Method of Frobenius.

    This method is attributed to the german mathematician Ferdinand Georg Frobenius (1849-1917 ).  Assume that    is regular singular point of the differential equation

        .  
    
A Frobenius series (generalized Laurent series) of the form  

where    can be used to solve the differential equation.  The parameter must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of   is zero.  This is called the indicial equation.  Next, a recursive equation for the coefficients is obtained by setting the coefficient of    equal to zero. Caveat. There are some instances when only one Frobenius solution can be constructed.

Definition (Indicial Equation).  The parameter in the Frobenius series is a root of the indicial equation  

        .

Assuming that the singular point is  ,  we can calculate    as follows:

        
and
        

Definition of    We state the following definition of  

        .  

The exponents of the singularity are the roots    of  .   

Derivation.

The Recursive Formula for    

    We are now in a position to derive the recursive formula for the sequence of coefficients    for the Frobenius series solution  

The recursive formula for computing    is

where

Derivation.

Starting with the differential equation  

Rewrite it in the form  

Multiply each term by the factor  .   

Use series for all the terms  .  

Make the substitutions  ,   and regroup the second and third terms in the D. E. as follows.  

Making all the series substitutions we get  

Move the terms and   into the summations where they belong  

Recall the following Cauchy product form for multiplying two infinite series  

Multiply each side by of the above Cauchy products by   .  

Now substitute the above series into

Get

Which can be combined as follows

And simplified

And further simplified

From which we get the equations

for  n = 0,1,2,...

Now split off the last term in the sum corresponding to  k=n  and write this as follows

From which we easily get

Recall that    and   

          
        
so the above quantity can be written in the form  

From this we can easily obtain the desired recursion formula for computing .

for  n = 1,2,3,...

Caveat.  Although the general formula above can easily be used in a recursive program it is slow because of the recursion depth that occurs with each successive term in the sequence.  For many practical problems the series for    and    are short and the traditional methods of pencil and paper will create a simple recursion formula.

Caveat.  Assume that the roots of the indicial equation are and .  If    or if  the roots differ by an integer, then only one Frobenius series solution can be found and other methods such as reduction of order must be used to create the second solution.

(c) John H. Mathews 2004