Complex analysis: Maximum modulus principle

This lecture is part of an online undergraduate course on complex analysis.
We prove the maximum modulus principle, and use to to prove the fundamental theorem of algebra and to find the symmetries of the unit disk.
For the other lectures in the course see • Complex analysis

[@johnchessant3012]

Another amusing application of m.m.p. that I read in Tristan Needham's "Visual complex analysis":
Which point inside a square maximizes the product of distances to the four vertices? By m.m.p., it can't possibly be the center!

[@mathstrek]

At 3:53, you can only conclude: |f(z)| = |f(0)| for all z on C. Set f(z) = u + iv with u² + v² = c.
To proceed, differentiate to get uu_x + vv_x = 0 and uu_y + vv_y = 0. Then Cauchy-Riemann says u_x = v_y and u_y = -v_x which gives us (u² + v²)u_x = 0 after some manipulation. If c ≠ 0, this forces u_x = 0 and similarly u_y = 0; thus v_x = v_y = 0 as well.

[@rgicquaud]

Nice video as always. You can also say that if f is non constant you can write f(z) = a_0 + a_n z^n + h.o.t. with a_0 and a_n both non zero. Then setting z = \lambda v for some well chosen "direction" v and small lambda, you get that |f(\lambda v)| > |a_0| = |f(0)|.

[@tavishu]

At 13:19, shouldn’t the inequality be reversed? Because inside the r-disc, |z| = |f| /r .

[@Alex_Deam]

I thought that at first, but the wiki page on this proof is more explicit at this point i.e. note the g(z)

[@ster2600]

We need U to be connected right?

[@_photography_]

Yeah, you can have separate constant values on separate connected components

[@xcl9189]

15:34: where is this F(0)' =1/F inverse (0)' coming from ?

[@tingxiangzou1107]

I think from f compose f inverse is identity and the chain rule.

[@migarsormrapophis2755]

yeee

[@SG-kj2uy]

first comment!

[@cycklist]

Your mother must be so proud.