- #1

MrGandalf

- 30

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## Homework Statement

Determine the maximum value of [ilatex]|3z^2 - 1|[/ilatex] in the closed disk [ilatex]|z| \leq 1[/ilatex] in the complex plane. For what values of z does the maximum occur?

**The attempt at a solution**

There is a theorem that says that all maximum values of an analytic function in a disc occurs at the bound, so the max values will be on some point on the circle [ilatex]|z|=1[/ilatex]. We can easily see that the maximizing points are [ilatex]z = \pm i[/ilatex], and in those cases we get [ilatex]|3(i)^2 - 1| = |-3-1| = |-4| = 4[/ilatex].