The Frobenius method constructs solutions of a linear ordinary differential equation near a regular singular point by allowing a power series to be multiplied by a possibly nonintegral power of the displacement from the expansion point.
At an ordinary point 
, expand 
in a Taylor series about
. Commonly, the expansion point can
be taken as 
,
resulting in the Maclaurin series
 |
(1)
|
Plug 
back into the ODE and group the coefficients by power. Now, obtain a recurrence
relation for the 
th
term, and write the series expansion in terms
of the 
s.
Expansions for the first few derivatives are
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  | |
 |  |  |
(4)
|
 |  |  |
If 
is a regular singular point of the ordinary
differential equation
 |
(5)
|
translate the independent variable so that 
and seek a Frobenius
solution of the form
 |
(6)
|
where 
.
Its derivatives are
 |  |  |
(7)
|
 |  |  | |
 |  |  |
(8)
|
 |  |  |
(9)
|
Substitute into the ordinary differential equation and group by power to obtain a recurrence
relation for the coefficients 
. Setting the coefficient
of the lowest power to zero gives the indicial
equation, which determines the allowed values of 
.
As an example, consider the Bessel differential equation
 |
(10)
|
 |
(11)
|
The indicial equation, obtained by setting 
, is then
![a_0[k(k-1)+k-m^2]=a_0(k^2-m^2)=0.](/images/equations/FrobeniusMethod/NumberedEquation6.svg) |
(12)
|
Since 
,
, so 
. Take 
and choose 
. The next coefficient gives
 |
(13)
|
(so 
)
and
![[a_nn(2m+n)+a_(n-2)]x^(m+n)=0](/images/equations/FrobeniusMethod/NumberedEquation8.svg) |
(14)
|
for 
,
3, ..., so
 |
(15)
|
for 
.
With ![a_0=1/[2^mGamma(m+1)]](/images/equations/FrobeniusMethod/Inline49.svg)
,
where 
is the gamma function, substitution into (6)
gives the Bessel function of the first
kind
 |
(16)
|
For noninteger 
,
and 
are linearly independent
solutions on 
.
For integer 
, however,
 |
(17)
|
and a second linearly independent solution is the Bessel function of the second
kind 
.
Fuchs's theorem guarantees at least one Frobenius solution at a regular singular point. Such a solution need not be an ordinary power series or a Laurent series, since its leading exponent can be nonintegral. If the roots of the indicial equation differ by an integer, including zero, a second linearly independent solution may contain a logarithm.
