1. Limited time only! Sign up for a free 30min personal 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!

Proof that e^z is not a finite polynomial

  1. Mar 31, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z.


    3. The attempt at a solution

    The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least one or more roots. e^z can never equal zero for hence this is a contradiction.

    Is it okay for me to apply the fundamental theorem of algebra like this or am I kind of using a bit too much machinery here?
     
  2. jcsd
  3. Mar 31, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Your proof is wrong. Indeed: [itex]e^{2\pi i}=0[/itex], so the function DOES have a root!!
     
  4. Mar 31, 2012 #3
    Wait, what? [itex]e^{i2\pi} = 1[/itex]
     
  5. Mar 31, 2012 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Woow, I'm stupid today. :cry:

    I'm sorry, your proof is alright!! I obviously need to get some sleep.
     
  6. Mar 31, 2012 #5
  7. Mar 31, 2012 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    That looks like too much machinery (at least for my tastes); I would rather just argue that a polynomial of degree n has (n+1)st derivative = 0 identically, and ask whether that can happen for exp(z).

    RGV
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof that e^z is not a finite polynomial
  1. Polynomial Proofs (Replies: 4)

  2. Polynomials Proof (Replies: 1)

  3. Polynomial Proof (Replies: 6)

Loading...