The question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm familiar with the branch cut for the principal branch of the logarithm given by the slit plane $\mathbb{C}- (-\infty, 0)$ where $-\pi \leq \theta < \pi$, with $\theta$ the argument of $z$.
[Math] Determining a branch of logarithm
complex-analysis
Best Answer
Define for each $z\in\mathbb{C}$, $$Arg(z)=\theta,\ \ \ \ \ \text{such that}\ \ \theta\in[-2\pi,0)\ \ \text{and}\ \ |z|e^{i\theta}=z.$$
Now define, $$f(z)=ln|z|+Arg(z)i\ \ \ \ \ z\in\mathbb{C}.$$
Function $f$ is called a branch of the Logarithm, since $f$ is analytic on $\mathbb{C}-[0,\infty)$ (quite literally we can prove $\lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}$ exists for every $z_0\in\mathbb{C}-[0,\infty)$) and (more importantly), $$e^{f(z)}=z\ \ \ \ \ \forall\ \ z\in\mathbb{C}-[0,\infty).$$ Likewise, $$f(-i)=\frac{-\pi}{2}i,$$ as desired.
Understanding Logarithms really boils down to understanding the terminology. So lets be clear, a logarithm (or branch of the logarithm if you want) is simply an analytic function, $f$, such that $e^{f(z)}=z$ for every $z\in A$, where $A$ is some region that $f$ is analytic on. There are many many functions (logarithms) that satisfy this property, in particular are the logarithms that remove a ray from the complex plane, i.e. $$f(z)=ln|z|+Arg(z)i.$$ This particular $f$ happens to be analytic and $e^{f(z)}=z$ for all $z\in \mathbb{C}-Ray$. You choose the Ray, by requiring that $Arg(z)\in (A,A+2\pi)$ for some $A$.
Not all logarithms are like $f$ (the branches of the logarithm). For example, define,
$$g(z)=ln|z|+(Arg(z)+2\pi n)i,$$ for $$Arg(z)+2\pi n<|z|<Arg(z)+2\pi (n+1)\ \ \text{and}\ \ Arg(z)\in[0,2\pi).$$
Function $g(z)$ is defined on a spiral-ish region (unless I made a terrible error). It is analytic where it is defined and most importantly $e^{g(z)}=z$ for every $z$ where it is defined. Hence, $g(z)$ is a logarithmic function. However, $$g(2)=ln|2|\ \ \ \text{and}\ \ \ g(7)=ln|7|+2\pi i,$$ and unusual property for branches of the logarithm.