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!
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.
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)|.
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!
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.
The average of points on the circle lies _within_ the circle unless you average over a constant.
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)|.
At 13:19, shouldn’t the inequality be reversed? Because inside the r-disc, |z| = |f| /r .
I thought that at first, but the wiki page on this proof is more explicit at this point i.e. note the g(z)
15:34: where is this F(0)' =1/F inverse (0)' coming from ?
I think from f compose f inverse is identity and the chain rule.
We need U to be connected right?
Yeah, you can have separate constant values on separate connected components
the best
yeee
first comment!
Your mother must be so proud.