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Other definition for fourier series

  1. Jan 24, 2014 #1
    Is correct to define fourier series like:

    [tex]f(t)=\sum_{k=0}^{\infty}a_k \cos \left (\frac{2 \pi k t}{T} \right ) + b_k \sin \left (\frac{2 \pi k t}{T} \right )[/tex]

    Where ak and bk:

    [tex]a_k=\frac{1}{T} \int_{-T}^{+T} f(t) \cos \left (\frac{2 \pi k t}{T} \right ) dt[/tex]

    [tex]b_k=\frac{1}{T} \int_{-T}^{+T} f(t) \sin \left (\frac{2 \pi k t}{T} \right ) dt[/tex]

    ?
     
  2. jcsd
  3. Jan 24, 2014 #2
    No. You are counting the period twice.
    If this were true, could you expand something like [itex]\cos \left (\frac{\pi t}{T} \right )[/itex]?
     
  4. Jan 25, 2014 #3
    I don't understand your answer
     
  5. Jan 27, 2014 #4
    I take this topic to introduce another question: in wikipedia, I found others difinitions to fourier series:

    [tex]f(t)=A_0+\sum_{n=1}^{\infty } A_n cos\left ( \frac{2 \pi n t}{T}-\phi_n \right )[/tex]
    where:

    ##A_0 = \frac{1}{2}a_0##
    ##A_n = \sqrt{a_{n}^{2}+b_{n}^{2}}##
    ##\phi_n = tan^{-1}\left ( \frac{b_n}{a_n} \right )##


    and:

    [tex]f(t)=\gamma_0+2\sum_{n=1}^{\infty } \gamma_n cos\left ( \frac{2 \pi n t}{T}+\varphi_n \right )[/tex]
    where:

    ##\gamma_0 = c_0##
    ##\gamma_n = abs(c_n)##
    ##\varphi_n = arg(c_n)##



    I'd like to know if ##\varphi_n## is or isn't equal to ##\phi_n## ?
     
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