For not an integer the recursion relation for the Bessel function generates something very similar to factorials. These quantities are most easily expressed in something called a Gamma-function, defined as
(10.16) |
Some special properties of function now follow immediately:
The first term is zero, and we obtain
(10.18) |
From this we conclude that
(10.19) |
Thus for integer argument the function is nothing but a factorial, but it also defined for other arguments. This is the sense in which generalises the factorial to non-integer arguments. One should realize that once one knows the function between the values of its argument of, say, 1 and 2, one can evaluate any value of the function through recursion. Given that we find
(10.20) |
Question: Evaluate
,
,
.
Answer:
,
,
.
We also would like to determine the function for . One can invert the recursion relation to read
(10.21) |
.
What is for ? Let us repeat the recursion derived above and find
(10.22) |
This works for any value of the argument that is not an integer. If the argument is integer we get into problems. Look at . For small positive
(10.23) |
Thus is not defined for . This can be easily seen in the graph of the function, Fig. 10.4 .
Finally, in physical problems one often uses ,
(10.24) |
This can be evaluated by a very smart trick, we first evaluate using polar coordinates
(See the discussion of polar coordinates in Sec. 7.1 .) We thus find
(10.26) |