Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
Canceling out the common factor x n1 , we obtained the required result (7.92).
…4† J 0
n…x† ˆ‰J n1 …x†J n‡1 …x†Š=2:
…7:93†
Proof: Adding (7.91) and (7.92) and dividing by 2x, we obtain the required
result (7.93).
If we subtract (7.91) from (7.92), Jn…x† 0 is eliminated and we obtain
xJ n‡1 …x†‡xJ n1 …x† ˆ2nJ n …x†
which is Eq. (7.90).
These recurrence formulas (or important identities) are very useful. Here are
some illustrative examples.
Example 7.2
Show that J 0
0…x† ˆJ 1 …x† ˆJ 1 …x†.
Solution:
From Eq. (7.93), we have
J0…x† 0 ˆ‰J 1 …x†J 1 …x†Š=2;
then using the fact that J n …x† ˆ…1† n J n …x†, we obtain the required results.
Example 7.3
Show that
J 3 …x† ˆ 8
x 2 1 J 1 …x† 4 x J 0…x†:
Solution:
Letting n ˆ 4 in (7.90), we have
J 3 …x† ˆ4
x J 2…x†J 1 …x†:
Similarly, for J 2 …x† we have
J 2 …x† ˆ2
x J 1…x†J 0 …x†:
Substituting this into the expression for J 3 …x†, we obtain the required result.
Example 7.4
Find R t
0 xJ 0…x†dx.
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