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I have several questions regarding the maximum modulus principle, but first let me interpret my understanding of this theorem:

  • Assuming we have some analytic, non-constant function $\;f:\Omega\subseteq\mathbb{C}\to \mathbb{C}$ where $\Omega$ is a domain, we state that there is no $z_0 \in \Omega$ such that $\left|f(z_0)\right|\ge\left|f(z)\right|$
  • On the contrary, if for an analytic function $\;f$ modulus $\left|f(z)\right|$ takes its maximum inside the domain $\Omega$ then $\;f$ is constant
  • Furthermore for a bounded domain $\Omega \;$ the maximum value of $\;\left|f\right|$ is taken on the boundary of $\Omega$. Formal: $\max\limits_{\overline\Omega}\left|f(z)\right|=\max\limits_{\partial\Omega}\left|f(z)\right|$

Now my questions:

  1. How can I physically interpret (or intuitionally understand) $\left|f(z)\right|$? What significant role does modulus of a complex function play?
  2. Why is it so important then to consider the maximum (or minimum) value of the modulus of a complex function?
  3. Wolfram Mathworld says (in the last sentence) that maximum modulus theorem is not always true on an unbounded domain. What does it mean? Does it mean that modulus of a non-constant function can take its maximum (minimum) inside an unbounded domain?
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  • $\begingroup$ In your first statement, you should add the phrase "for all $z \in \Omega$". As far as the third question, I'm quite sure it is referring to your third statement. If the domain is unbounded, then the maximum need not be attained on the boundary. $\endgroup$
    – Paul Hurst
    Commented Jul 31, 2014 at 22:51
  • $\begingroup$ In this context, I think that $|f(z)|$ should be thought of as an absolute value, or measurement of the magnitude of $f(z)$. It might be interesting to look at the properties of functions defined by $g(z) = |f(z)|^2$ ($g: \mathbb{C} \rightarrow \mathbb{R}$), where $f(z)$ is analytic. $\endgroup$
    – Paul Hurst
    Commented Jul 31, 2014 at 22:56

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