(985) |

where

Note that Neumann functions are allowed to appear in the above expression, because its region of validity does not include the origin (where ). The logarithmic derivative of the th radial wavefunction just outside the range of the potential is given by

(987) |

where denotes , etc. The above equation can be inverted to give

Thus, the problem of determining the phase-shift is equivalent to that of determining .

The most general solution to Schrödinger's equation inside the range of the potential ( ) that does not depend on the azimuthal angle is

(989) |

where

(990) |

and

The boundary condition

ensures that the radial wavefunction is well-behaved at the origin. We can launch a well-behaved solution of the above equation from , integrate out to , and form the logarithmic derivative

(993) |

Because and its first derivatives are necessarily continuous for physically acceptible wavefunctions, it follows that

(994) |

The phase-shift is obtainable from Equation (988).