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Math Help - not-so-simple equation to solve

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    22

    not-so-simple equation to solve

    Dear all,
    I have a tricky problem; I need to solve this equation for N

       \binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N\ln\frac{r}{L} =0<br />

    Is it possible in some way?
    The only constraints are R \leqslant L, R,L>0

    Thanks a lot
    Simo
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Going gets tough, tough resort to numerical methods. The following is Mathematica code to calculate the value of n for the supplied values of R and L. First out is the expression for the sum, second out is the numerically calculated value of n which makes the sum equal to zero, and the third out is a back-substitution of the the root for a check.

    Code:
    In[234]:=
    R = 3; 
    L = 5; 
    formula1 = Sum[Binomial[R, r]*(-1)^(R - r)*
        (r/L)^n*Log[r/L], {r, 1, R}]
    root = n /. FindRoot[formula1 == 0, 
        {n, 1.2}]
    formula1 /. n -> root
    
    Out[236]= -(3/5)^n Log[5/3] + 3 (2/5)^n Log[5/2] - 3 5^-n Log[5]
    
    Out[237]= 1.38015
    
    Out[238]= -5.55112*10^-17
    Attached Thumbnails Attached Thumbnails not-so-simple equation to solve-sumroot.jpg  
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  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
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    Ok,
    I already have a numerical solution like that, but it is not interesting for me.
    Is not possible to find an analytical solution, without resort to numerical methods?
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