3
$\begingroup$

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 $z$.

$\endgroup$
1
  • $\begingroup$ My guess would be that perhaps the mapping is $z \mapsto \frac{\pi z}{2}$? $\endgroup$
    – Libertron
    Mar 25 '14 at 0:59
7
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.