Why does the scattering length it take the form it does?

A brief section on S-wave scattering is found here - http://www.int.washington.edu/users/mjs5/Class_560/lec560_2/node2.html it discusses the physical significance of the cot term. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I’ll summarize and try not to plagiarize at the same time. The cot term, as I’m sure you know, just comes from the mathematics. Many years ago others also questioned the presence of the cot term. They went on to show mathematically that using a power series expansion, it represents correctly the limits of the scattering amplitude. Also, it can be used to calculate the bound states as well.