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!

Simplifying an expression involving complex exponentials

  1. Nov 7, 2012 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data

    Simplify the following expression:
    \sum_{ \alpha_1 + \cdots + \alpha_n = k} e^{i(\alpha_1 \theta_1 + \cdots + \alpha_n \theta_n)}

    2. Relevant equations

    [tex]\alpha_n = k - \sum_{j=1}^{n-1} \alpha_j[/tex]
    [tex]0 = \sum_{j=1}^{n} \theta_j[/tex]

    3. The attempt at a solution

    I am trying to simplify the expression above into something a little more manageable. This expression comes from determining the characters of irreps of SU(n), so if that gives anyone ideas on how to simplify this, then any advice would be much appreciated.
  2. jcsd
  3. Nov 8, 2012 #2
    I guess the first step I would do is to expand the sum of states over all allowed values of {α12,...}. So instead of having 1 sum, try writing it in the "traditional" form with n-1 sums;
    [tex] \sum_{\alpha_1 = 0}^k \sum_{\alpha_2 = ...} ... \sum_{\alpha_{n-1} = ...} [/tex]
    There's no sum over the last alpha because you already know it's value.

    Then "all" you need to do is to do geometric series over and over again.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Simplifying an expression involving complex exponentials