In the book these subjects are discussed on pp. 16-19. Here we shall give a more detailed discussion of the two solutions in the case where the index is an integer. In particular, this leads to the Neumann function.

The power series solution of the Bessel equation (1-52) should satisfy the
recursion relation (1-54). In the coefficients *c*_{n}, *n* is an even integer,
so we take *n*=2*k*. Then

(46) |

To conform to standard notation we replace the index

and

These functions are called Bessel functions of index and , respectively. The general solution of the Bessel equation is therefore

(49) |

where the

This solution breaks down when ,
where *n* is an integer. The reason
is that for
*k*=0,1,2,...,*n*-1 the function
has poles,
and therefore there are no contributions to the sum in (48) for
these values of *k*. Hence

(50) |

Introducing the new summation variable

(51) |

Thus

The reader can easily check that this function satisfies the Bessel equation. Using the -function introduced in the last section, we get

and

Using the result (39) in the last section we get from (53)

(55) |

where

(56) |

To treat the case where we need to use results derived in the previous section. By means of (34) we easily deduce that

To obtain the first of these equations we have computed near the pole, using (34). The values

= | |||

+ | (58) |

Collecting results we finally get by use of the definition (52)

= | |||

- | (59) |

We see that near

(60) |

where

(61) |

We mention that the Neumann function in general is defined by

By means of l'Hospital's rule it is easily seen that this definition agrees with (52) in the case .