where . The dummy variable of integration has been changed in order to highlight the fact that is a function of only. Using a variant of the convolution theorem and the fact that the basis functions are real enables the integral to be rewritten as

Using equation (9a) we obtain

(12) |

Introducing differential operators , obtained from by making the replacement

where in Cartesian coordinates, equation (13) becomes

where we have used the fact that the integrand is an even function of for all values of and to change the limits of the integral. From equation (14) no longer appears manifestly symmetric with respect to swapping and . Nonetheless, it still is because under the swap , and .

The three spherical Bessel functions in equation (14)
can all be expressed in terms of trigonometric functions and algebraic
powers of the argument, using the recursion rules
(40,41, Appendix). The product of three
trigonometric functions can always be expressed as a sum of four
trigonometric functions with different arguments, using well-known
identities. The result is to split the integrand up into terms of the
following form:

(15) | |||

These terms are individually singular and generally possess a pole of order on the real axis at and cannot be integrated. However, since we are integrating finite well-behaved functions we know that the total integrand cannot contain any non-integrable singularities. Therefore we can add extra contributions to each term to cancel all the singularities except simple poles, and all these extra terms must cancel when the terms are added together to obtain the whole integrand.

We shall evaluate the integrals using the calculus of residues so
that the general integral to be performed is

(16) |

This integrand now has simple poles lying on the contour of integration at . The residues of these poles are

(18) | |||

Summing the residues to perform the Cauchy principal value integrals, and taking real or imaginary parts as appropriate, we obtain the following results:

where

(21) |

The result for is obtained by summing the results in equations (19,20,22, 23) for all the terms in the expansion of the integrand (14) and then operating with the differential operators .

A second special case occurs when
,
and in this case it is simplest to perform the integral
(10) in real-space using the generalised orthogonality
relation for spherical Bessel functions (43) when
.

(24) |

(25) |