1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex analysis - maximum/modulus principle

  1. May 3, 2006 #1
    Suppose that f is analytic on a domain D, which contains a simple closed curve lambda and the inside of lambda. If |f| is constant on lambda, then either f is constant or f has a zero inside lambda ...
    i am supposed to use maximum/modulus principle to prove it ...

    here is my take:
    if f is constant, i dont see a reason why |f| wouldnt be constant :)

    if f is not constant, then the max/min principle applies ...
    meaning |f| can not have any local max/min on D
    now i am lost at this point ...

    i would also like a little clarification on what f is constant on lambda means, because the way i see it .. .if lambda is a closed loop (say a circle), then how can f be always increasing /decreasing ...? maybe i am misinterpreting the problem
  2. jcsd
  3. May 3, 2006 #2


    User Avatar
    Science Advisor
    Homework Helper

    if your curve lambda is a(t), t from 0 to 1 then |f| constant on lambda means |f(a(t))|=K for some constant K and all t. Normally, |f(a(t))| can wobble all over the place as it winds around the loop, it can't be increasing on all of [0,1].

    if f is a constant on some set, then any function of f is a constant, specifically it's modulus.

    What happens if you have no zero? Consider 1/f.
  4. May 3, 2006 #3
    thats what i thought ....

    but i still have no insight about f having any zero inside lambda ....

    like i kinda see it visually ... like i know it makes sense ... but i got no clue how to "prove" it
  5. May 3, 2006 #4


    User Avatar
    Science Advisor
    Homework Helper

    If there's no zero of f, you can apply maximum modulus principle to both f and 1/f. What happens?
  6. May 3, 2006 #5
    if i apply max/min modulus to both f and 1/f, then it means both f and 1/f dont have a local max/min ... so what does it tell me?
    i'm sorry ... i dunno if it's lack of confidence or what .. but i still dont see it
  7. May 3, 2006 #6


    User Avatar
    Science Advisor
    Homework Helper

    If you have minimum modulus principle as well, then you can forget about 1/f.

    Apply min/max modulus to f then. What does this tell you about |f| on the interior compared to it's values on the boudnary? (remember |f| is a continuous function, lambda+interior is a closed and boundeed set)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Complex analysis - maximum/modulus principle
  1. Complex Analysis (Replies: 4)

  2. Complex analysis (Replies: 5)

  3. Complex analysis (Replies: 10)