0
$\begingroup$

I'm struggling to understand why the maximum modulus principle holds. Why is it that a holomorphic function that is nonconstant in an open subset of $\mathbb{C}$ cannot have a local maximum inside the subset? Intuitively, why are no such curves with local maxima not holomorphic?

More specifically, I've been trying to figure out why a holomorphic function on compact complex connected curve must be constant. This directly follows from the maximum modulus principle because there must be a maximum due to compactness, but I don't understand why either of these properties hold.

$\endgroup$
  • 1
    $\begingroup$ It is a consequence of the “open mapping theorem” (which you might find more intuitive or not :) $\endgroup$ – Martin R Jan 12 at 21:41
1
$\begingroup$

If $f$ is holomorphic, it's real and imaginary part are harmonic, i.e. they satisfy Laplace's equation. This is the equation satisfied by a stretched membranes. Such a physical configuration intuitively cannot have bumps. That is a cool way to see it.

$\endgroup$
  • $\begingroup$ I'm sorry, I'm not familiar with Laplace's equation. Can you clarify what you mean by stretched membranes? $\endgroup$ – Alvin Chen Jan 12 at 21:45
  • $\begingroup$ if $u=Re f$, then $\nabla^2 u=0$ (Laplace's equation, en.wikipedia.org/wiki/Laplace%27s_equation). This is nothing more than a rewriting of Cauchy-Riemann conditions. And it is the equation that an actual physical membrane follows. For instance consider a drum, by membrane I mean thin layer of material you hit to produce sound. if you imagine for instance chopping off a bit of the side of the drum, the resting form of the membrane will be the solution of Laplace's equation. $\endgroup$ – David Jaramillo Jan 12 at 21:50
  • 1
    $\begingroup$ I realized I could also mention that for this functions, the value they take is the average of points around it, so that's another intuitive argument for the maximum modulus principle. I guess this is discussed also in whatever book you are using, but if you are looking for references, Chapter 10 of Ulrich's in complex analysis should be a good place to look $\endgroup$ – David Jaramillo Jan 12 at 21:54

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.