Theory and Proof

for

Frobenius Series Solution of a Differential Equation

Background.

    Consider the second order linear differential equation  
    
(1)        [Graphics:Images/FrobeniusSeriesProof_gr_1.gif].

Put this equation in the form   [Graphics:Images/FrobeniusSeriesProof_gr_2.gif],  then use the substitutions  [Graphics:Images/FrobeniusSeriesProof_gr_3.gif]  and  [Graphics:Images/FrobeniusSeriesProof_gr_4.gif]  and rewrite the differential equation (1) as follows  

(2)        [Graphics:Images/FrobeniusSeriesProof_gr_5.gif].

 

Definition (Analytic).  The functions [Graphics:Images/FrobeniusSeriesProof_gr_6.gif] and [Graphics:Images/FrobeniusSeriesProof_gr_7.gif] are analytic at  [Graphics:Images/FrobeniusSeriesProof_gr_8.gif]  if they have Taylor series expansions with radius of convergence  [Graphics:Images/FrobeniusSeriesProof_gr_9.gif]  and   [Graphics:Images/FrobeniusSeriesProof_gr_10.gif],  respectively.  That is  

        [Graphics:Images/FrobeniusSeriesProof_gr_11.gif]  which converges for  [Graphics:Images/FrobeniusSeriesProof_gr_12.gif]
    and
        [Graphics:Images/FrobeniusSeriesProof_gr_13.gif]  which converges for  [Graphics:Images/FrobeniusSeriesProof_gr_14.gif].

 

Definition (Ordinary Point).  If the functions  [Graphics:Images/FrobeniusSeriesProof_gr_15.gif] and [Graphics:Images/FrobeniusSeriesProof_gr_16.gif]  are analytic at [Graphics:Images/FrobeniusSeriesProof_gr_17.gif], then the point  [Graphics:Images/FrobeniusSeriesProof_gr_18.gif]  is called an ordinary point of the differential equation  

        [Graphics:Images/FrobeniusSeriesProof_gr_19.gif].

Otherwise, the point  [Graphics:Images/FrobeniusSeriesProof_gr_20.gif]  is called a singular point of the differential equation (1).

 

Definition (Regular Singular Point).  Assume that  [Graphics:Images/FrobeniusSeriesProof_gr_21.gif]  is a singular point of (1) and that  [Graphics:Images/FrobeniusSeriesProof_gr_22.gif] and  [Graphics:Images/FrobeniusSeriesProof_gr_23.gif]  are analytic at  [Graphics:Images/FrobeniusSeriesProof_gr_24.gif].  They will have Maclaurin series expansions with radius of convergence [Graphics:Images/FrobeniusSeriesProof_gr_25.gif] and  [Graphics:Images/FrobeniusSeriesProof_gr_26.gif], respectively.  That is  

        [Graphics:Images/FrobeniusSeriesProof_gr_27.gif]  which converges for  [Graphics:Images/FrobeniusSeriesProof_gr_28.gif]
    and
        [Graphics:Images/FrobeniusSeriesProof_gr_29.gif]  which converges for  [Graphics:Images/FrobeniusSeriesProof_gr_30.gif]

Then the point  [Graphics:Images/FrobeniusSeriesProof_gr_31.gif]  is called a regular singular point of the differential equation (1).

Remark.  This all boils down to the idea that   [Graphics:Images/FrobeniusSeriesProof_gr_32.gif]  and   [Graphics:Images/FrobeniusSeriesProof_gr_33.gif] both have removable singularities at  [Graphics:Images/FrobeniusSeriesProof_gr_34.gif].

 

 

Method of Frobenius.

    This method is attributed to the german mathematician Ferdinand Georg Frobenius (1849-1917 ).  Assume that  [Graphics:Images/FrobeniusSeriesProof_gr_35.gif]  is regular singular point of the differential equation

        [Graphics:Images/FrobeniusSeriesProof_gr_36.gif].  
    
A Frobenius series (generalized Laurent series) of the form  

[Graphics:Images/FrobeniusSeriesProof_gr_37.gif]

where  [Graphics:Images/FrobeniusSeriesProof_gr_38.gif]  can be used to solve the differential equation.  The parameter [Graphics:Images/FrobeniusSeriesProof_gr_39.gif] must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of  [Graphics:Images/FrobeniusSeriesProof_gr_40.gif] is zero.  This is called the indicial equation.  Next, a recursive equation for the coefficients is obtained by setting the coefficient of  [Graphics:Images/FrobeniusSeriesProof_gr_41.gif]  equal to zero.
Caveat.
There are some instances when only one Frobenius solution can be constructed.

 

Definition (Indicial Equation).  The parameter [Graphics:Images/FrobeniusSeriesProof_gr_42.gif] in the Frobenius series is a root of the indicial equation  

        [Graphics:Images/FrobeniusSeriesProof_gr_43.gif].

Assuming that the singular point is  [Graphics:Images/FrobeniusSeriesProof_gr_44.gif],  we can calculate  [Graphics:Images/FrobeniusSeriesProof_gr_45.gif]  as follows:

        [Graphics:Images/FrobeniusSeriesProof_gr_46.gif]
and
        [Graphics:Images/FrobeniusSeriesProof_gr_47.gif].

 

Definition of  [Graphics:Images/FrobeniusSeriesProof_gr_48.gif]  We state the following definition of  [Graphics:Images/FrobeniusSeriesProof_gr_49.gif]

        [Graphics:Images/FrobeniusSeriesProof_gr_50.gif].  

The exponents of the singularity are the roots  [Graphics:Images/FrobeniusSeriesProof_gr_51.gif]  of  [Graphics:Images/FrobeniusSeriesProof_gr_52.gif].   

Derivation.

 

The Recursive Formula for  [Graphics:Images/FrobeniusSeriesProof_gr_78.gif]  

    We are now in a position to derive the recursive formula for the sequence of coefficients  [Graphics:Images/FrobeniusSeriesProof_gr_79.gif]  for the Frobenius series solution  

[Graphics:Images/FrobeniusSeriesProof_gr_80.gif]

The recursive formula for computing  [Graphics:Images/FrobeniusSeriesProof_gr_81.gif]  is

[Graphics:Images/FrobeniusSeriesProof_gr_82.gif]

where

[Graphics:Images/FrobeniusSeriesProof_gr_83.gif]

Derivation.

 

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(c) John H. Mathews 2004