Calculate the derivate

1. Jul 7, 2004

Feynman

:surprise:
Hello,
I've to calculate the derivate of :
$$\displaystyle \sigma_{N}(t):=\sum_{i=1}^{N}g_{i}(t)$$
and $$g_{i}(t)$$ verify the differential equation:

$$\displaystyle \frac{d g_i}{d t}(t) = \sum_{k=1}^{i-1} \frac{1}{k}K(k,i-k) g_{i-k}(t) g_k(t) - \sum_{l=1}^{\infty} \frac{1}{l}K(l,i) g_i(t)g_l(t)$$.

I've to justify:
$$\displaystyle \partial_{t}\sigma_{N}(t):= - \sum_{i,j\leqN;i+j>N}\frac{1}{j}K(j,i) g_i(t)g_l(t)$$
Thanks.