**K.E. Schmidt
Department of Physics and Astronomy
Arizona State University
Tempe, AZ U.S.A.
**

Many of you are having trouble taking the limits of a finite system to infinity and converting the sums to the corresponding integrals especially when Bessel functions are involved.

Let's look at examples of taking the limit of two simple systems as their size goes to infinity and converting the corresponding sum of basis functions to an integral. We covered the simplest example in class, the usual coulomb Green's function for free space,

G(,') = ,
| (1) |

In the usual case, we imagine a cubic box of side
*L* ,
with periodic boundary conditions. The Laplacian separates in
cartesian coordinates, and we need to solve the three equations

+ k_{i}^{2}f_{n}(x_{i}) = 0 .
| (2) |

= exp(i')
| (3) |

= (m + n + o) ,
| (4) |

The Green's function is

G(,') = 4
| (5) |

G(,') =
| (6) |

In the limit of
*L* for any nonzero *q* , the integers
*m* , *n* , *o* go to infinity, and the spacing between adjacent
*q* values goes to zero. This means we can convert the sum to an
integral, and for sums spaced by 1,

dn
| (7) |

G(,') = dn dm do .
| (8) |

dn
| = |
dq_{x} = dq_{x}
| |

dm
| = |
dq_{y} = dq_{y}
| |

do
| = |
dq_{z} = dq_{z}
| (9) |

dn dm do = d^{ 3}q .
| (10) |

G(,') = d^{ 3}q
| (11) |

G(,') = dq .
| (12) |

G(,') = du ,
| (13) |

The last integral

I = du
| (14) |

I =
| (15) |

G(,') =
| (16) |

Now that we did it the easy way, let's use a harder method that may
help in taking other limits. Let's again calculate the free
space Green's function, but let's take the limit using Dirichlet
boundary conditions on a sphere of radius *R* that we will let
go to infinity. Separating variables as in class, the eigenfunctions
of are

() = j_{l}(q_{lo}r)Y_{lm}(,)
| (17) |

The Green's function is

G(,') = 4j_{l}(q_{lo}r)j_{l}(q_{lo}r')Y_{lm}(,)Y^{ *}_{lm}(',') ,
| (18) |

In the limit of
*R* for any nonzero *q* value,
the integer *o* must go to infinity, and the spacing between
adjacent *q* values goes to zero. This means the the *o* sum
can be converted to an integral. Just as in the cartesian case,
we need to calculate *do*/*dq*_{l} to change to an integration over *q*_{l} .
Since *o* goes to for any finite *q*_{l} value, the Bessel
function zeroes can be evaluated for large argument, since the small
arguments contribute an amount of measure zero as
*R* .

The assymptotic expansion of *j*_{l}(*x*) is

j_{l}(x) sin(x - l/2) ,
| (19) |

q_{lo} = (l + 2o)
| (20) |

G(,') = 8dq_{l}j_{l}(q_{l}r)j_{l}(q_{l}r')Y_{lm}(,)Y^{ *}_{lm}(',') .
| (21) |

G(,') = 8dqj_{l}(qr)j_{l}(qr')Y_{lm}(,)Y^{ *}_{lm}(',') .
| (22) |

Rather than perform the integration in Eq. 22, we can verify that this is the correct expression by comparing with our previous result. Jackson Eq. 10.43 gives the expansion of a plane wave in terms of spherical Bessel functions,

exp(i ) = 4i^{ l}j_{l}(kr)Y_{lm}(,)Y_{lm}^{*}(,)
| (23) |

We can therefore write

exp(i [ - ']) = (4)^{2}i^{ l}j_{l}(qr)Y_{lm}(,)Y_{lm}^{*}(,)(- i)^{l'}j_{l'}(qr')Y^{ *}_{l'm'}(',')Y_{l'm'}(,)
| (24) |

It is also possible to do the integrals, but we know the result
that must occur, and that means the integrals will be somewhat ugly.
For example for *l* = 0 , the integral can be done readily with contour
integration similarly to the contour integral above to give

dq
| = | ||

= | (25) |

The other integrals could be done similarly, but this is a hard way of getting the standard result.