Cumulant Expansion: How Does It Work?

[aaaa202]

Can anyone tell me how a cumulant expansion, like the one on the picture, works? If I take the logarithm of the expansion in eq. (29-26) I don't get what is stated below. What should I do to get that?

 

[Svein]

If I take the logarithm of the expansion in eq. (29-26) I don't get what is stated below. What should I do to get that?

I also don't agree with the transfer from (29-26) to (29-27). Either there is something very wrong with the statement or there is something going on backstage that we are not shown.

 

[micromass]

You need to verify that the derivatives of $\log(G(k))$ evaluated at $k=0$ are exactly the Ξ_r
For example: $\frac{d}{dk}\log(G(k)) = \frac{G^\prime(k)}{G(k)}$. Remember that $G(0) = 1$ and $G^\prime(0) = <Y>$. Hence the derivative evaluated at $0$ is $<Y>$, as desired. I will let you compute the other Ξ_r.