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Bessel function summation

  1. Dec 30, 2012 #1
    1. The problem statement, all variables and given/known data
    What is easiest way to summate
    [tex]\sum^{\infty}_{n=1}J_n(x)[i^n+(-1)^ni^{-n}][/tex]
    where ##i## is imaginary unit.


    2. Relevant equations



    3. The attempt at a solution
    I don't need to write explicit Bessel function so in sum could stay
    [tex]C_1J_(x)+C_2J_2(x)+...[/tex]
    Well I see that terms in the sum will be
    [tex]2iJ_1(x)-2J_2(x)+...[/tex]
    But I search for more sofisticated solution. Is there any way to sum this using ##i^1=i##,##i^2=-1##,##i^3=-i##,##i^4=1##?
     
  2. jcsd
  3. Dec 30, 2012 #2

    lurflurf

    User Avatar
    Homework Helper

    cute

    first i^n+(-1)^n i^-n=2 i^n

    then split into sums over

    2k
    2k+1

    Which have nice well know sums involving sin(x) and J0(x)
     
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