10.5 Properties of Bessel functions
Bessel functions have many interesting
properties:
Let me prove a few of these. First notice
from the definition that
is even or odd if
is even or odd,
|
(10.40) |
Substituting
in the definition of the Bessel function gives
if
, since in that case we have the sum of positive powers
of
, which are all equally zero.
Let’s look at
:
Here we have used the fact that since
,
[this can also be proven by defining a recurrence
relation for
]. Furthermore we changed summation variables to
.
The next one:
Similarly
The next relation can be obtained by
evaluating the derivatives in the two equations above, and
solving for
:
Multiply the first equation by
and the second one by
and add:
After rearrangement of terms this leads to
the desired expression.
Eliminating
between the equations gives (same multiplication, take
difference instead)
Integrating the differential relations
leads to the integral relations.
Bessel function are an inexhaustible subject
– there are always more useful properties than one knows.
In mathematical physics one often uses specialist books.