Find a maximum of complex function

Question

I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the function $f:z \mapsto |z^3 + 2iz|$ from $\mathbb C$ to $\mathbb C$

$$ \large { \displaystyle \max_{z \in {\mathbb C},|z| \leq 1} |z^3 +2i z |} $$ Since : $$(\forall z \in \Delta) \quad f(z) \leq 3 $$ is obtained using triangular inequality, we can yet try to find some $z_0 \in {\mathbb C}$ such that $f(z_0)=3$

Does anybody have an idea?

Thanks.

That's a much better problem statement. It tells us that you have worked quite a bit on the problem, and already know a lot about it. Maybe with my hint you can polish it off. — Gerry Myerson, Jun 09 '12 at 13:04
@Steve D: I did not want to talk too much for my English (even beginners). @ Paul Slevin,Gerry Myerson, draks : Thank you for your reception and your instructions for using this nice site by which I edited my post. (and excuse my english) — Mohamed, Jun 09 '12 at 13:07

score 5 · Accepted Answer · answered Jun 09 '12 at 13:03

5

Hint: when do you get equality in the triangle inequality?

answered Jun 09 '12 at 13:03

Gerry Myerson

179,216

@ Gerry Myerson : When the two terms are directely proportional. I well see this . – Mohamed Jun 09 '12 at 13:13
@ Gerry Myereson: Thank's for hint. Using it, i find that $z_0=\frac{\sqrt 2}{2} (1+i)$ gives $f(z_0)=3$ – Mohamed Jun 09 '12 at 13:27
Well done! ${}$ – Gerry Myerson Jun 09 '12 at 23:56

score 5 · Answer 2 · answered Jun 09 '12 at 13:26

5

If you've begun a study of complex functions, you may have seen the Maximum Modulus Principle. Since $z^3 + 2iz$ is a polynomial and entire (analytic in the complex plane), the maximum of $|z^3 + 2iz|$ you seek must occur on the boundary of the unit disk. Gerry's Hint then quickly points you in the right direction!

answered Jun 09 '12 at 13:26

hardmath

37,015

Ah, drat, you type faster +1 – huon Jun 09 '12 at 13:28

score 3 · Answer 3 · answered Jun 09 '12 at 13:27

3

Hint: $z^3 + 2iz$ is differentiable on $\mathbb{C}$ (i.e. holomorphic) so you can apply the maximum modulus principle, and deduce that the maximum of $f$ lies on the boundary of $\Delta$, which has a simple parametrisation, so you can use standard techniques from real one-variable calculus to find the maximum.

(A slightly different approach to Gerry Myerson's, much more complicated in this case but also far more general.)

answered Jun 09 '12 at 13:27

huon

797

@ hardmath and dbaupp : Thank you for the idea of maximum module principle.I was looking for a suitable method for beginning students. It turns out that the Pricipe gives the idea already! Thank you! – Mohamed Jun 09 '12 at 13:37

Find a maximum of complex function

3 Answers3