Results 1 to 2 of 2

Math Help - Fourier series - complex exponential form

  1. #1
    Member Mollier's Avatar
    Joined
    Nov 2009
    From
    Norway
    Posts
    234
    Awards
    1

    Fourier series - complex exponential form

    Hi,

    I was taking a break from numerical analysis and had a sneak peak at Fourier Series.
    I want to rewrite the trigonometric form into the form with complex exponentials.
    I have,

    f(t) = \sum^n_{k=0}a_ksin(2\pi kt) + b_kcos(2\pi kt),

    and rewrite it using,

    sin(2\pi kt) = \frac{e^{2\pi ikt}-e^{-2\pi ikt}}{2i},

    and,

    cos(2\pi kt) = \frac{e^{2\pi ikt}+e^{-2\pi ikt}}{2} to get,

    f(t) = \sum^n_{k=0} \frac{e^{2\pi ikt}(a_k+ib_k)-e^{-2\pi ikt}(a_k-ib_k)}{2i}.

    Not sure where to go from here. I see that there is come complex conjugate action in the numerator...
    Any hints?

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Jan 2009
    Posts
    56
    Your'e basically done, if a_k and b_k are real coefficients then write c_k=sgn(k)(a_k+ib_k) and c_0=b_0 and you get a summation from -n to n.

    If the coefficients aren't real, just write a_k= s_k+ir_k and b_k=d_k+it_k.
    you get that  a_k+ib_k= s_k+id_k-t_k+ir_k
     a_k-ib_k= s_k-id_k+t_k+ir_k

    And you can do the same thing as with the real part, just replacing a_k and b_k with s_k and d_k and also with t_k and r_k.

    Cheers!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex number in exponential form
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 17th 2010, 07:59 PM
  2. Complex Fourier Series & Full Fourier Series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 9th 2009, 06:39 AM
  3. Complex number(exponential form)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 8th 2009, 03:35 AM
  4. Complex Exponential Form
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 31st 2009, 11:10 PM
  5. complex number exponential form
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: February 21st 2009, 10:42 AM

Search Tags


/mathhelpforum @mathhelpforum