If is an ordinary point of the Ordinary Differential Equation, expand in
a Taylor Series about , letting
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
If is a regular singular point of the Ordinary Differential Equation,
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
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
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.
See also Fuchs's Theorem, Ordinary Differential Equation
Arfken, G. ``Series Solutions--Frobenius' Method.'' §8.5 in Mathematical Methods for Physicists, 3rd ed.
Orlando, FL: Academic Press, pp. 454-467, 1985.
© 1996-9 Eric W. Weisstein