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!

Half-range sine expansion

  1. Sep 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Hi all.

    I have to find the half-range sine expansion of a function f(x) = 1 for 0<x<2. My question is: This function is not piecewise smooth, so why does the book ask me to do this?
  2. jcsd
  3. Sep 20, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Hi Niles! :smile:

    It looks piecewise smooth to me …

    what definition of "piecewise smooth" are you using? :smile:
  4. Sep 21, 2008 #3
    That f and f' must be piecewise continuous, i.e. that there are a finite number of discontinuities, where the limits exists.

    f is "piecewise" continuous, so that is OK. But f' is defined on the interval 0 < x < 2 with no discontinuities?
    Last edited: Sep 21, 2008
  5. Sep 21, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    f' = 0 on the interval 0 < x < 2 … that's continuous, isn't it? :wink:
  6. Sep 21, 2008 #5
    Yes, but if there are no discontinuities for either f or f', then f is simply just smooth? Does this count as f being piecewise smooth?
  7. Sep 21, 2008 #6
    That's right. In this case, there are two pieces, and in each piece, f and f' are both continuous.
  8. Sep 21, 2008 #7
    According to my book, if a function f is defined on an interval 0 < x < p, and if f is piecewise smooth, then it has a sine series expansion.

    In this case f is defined on 0 < x < 2. but it is only smooth - not piecewise smooth - on this interval?
  9. Sep 21, 2008 #8


    User Avatar
    Science Advisor
    Homework Helper

    Everything that's smooth is a fortiori (all the more so) piecewise smooth! :biggrin:
  10. Sep 21, 2008 #9
    So "smooth > piecewise smooth"?

    I just thought that there had to be some discontinuities.
  11. Sep 21, 2008 #10


    User Avatar
    Staff Emeritus
    Science Advisor

    Generally speaking "piecewise smooth" does include "smooth". But here, since you are talking about a "half-range sine expansion", you are assuming that the function is odd (as is sin(nx) for all x). That is, the full range is -2 to 2 and f(x)= -1 for -2< x< 0, 1 for 0< x< 2 and is periodic over all real numbers with period 4. That is surely "piecewise" smooth.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Half-range sine expansion