Exponential form of a Fourier series

Discussion in 'Math' started by amilton542, Nov 27, 2014.

  1. amilton542

    Thread Starter Active Member

    Nov 13, 2010
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    Does anyone know why the exponential form of a Fourier series works from - infinity to +infinity?

    I'm using Advanced Engineering Mathematics by Kreyszig but I find his derivations are a bit "arty farty" if that makes sense. I don't like them. He omits too much.
     
  2. WBahn

    Moderator

    Mar 31, 2012
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    Your question is ambiguous.

    Are you talking about -∞ to +∞ in the time domain as opposed to 0 to +∞ in the time domain for the typical Laplace transform?

    If so, the reason is that the typical Laplace transform that we usually use is the one-sided Laplace transform and it blows up for t<0. There is the two-sided Laplace transform that works over all time, but it is quite a bit more troublesome to use.
     
  3. studiot

    AAC Fanatic!

    Nov 9, 2007
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    There's no magic about this

    Application of the Euler formulae connecting sin and cos with a complex exponential (did you mention it was complex?) yields

    f(x) = {c_0} + \sum\limits_1^\infty  {\left( {{c_n}{e^{inx}} + {k_n}{e^{ - inx}}} \right)}

    The summation runs from n=1 to n = infinity.

    Put that into the first term in the sum you get all the terms from 1 to infinity.

    Put it into the second and you get all the terms from minus infinity to -1,
    The additional term for n=0 when the summationlimits are changed comes out of the constant before the old summation, which is then absorbed into the new summation.

    If you like you can decompose it thus


    f(x) = \sum\limits_{ - \infty }^{ - 1} {\left( {{c_n}{e^{inx}}} \right)}  + {c_0}{e^{0ix}} + \sum\limits_1^\infty  {\left( {{c_n}{e^{inx}}} \right)}


    since both expressions have the same form these can be combined into a single expression running from minus infinity to plus infinity.

    f(x) = \sum\limits_{ - \infty }^\infty  {{c_n}} {e^{inx}}

    Which is the exponential form you are asking about.
     
    Last edited: Nov 28, 2014
  4. WBahn

    Moderator

    Mar 31, 2012
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    My apologies -- I thought you were asking about the Fourier transform, not the Fourier series. Studiot gave a good answer.
     
  5. amilton542

    Thread Starter Active Member

    Nov 13, 2010
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    Yeah I get it now. Change the summation limits of the exponential with the negative exponent then it becomes positive. Another thing I was a bit skeptical about was how the latter expression "masks" the constant but I see what you've done now.

    O.K thanks.
     
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