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Fourier Series

  1. Mar 16, 2005 #1
    I'm a little confused about the difference between the half range fourier series and the full range fourier series. What is the difference between the two in an odd function like f(x)=x and an even function like f(x)=x^2 ? Maybe an example to clear things up. Thank you.
     
  2. jcsd
  3. Mar 17, 2005 #2

    matt grime

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    Given a function from [0,1] to R where f(0)=0, there are several ways to make from it a periodic function on some interval. The simplest is to repeat the function by setting f(x+1)=f(x). However, there is another option, and one that cuts your work in half.

    We extend the function to [-1.1] by

    1) refelcting in the y axis, ie set f(-x) = f(x)

    2) rotate 180 degrees about the origin by setting f(-x)=-f(x)

    then repeat these to get a periodic function.

    The first is even on the interval [-1,1] so it only has cosines in its fourier series, the second is odd so only has sines.

    So, take f(x)=x on [0,1] if we extend it to an even funciton on [-1,1] then we get |x|, if we extend to an odd function we just get x. The first has a fourier series only using cosines, the second only using sines.
     
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