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.

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  

Now look at the first term in each of the series, multiply and add as indicated to obtain the term with the lowest power of  x.  Hint. It is found by changing all of the upper limits in the summations from   to .  

Mathematica can be of assistance for this computation by changing the upper limit in the summations from   to .  

The result is

This equation is equivalient to    and since    we can cancel it and the common factor    to get

        .
        
We have arrived at the indicial equation which was our goal.

The roots of    are denoted  ,  and could be real or complex.

(c) John H. Mathews 2004