Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics. I

A general program is being developed for the systematic and careful evaluation of Green functions in pion-nucleon problems. This paper is the first in a series of projects involving integral method for the calculation of the pion and nucleon propagators. A method is presented for evaluating efficiently and accurately integrals of the form integral r m chi l1 (k 1 r) chi l2 (k 2 r). . . chi ln (k n r) dr, where m and n are arbitrary integers and chi l (x) can be either a spherical Bessel function, j l (x), or a spherical Neumann function, n l (x). The range of integration depends upon the particular problem encountered. The prototype integral studied is I ll'L (kk'P) identical to integral 0 infinity r 2 j l (kr) j l' (k'r)j L (pr)dr whose integrand for large r has a slowly decreasing oscillatory behaviour. Rapid convergence is ensured by rotating, in the complex plane, the upper part of this integral, giving an integrand which decreases exponentially. A scaling formula is used to evaluate I ll'L (kk'p) for very small or very large values of the momenta. Also, it is shown that, if l, l' and L satisfy triangular inequalities and if l+l'+L is even, then k, k' and p must also satisfy triangular inequalities, which is the condition required by the vector delta function delta (k+k'-p). Finally the authors present sum rules and integral relations for I ll'L (kk'p).