# Ursell function

In statistical mechanics, an Ursell function or connected correlation function, is a cumulant of a random variable. It is also called a connected correlation function as it can often be obtained by summing over connected Feynman diagrams (the sum over all Feynman diagrams gives the correlation functions).

If X is a random variable, the moments sn and cumulants (same as the Ursell functions) un are related by the exponential formula:

$\operatorname E(\exp(zX)) = \sum_n s_n \frac{z^n}{n!} = \exp\left(\sum_n u_n \frac{z^n}{n!} \right)$

(where E is the expectation). Formally, the quantities un are functions of the random variable X, in the same way as the expectation:

$u_n= u_n(X).$

The Ursell functions for multivariate random variables are defined analogously to the above, and in the same way as multivariate cumulants.[1]

The function was named after Harold Ursell, who introduced it in 1927.

## References

1. ^ Shlosman, S. B. (1986). "Signs of the Ising model Ursell functions". Communications in Mathematical Physics 102 (4): 679–686. Bibcode:1985CMaPh.102..679S. doi:10.1007/BF01221652. edit