0
$\begingroup$

I took only a first course of complex analysis.

I have learned some multi-valued functions which consistently extend the classical real functions such as Log and Trigonometric functions.

However, I don't see any advantage of considering all the braches. I think it's really convenient to consider $(-\pi,\pi]$ branch so that I can treat those multi-valued function as single-valued functions.

Moreover, it's written in wikipedia that if one knows Riemann Surface, there is no need to separate branches.

Is it okay to just not consider all branches , but to focus on the Principal values?

$\endgroup$
1
$\begingroup$

$\log(z)$ is a smooth multivalued function; it is everywhere differentiable. This is a rather nice feature for doing calculus, complex analysis in particular, because many theorems and calculations only work for continuous or differentiable functions.

For example, $\log(z)$ is an antiderivative of $1/z$ on the whole of the complex plane excluding the origin. $\mathop{\mathrm{Log}}(z)$ is not.

Sometimes, discontinuity is a bigger pain than multivaluedness.

Sometimes you can even have your cake and eat it too: sometimes you work in a domain that intersects the negative real axis, but does not surround the origin. In such a case, you can pick a branch cut for $\log(z)$ in which the logarithm is single-valued and differentiable. You can't do that if you only allow yourself to ever use the principal value.

$\endgroup$
  • $\begingroup$ Do you mean "a" branch cut of $log(z)$ by an antiderivative of $1/z$? $\endgroup$ – Rubertos Sep 13 '14 at 6:27
  • $\begingroup$ And what do you mean $log(z)$ is differentiable everywhere? Do you mean "we can choose a branch of $log(z)$ which differentiable at a given point"? $\endgroup$ – Rubertos Sep 13 '14 at 6:29
  • $\begingroup$ @Rubertos: It need to be filled in with a suitable definition of derivative for a multi-valued function. There are surely standard ways to go about this, although I can't say I've encountered it. Probably something to do with sheaves and/or Riemann surfaces. One natural definition that does spring to mind is to define the derivative as the multivalued function whose values at any point are given by looking at every branch of the original function and taking the derivative on the branch. In the case of $\log(z)$, you have that the derivative on every branch is $1/z$. $\endgroup$ – Hurkyl Sep 13 '14 at 6:43
  • $\begingroup$ But really, at my depth of knowledge of complex analysis, the "derivative of a multi-valued function" is a motivational idea rather than a precise idea, and gets turned into a precise statement in an ad-hoc fashion as I'm working on a problem and I want to use that idea. $\endgroup$ – Hurkyl Sep 13 '14 at 6:45
  • $\begingroup$ Now I clearly got it. Thank you very much :) $\endgroup$ – Rubertos Sep 13 '14 at 6:46

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.