Augmented spherical Neumann function

We note that the augmented spherical Neumann Ntjl(r/St) and Bessel J (r) functions are in this case defined from a tail of radial dependence (r/S ) rather than (r/S) as in the energy-dependent orbital (8.1). The addition theorem (6.13,8.7) which represents the expansion in the sphere at q of the tail of the muffin-tin orbital centred at q must therefore be changed to include the correct radial dependences, i.e. (r/St). From (6.13,8.7) we find... 

It now remains to define the tail N icr) of the augmented muffin-tin orbital in a suitable form. Here we recall that the original spherical Bessel and Neumann functions obey the expansion theorem (5.14), and it is therefore natural to require that the augmented functions and also satisfy this theorem. Hence, we are led to the definition... 

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