1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Need help in explaining the calculation of n-th power of a summation

  1. Apr 29, 2010 #1

    I got to the following equation while going through a book. I can't figure out how the second line comes from the first. Can anyone please help me understand?

    (1/2*\sum_{q=-Q}^Q V_s,q .H(w_q) .exp(iw_q t))^n is written as,

    1/2^n * \sum_{q1=-Q}^Q \sum_{q2=-Q}^Q ....... \sum_{qn=-Q}^Q V_s,q1 . V_s,q2 .... V_s,qn .H(w_q1).H(w_q2).....H(w_qn).exp(i(w_q1 +w_q2 +....w_qn).t)

    Please forgive my poor latex style writing.
  2. jcsd
  3. May 2, 2010 #2
    Well, I can forgive it, but unfortunately I can't read it.
  4. May 2, 2010 #3
    [itex](1/2*\sum_{q=-Q}^Q V_{s,q} .H(w_q) .exp(iw_q t))^n[/itex]

    is written as,

    [itex]1/2^n * \sum_{q_1=-Q}^Q \sum_{q_2=-Q}^Q ....... \sum_{q_n=-Q}^Q V_{s,q_1} . V_{s,q_2} .... V_{s,q_n} .H(w_{q_1}).H(w_{q_2}).....H(w_{q_n}).exp(i(w_{q_1} +w_{q_2} +....w_{q_n}).t)[/itex]

    The above is hopefully closer. I don't know if its just me, but I see quite a few posts with unprocessed Latex.
    Last edited: May 2, 2010
  5. May 2, 2010 #4
    And if you have

    [itex](A\sum_{i=r}^s a_i)^n=A^n(a_r+a_{r+1}\dots a_s)(a_r+a_{r+1}\dots a_s)\dots (a_r+a_{r+1}\dots a_s)\text{ (to }n\text{ factors})[/itex]

    you get

    [itex]A^n(\Sigma a_{i_1}a_{i_2}\dots a_{i_n}})[/itex]

    where the sum on the second line runs through all different ways of selecting each [itex]a_{i_k}[/itex] from the corresponding bracket on the rhs of the first line.

    This can be written as

    [itex]A^n\sum_{i_1=r}^s\sum_{i_2=r}^s\dots \sum_{i_n=r}^s a_{i_1}a_{i_2}\dots a_{i_n}}[/itex]

    and as the indices run from r to s in each of the multiple sums, all ways of selecting [itex]a_{i_1}a_{i_2}\dots a_{i_n}[/itex] will occur exactly once.

    With appropriate values inserted from your example this gives you the transition between the first and second lines.
  6. May 3, 2010 #5
    Makes Perfect sense. Thank you so much for the explanation :D and special thanks for forgiving my latex :P
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook