An example of determining the conditions of existence of the various integrals. Existence of the integrals depends upon the behavior of the singularities appearing at the edges of the region of integration.
Consider the single pair overlap integral 
, where 
are as in the definitions for theorem 14a, with the minor change that 
contains all m indices, which may be made without loss of generality. It will be shows that the conditions for existence of this integral are 
, 
, 
and 
. Write the integral as
That , , follows immediately from the fact that 
exists iff and the fact that any 
may independently be near zero for this particular overlap case. Now, either 
or 
may also be near zero. We consider the first case, in which is near zero, and use symmetry to supply the result for the second case. Letting 
and 
, rewrite equation 9.137 in a form that isolates as
Each of the three integrals over 
in equation 9.138 may be done in closed form. Do these integrals using theorem 9a and induction to find
Apply the binomial theorem in equation 9.139 to expand two of the three factors in the integrand, 
and 
, in series. Using these series, note that each term in the series for will contribute the same power of x after the integration over y, while the terms of the series for contribute increasing powers of x after the integration over y. Note also that if the lowest-power-of-x term is integrable over x in a region containing 0, then all of the terms are. Thus, the worst case occurs with the constant term from the binomial series for . After integration over y with this constant term, and considering the small x region of integration, we are left with the integral over x given by
where 
, and C is a constant. This integral exists for . This, symmetry, and the first condition (given by , ) establish the result. The method for more complicated overlap structures is also indicated by this discussion.
The discussion above is of interest in another way: it provides a general method for finding multiple overlap integrals without the use of transform theory.