# not-so-simple equation to solve

• Nov 28th 2008, 05:39 AM
Simo
not-so-simple equation to solve
Dear all,
I have a tricky problem; I need to solve this equation for $\displaystyle N$

$\displaystyle \binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N\ln\frac{r}{L} =0$

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

Thanks a lot
Simo
• Nov 28th 2008, 06:54 AM
shawsend
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
• Nov 28th 2008, 07:07 AM
Simo
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?