I was reading a book on complex analysis, and at some point during the chapter about the complex logarithm, the author stated:
because in the complex numbers, the function $\exp:\mathbb C\to \mathbb C$ is periodic (for all $z\in \mathbb C, \exp (z + 2i\pi) = \exp z$) it's not possible to define an inverse function withought changing the domain.
I stop reading and tried to do it myself, and this is what I thought of. Let $A$ be the set $\{a+bi:-\pi <b\leq \pi\}:$
Then, if we restrict $\exp$ to this domain, we have a bijection. Then, we can define the logarithm as: $$\log_A :\mathbb C \to A\ \ \ \ (1)$$
as: $\log(z) = ln(|z|) + i \arg(z)$, where $\arg(z)\in (-\pi, \pi]$
But what they actually did is:
Let $L\phi = \{t(\cos(\phi),\sin(\phi)):0\geq t \in \mathbb R\}\subseteq \mathbb R^2 \simeq \mathbb C$, where $0 \leq \phi < 2\pi$, and let $$\mathcal{D}_\phi = \mathbb C \setminus L_\phi$$ For all $z\in\mathcal{D}_\phi $ there is an unique argument of $\arg _\phi z$ of $z$ such that $\phi < \arg_\phi z < \phi + \pi$. Thus, we can define a function called a branch of the logarithm: $$\log_:\mathcal{D}_\phi\to \mathbb C\ \ \ \ (2)$$, defined by: $$\log z = \ln |z| + i \arg_\phi z$$ The main branch of the logarithm is defined on the domain $\mathcal{D}_0$, which we can get by subtracting the set $\{(x,0): x \leq 0\}$ from $\mathbb C$. For the main branch, we have that $-\pi < \arg_0 z < \pi$
There are some things I don't understand about this definition of the logarithm:
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If the main branch of the logarithm is defined in $\mathcal{D}_0$, shouldn't we have that: $0 < \arg_0 z < 2\pi$? Shouldn't the main branch be defined on $\mathcal{D}_{-\pi}$? Is this some kind of typo or did I miss something?
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Why define the logarithm as the author did in (2) in the first place? Isn't the definition in (1) more practical and straightforward?
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According to definition (2), the main branch of the logarithm isn't defined on $-1$ for example, however, there are complex numbers such that $\exp (z) = -1$, and using the definition (1), we can calculate the logarithm of those numbers. Why exclude a hole line from the complex plane? By Doing that we exclude the possibility of calculating the logarithm for many complex numbers. That doesn't make much sense to me.
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In order for the exponential function to be an inverse of the main branch logarithm defined in (2), we need to restrict $\exp$ to the domain $B=\{a+bi:b\neq \pi + 2k\pi, k \in \mathbb N \}$, in order to avoid that the output of exponential sits in the negative real axis, and even if we do this restriction, we still didn't get rid of the periodicity of the exponential function, and we still need to restrict it to $A$, All this definition with the branches just seems way less practical than just restricting the $\exp$ function to $A$ in the first place and defining the complex logarithm as in (1).
Why is definition $(2)$ used? What are the advantages over the definition $(1)$?
Best Answer
It does look like a typo in their inequality $$\phi < \arg_\phi z < \phi + \pi $$ From the context of the quoted passage, that inequality should instead be $$\phi - \pi < \arg_\phi z < \phi + \pi $$ which is consistent with the last inequality of that quotation namely $$-\pi < \arg_0 z < \pi $$
Regarding your question about excluding the "hole line", in order to use tools of complex analysis one wants the domain of one's function to be an open subset of the complex plane on which the function is continuous (at the very least; of course one usually also wants the function to be differentiable over $\mathbb C$). This works using $\mathcal D_0 = \mathbb C - L_0$, but if you use $\mathcal D_0 = \mathbb C - \{0\}$ (or $\mathbb C$) then the function $\arg_0(z)$ is discontinuous at every point on $L_0$, and therefore the branch of the logarithm defined using $\arg_0(z)$ is also discontinuous on $L_0$.
Regarding (2), one gets more flexibility using (2) than using (1). Sometimes one wants a branch of the logarithm with a different domain $\mathcal D_\phi$ that includes the line $L_0$, and (2) gives you that (at the expense of deleting some other $L_\phi$). Other times one might want a different branch of the logarithm with the exact same domain $\mathcal D_0$, and one gets this using $\mathcal D_{2\pi}$ or $\mathcal D_{4\pi}$ or any set of the form $\mathcal D_{2\pi n}$ for different integers $n \in \mathbb Z$.
The point here is that by allowing the possibility of many different branches of the logarithm, one is able to find a differentiable inverse of the exponential function near an arbitrary point of the complex plane $z \in \mathbb C - \{0\}$, at the expense of restricting the exponential to a horizontal strip before inverting. Using $\arg_0(z)$ only lets you find a differentiable inverse to the exponential on the rather limited portion of the complex plane where the imaginary part is restricted to the horizontal strip $(-\pi,+\pi)$.