The linear differential equation

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

due to Frobenius.\, Since the parameter
is indefinite, we may regard
as distinct from 0.
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
![{\displaystyle {\begin{matrix}{\begin{cases}{[}r^{2}-p^{2}{]}a_{0}=0,\\{[}(r+1)^{2}-p^{2}{]}a_{1}=0,\\{[}(r+2)^{2}-p^{2}{]}a_{2}+a_{0}=0,\\\qquad \qquad \ldots \\{[}(r+k)^{2}-p^{2}{]}a_{k}+a_{k-2}=0.\end{cases}}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/184c51557d04c4f72014dd321e78930139bb19c9)
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
![{\displaystyle {\begin{matrix}y_{1}:=a_{0}x^{p}\left[1\!\!{\frac {x^{2}}{2(2p\!+\!2)}}\!+\!{\frac {x^{4}}{2\!\cdot \!4(2p\!+\!2)(2p\!+\!4)}}\!-\!{\frac {x^{6}}{2\!\cdot \!4\!\cdot \!6(2p\!+\!2)(2p\!+\!4)(2p\!+\!6)}}\!+-\ldots \right]\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b09fe2d53e7008cc8e68bbac076630f231d30a5)
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
:
![{\displaystyle {\begin{matrix}y_{2}:=a_{0}x^{-p}\!\left[1\!-\!{\frac {x^{2}}{2(-2p\!+\!2)}}\!+\!{\frac {x^{4}}{2\!\cdot \!4(-2p\!+\!2)(-2p\!+\!4)}}\!-\!{\frac {x^{6}}{2\!\cdot \!4\!\cdot \!6(-2p\!+\!2)(-2p\!+\!4)(-2p\!+\!6)}}\!+-\ldots \right]\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92f274b6d4a3592813380d9ab416250752aef090)
The power series of (5) and (6) converge for all values of
and are linearly independent (the ratio
tends to 0 as\,
).\, With the appointed value
the solution
is called the \htmladdnormallink{Bessel function {http://planetphysics.us/encyclopedia/BesselEquation2.html} 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 formula 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 of
.\, It can be shown that the other solution of (1) ought to be searched in the form\,
.\, Then the general solution is\,
.\\
Other formulae
The first kind Bessel functions of integer order have the generating function
:

This function has an essential singularity at\,
\, but is analytic elsewhere in
; thus
has the Laurent expansion in that point.\, Let us prove (8) by using the general expression
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 gives
This means that
has the Laurent expansion (8).
By using the generating function, one can easily derive other formulae, e.g.
the integral representation of the Bessel functions of integer order:
Also one can obtain the addition formula
and the series representations of cosine and sine: