Investigation of the formula for the average of two S-matrix elements in compound nucleus reactions

Recently the problem of calculating the average of two S-matrix elements describing compound nucleus reactions was solved by using the technique of generating functions and grassmann integration. In this paper we analyze the solution, a three-dimensional integral, both analytically and numerically. We study the expansion of the integral in powers of the transmission coefficients and the expansion in inverse powers of the transmission coefficients. The former one, which also contains logarithmic terms, is shown to agree with previous work. The latter one is identical to the expansion obtained with the help of the replica trick. After a suitable change of integration variables the integrand becomes finite everywhere in its domain and numerical integration is without any problem. Earlier Monte Carlo calculations are confirmed. The average of two S-matrix elements at the same energy obtained by a Mexican group using general properties of the S-matrix and the maximum entropy principle is shown to coincide with the microscople result.