Theory and Proof

for

Frobenius Series Solution of a Differential Equation

Background.

Consider the second order linear differential equation

(1)        .

Put this equation in the form   ,  then use the substitutions    and    and rewrite the differential equation (1) as follows

(2)        .

Definition (Analytic).  The functions and are analytic at    if they have Taylor series expansions with radius of convergence    and   ,  respectively.  That is

which converges for
and
which converges for  .

Definition (Ordinary Point).  If the functions   and   are analytic at , then the point    is called an ordinary point of the differential equation

.

Otherwise, the point    is called a singular point of the differential equation (1).

Definition (Regular Singular Point).  Assume that    is a singular point of (1) and that   and    are analytic at  .  They will have Maclaurin series expansions with radius of convergence and  , respectively.  That is

which converges for
and
which converges for

Then the point    is called a regular singular point of the differential equation (1).

Remark.  This all boils down to the idea that     and    both have removable singularities at  .

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.

(c) John H. Mathews 2004