in which is a constant (non-negative if it is real), is called the Bessel’s equation. We derive its general solution by trying the series form
We substitute (2) and the derivatives of the series in (1):
Thus the coefficients of the powers , , and so on must vanish, and we get the system of equations
The last of those can be written
Because , the first of those (the indicial equation) gives , i.e. we have the roots
Let’s first look the the solution of (1) with ; then , and thus
From the system (3) we can solve one by one each of the coefficients , , and express them with which remains arbitrary. Setting for the integer values we get
(where ). Putting the obtained coefficients to (2) we get the particular solution
In order to get the coefficients for the second root we have to look after that
or . Therefore
where is a positive integer. Thus, when is not an integer and not an integer added by , we get the second particular solution, gotten of (5) by replacing by :
the solution is called the Bessel function of the first kind and of order and denoted by . The similar definition is set for the first kind Bessel function of an arbitrary order (and ). For the general solution of the Bessel’s differential equation is thus
where with .
The explicit expressions for are
which are obtained from (5) and (6) by using the last for gamma function.
E.g. when the series in (5) gets the form
Thus we get
analogically (6) yields
and the general solution of the equation (1) for is
In the case that is a non-negative integer , the “+” case of (7) gives the solution
but for the expression of is , i.e. linearly dependent on . It can be shown that the other solution of (1) ought to be searched in the form
. Then the general solution is .
The first kind Bessel functions of integer order have the generating function :
of the coefficients of Laurent series. Setting to this , , gives
The paths and go once round the origin anticlockwise in the -plane and -plane, respectively. Since the residue of in the origin is , the residue theorem (http://planetmath.org/CauchyResidueTheorem) gives
This that has the Laurent expansion (8).
By using the generating function, one can easily derive other formulae, e.g. the of the Bessel functions of integer order:
Also one can obtain the addition formula
and the series of cosine and sine:
- 1 N. Piskunov: Diferentsiaal- ja integraalarvutus kõrgematele tehnilistele õppeasutustele. Kirjastus Valgus, Tallinn (1966).
- 2 K. Kurki-Suonio: Matemaattiset apuneuvot. Limes r.y., Helsinki (1966).
|Date of creation||2013-03-22 16:34:57|
|Last modified on||2013-03-22 16:34:57|
|Last modified by||pahio (2872)|
|Synonym||Bessel’s differential equation|