $f$ is called a distinguished logarithm of $g$.

A proof of this result may be found in Sato, K. Levy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. pp. 33-34. Distinguished logarithms are used in the analysis of characteristic functions and the study of infinitely divisible measures in Probability.