Why the K-matrix is introduced

R has the advantage of being real and meromorphic, but the disadvantage that it depends on specific choices of a and b. In reality, this is not always as great a handicap as it may seem. In fact, it may provide useful numerical methods by which R can be computed. However, it is a disadvantage if one seeks to establish general properties or theorems, which by their nature should be exact results. 

On the other hand, S is independent of a and b. The problem is that, in general, it does not have a simple pole structure. It is always possible to write S as a. sum of poles  

It is a theorem that, if R is a meromorphic function, then the expression (AR + B)/(CR + D), where A, B, C and D are constant is also meromorphic. Hence K is real, independent of a and b and meromorphic (i.e. it possesses a simple pole structure).2 

2 Strictly, K, although real, is not completely meromorphic, because it has a branch point at k — 0. This, however, is immaterial for the present discussion. Note also that the condition for K to be meromorphic in the range of interest requires that tan ka be at least approximately constant. 

In this sense, K is more fundamental than R. For example, if b is chosen as above to make A = 0, then all E coincide with the resonance energies in S. It follows that the xa equal the resonance wavefunctions, at least for r o. Now consider the dependence of the theory on a, which can normally be omitted since ka is usually small (see, e.g., [370]) and imagine o to be unphysically large. The dependence of R on a is then simple and explicit in this case, instead of (8.12), 

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