Frobenius method

This method enables one to compute a fundamental system of solutions for a holomorphic differential equation near a regular singular point (cf. also Singular point).

Suppose one is given a linear differential operator

(a1)

where for and some , the functions

(a2)

are holomorphic for and (cf. also Analytic function). The point is called a regular singular point of . Formula (a1) gives the differential operator in its Frobenius normal form if .

The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation

(a3)

in the domain near the regular singular point at . Here, , and for an equation in normal form, actually . The cut along some ray is introduced because the solutions are expected to have an essential singularity at .

The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation

(a4)

where the differential operator is made from (a1) by retaining only the leading terms. The Euler–Cauchy equation can be solved by taking the guess with unknown parameter . One gets with the indicial polynomial

(a5)

In the following, the zeros of the indicial polynomial will be ordered by requiring

It is assumed that all roots are different and one denotes their multiplicities by .

The method of Frobenius starts with the guess

(a6)

with an undetermined parameter . The coefficients have to be calculated by requiring that

(a7)

This requirement leads to and

(a8)

as a recursion formula for for all . Here, are polynomials in of degree at most , which are given below.

The easy generic case occurs if the indicial polynomial has only simple zeros and their differences are never integer valued. Under these assumptions, the functions

are a fundamental system of solutions of (a3).

Complications.

Complications can arise if the generic assumption made above is not satisfied. Putting in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients . These solutions are rational functions of with possible poles at the poles of as well as at .

The poles are compensated for by multiplying at first with powers of and differentiation by the parameter before setting .

Since the general situation is rather complex, two special cases are given first. Let denote the set of natural numbers starting at (i.e., excluding ). Note that neither of the special cases below does exclude the simple generic case above.

All solutions have expansions of the form

The leading term is useful as a marker for the different solutions. Because for and , all leading terms are different, the method of Frobenius does indeed yield a fundamental system of linearly independent solutions of the differential equation (a3).

Special case 1.

For any , the zero of the indicial polynomial has multiplicity , but none of the numbers is a natural number.

In this case, the functions

are linearly independent solutions of the differential equation (a3).

Special case 2.

Suppose .

Then the functions

all with and , are linearly independent solutions of the differential equation (a3). The solution for may contain logarithmic terms in the higher powers, starting with .

Special case 3.

Let and let be a zero of the indicial polynomial of multiplicity for .

In this case, define to be the sum of those multiplicities for which . Hence,

The functions

with and , are linearly independent solutions of the differential equation (a3).

The method looks simpler in the most common case of a differential operator

(a9)

Here, one has to to assume that to obtain a regular singular point. The indicial polynomial is simply

Only two special cases can occur:

1) . The functions

are a fundamental system.

2) . The functions

with in the second function, are two linearly independent solutions of the differential equation (a9). The second solution can contain logarithmic terms in the higher powers starting with .

The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. also Fuchsian equation). A similar method of solution can be used for matrix equations of the first order, too. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers.

Computation of the polynomials .

In the guess

the coefficients have to be calculated from the requirement (a7). Indeed (a1) and (a2) imply

Here, are polynomials of degree at most determined by setting

Because of (a7), one finds and the recursion formula (a8).

References

[a1]	R. Redheffer, "Differential equations, theory and applications" , Jones and Bartlett (1991)
[a2]	F. Rothe, "A variant of Frobenius' method for the Emden–Fowler equation" Applicable Anal. , 66 (1997) pp. 217–245
[a3]	D. Zwillinger, "Handbook of differential equations" , Acad. Press (1989)