Results 1 to 4 of 4

Thread: Help!!! Fourier

  1. #1
    Newbie Quantum_Alpha's Avatar
    Joined
    Sep 2007
    Posts
    3

    Exclamation Help!!! Fourier

    My def. of FT is
    $\displaystyle \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt$, T= period

    * How to calculate Fourier transform for
    $\displaystyle f1(t)= e^{-t},$ for $\displaystyle t >= 0$; and $\displaystyle = e^t$ for $\displaystyle t<0$ (Which is the period here???)

    *$\displaystyle f2(t)=2,$ for $\displaystyle -1/2<t<1/2 $, this gives me $\displaystyle \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty$ I don't remember what to do with that...
    Maple says that FT in this case is $\displaystyle 2\pi\delta(\pi)$.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    10
    Quote Originally Posted by Quantum_Alpha View Post
    My def. of FT is
    $\displaystyle \int_{-\infty}^{\infty}f(t)e^{\frac{i2\pi t}{T}}dt$, T= period

    * How to calculate Fourier transform for
    $\displaystyle f1(t)= e^{-t},$ for $\displaystyle t >= 0$; and $\displaystyle = e^t$ for $\displaystyle t<0$ (Which is the period here???)

    *$\displaystyle f2(t)=2,$ for $\displaystyle -1/2<t<1/2 $, this gives me $\displaystyle \frac{-e^{\alpha t}}{\alpha} |_{-\infty}^{\infty}=\infty$ I don't remember what to do with that...
    Maple says that FT in this case is $\displaystyle 2\pi\delta(\pi)$.
    This gives me the impression that you are confusing Fourier series with Fourier transforms.

    If a function is periodic, with period T, then it has a Fourier series, in which the n-th (complex) Fourier coefficient is $\displaystyle \hat{f}(n) = \frac1{T}\int_0^Tf(t)e^{i2\pi t/T}dt$.

    Fourier transforms are for functions (non-periodic) that are integrable over the whole real line, and the Fourier transform is a function, defined by $\displaystyle \hat{f}(w) = \int_{\infty}^{\infty}f(t)e^{-iwt}dt$.

    For the two functions f_1 and f_2, the integrals for the Fourier transforms are
    $\displaystyle \hat{f}_1(w) = \int_{-\infty}^0e^{t}e^{-iwt}dt + \int_0^{\infty}e^{-t}e^{-iwt}dt$,
    $\displaystyle \hat{f}_2(w) = 2\int_{-1/2}^{1/2}e^{-iwt}dt$.
    Last edited by Opalg; Dec 22nd 2007 at 11:42 AM. Reason: correcting error in formula for Fourier coefficient
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie Quantum_Alpha's Avatar
    Joined
    Sep 2007
    Posts
    3
    Ok, I will resolve these integrals, I hope not to have problems with the $\displaystyle \infty$'s
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie Quantum_Alpha's Avatar
    Joined
    Sep 2007
    Posts
    3
    Gives me $\displaystyle \hat{f}_1(w)=\frac{2}{1-iw}$, and $\displaystyle \hat{f}_2(w) = -\frac{2}{iw}(e^{\frac{-iw}{2}}-e^{\frac{iw}{2}})$.
    Are they ok?

    (Oh! About period, it was because in my def. I have $\displaystyle w =2\pi/T$)
    Last edited by Quantum_Alpha; Dec 21st 2007 at 05:27 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Fourier
    Posted in the Calculus Forum
    Replies: 8
    Last Post: May 5th 2011, 09:08 PM
  2. fourier
    Posted in the Calculus Forum
    Replies: 9
    Last Post: Jun 7th 2010, 08:02 AM
  3. Complex Fourier Series & Full Fourier Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Dec 9th 2009, 05:39 AM
  4. from fourier transform to fourier series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Feb 1st 2008, 06:35 AM
  5. Fourier of e-|t|
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Oct 23rd 2007, 01:22 PM

Search Tags


/mathhelpforum @mathhelpforum