## Frobenius Method

If is an ordinary point of the Ordinary Differential Equation, expand in a Taylor Series about , letting

 (1)

Plug back into the ODE and group the Coefficients by Power. Now, obtain a Recurrence Relation for the th term, and write the Taylor Series 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)

solutions may be found by the Frobenius method or by expansion in a Laurent Series. In the Frobenius method, assume a solution of the form
 (6)

so that
 (7) (8) (9)

Now, plug back into the ODE and group the Coefficients by Power to obtain a recursion Formula for the th term, and then write the Taylor Series in terms of the s. Equating the term to 0 will produce the so-called Indicial Equation, which will give the allowed values of in the Taylor Series.

Fuchs's Theorem guarantees that at least one Power series solution will be obtained when applying the Frobenius method if the expansion point is an ordinary, or regular, Singular Point. For a regular Singular Point, a Laurent Series expansion can also be used. Expand in a Laurent Series, letting

 (10)

Plug back into the ODE and group the Coefficients by Power. Now, obtain a recurrence Formula for the th term, and write the Taylor Expansion in terms of the s.

Arfken, G. Series Solutions--Frobenius' Method.'' §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454-467, 1985.