Things Which Can Go Wrong with Complex Numbers - Comments

[micromass]

micromass submitted a new PF Insights post

Things Which Can Go Wrong with Complex Numbers

Continue reading the Original PF Insights Post.

 

[haruspex]

I thought Log(z) was defined to return the principal value, i.e. In the range $(-\pi,\pi]$, while log(z) is left as shorthand for the set of values which satisfy $z=e^w$. Thus $log(z)=Log(z)+2\pi n i$.

Likewise, $\sqrt .$ is defined to return a complex number with argument in the range $(-\pi,\pi]$. A difficulty here is that there is no corresponding shorthand (is there?) for the set of solutions to the square root operation.

It might be interesting to develop some generic rules for multivalued functions. E.g. If f() is such an operation, we might write {f(x)} for the set of values and F(x) for the principal value. If f distributes across multiplication (e.g. raising to a power, $(ab)^c=a^cb^c$) then we could write $F(ab)\in \{f(ab)\}\subseteq f(a)f(b)$.

 

[klotza]

If you'll pardon some self-promotion, one of the cool implications of the multi-valued negative logarithm is that it leads to the x^x spindle.

http://klotza.blogspot.com/2015/11/the-sophomores-spindle-all-about.html

 

[jasonRF]

I really like this insight. I wish I had seen a writeup like this 25 years ago while first learning about complex numbers; it wasn't until taking an elective in complex analysis my senior year of college that I finally started to get a handle on this. A link to this insight should become the standard reply to these kinds of questions that show up in the forums.

The use of Log versus log may be a little non-standard. I must admit that when I first skimmed the article I assumed the capital version was the principal branch, but I can never remember how people define the principle branch anyway ($-\pi \leq \theta < \pi$; $-\pi < \theta \leq \pi$; $0 \leq \theta < 2 \pi$, etc) so I always have to check how any given author defines it. When I fully read the insight the notation is clearly defined so I have no problem with it at all.

Great work!