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!

Homework Help: Analysis Proof: Sum of odd and even functions.

  1. Apr 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Show f(x) can be expressed as the sum of E(x) and an odd function O(x).

    f(x) is defined for all x (assume domain D symmetric about 0)

    [tex]f(x) = \frac{f(x)+f(-x}{2}[/tex]
    How does it look if [tex]f(x) = e^x[/tex]?

    2. Relevant equations

    3. The attempt at a solution

    So I got [tex]\frac{f(x)+f(-x)+f(x)-f(-x)}{2}[/tex]

    as my solution for that.

    From that I need to answer: How does it look if [tex]f(x) = e^x[/tex] ? I'm not sure what to do to show that.
  2. jcsd
  3. Apr 8, 2010 #2

    Char. Limit

    User Avatar
    Gold Member

    First: Is that a sum of odd and even functions?

    Second: Well, try sticking e^x in for f(x) and see what functions you get.
  4. Apr 8, 2010 #3
    I stuck in e^x for f(x) and got e^x. Is this supposed to represent something?
  5. Apr 8, 2010 #4

    Char. Limit

    User Avatar
    Gold Member

    You should get...

  6. Apr 8, 2010 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Well, since the two sides are equal I would hope you get ex back when you cancel everything. But notice without canceling what you get

    [tex]e^x = \frac{e^x+e^{-x}}{2} + \frac{e^x-e^{-x}}{2} = cosh(x)+sinh(x)[/tex] which is a way of writing [tex]e^x[/tex] as a sum of an odd and an even function
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook