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A few weeks ago we did Liouville's theorem, which states that any bounded complex function is continuous. Okay... (I can't really imagine the picture of a function which is bounded to be constant, e.g. sin(z) is bounded, at least it should, because the Real part is bounded by [-1,1] but, if I picture the sine function it is clearly

**not**constant. So I am confused).

And today we did the maximum principle, which is more general than Liouville's (but still a corollary). It states that if the function reaches a maximum (e.g. local), then the function is constant. The proof makes sense and all, it's pure logic.

But if I think about it graphically, it makes none.

So if anyone of you guys could help me understand this concept I would appreciate. Also, how would you apply the maximum principle to a complex function? Say, e^(z^2), which in the real setting reaches a maximum?

Thank you