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Fourier Series: Can even functions be changed to odd?

  1. Apr 27, 2013 #1
    When creating a Fourier series for a function [itex]f(x)[/itex], I consider whether the function is odd or even first. Yet, often these functions are in the positive region [itex] [0, L] [/itex].

    Since [itex]f(x)[/itex] is only defined in this region, can I change the function to get a desired parity? By example, my concern originated with the function [itex]f(x) = x \sin x[/itex]. This is an even function, but I could modify the function as [itex]f(x) = |x| \sin x [/itex] to make it odd while retaining the desired information in [itex] [0, L] [/itex].

    Can this be done? And are there problems with doing this?

  2. jcsd
  3. Apr 27, 2013 #2
    What do you mean with "retaining the desired information"? What exactly is it that you want to do?
  4. Apr 27, 2013 #3
    Meaning that [itex]x \sin x[/itex] on [itex][0,\pi][/itex] is the same as [itex]|x| \sin x[/itex] on [itex][0,\pi][/itex].

    The application is usually for solving heat equations. So the region is usually x = 0 to x = L in the problems I see. Again, another example: When I work out all the details of a solution by eigenfunction expansion based on homogeneous Dirichlet boundary conditions, I end up with a Fourier sine series equal to some function [itex]f(x)[/itex], where [itex]f(x)[/itex] is the initial condition u(x,0). But to have a Fourier sine series, [itex]f(x)[/itex] must be an odd function, right?
  5. Apr 27, 2013 #4
    OK. You're right then. The right way to find a sine series converging to ##f## is to extend it to an odd function on entire ##[-L,L]##. So you'll have to work with ##|x|\sin(x)## on ##[-\pi,\pi]##, like you suggested.
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