Fuchs' Theorem
If is an ordinary point of , there are two linearly independent solutions . If is a regular singular point, there is at least one solution with , converging in some neighborhood of .
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If is an ordinary point of the ODE
then there are two linearly independent solutions of the form
which converge for some neighborhood of . i.e. there is a Taylor series solution about .
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If is a regular singular point of the ODE, then there is at least one solution of the form
where can be real or complex, and [so that is unique]. The series converges for some neighborhood of .
This is called a Frobenius series.
Note that there is no guarantee of two linearly independent solutions in this case.
Reading part 2
The requirement makes the exponent unique, and may be real or complex.
Part 2 guarantees only at least one solution. Unlike part 1, there is no guarantee of two linearly independent solutions at a regular singular point: whether a second linearly independent Frobenius-type solution exists depends on the roots and of the indicial equation.
Both parts give series that converge for some neighborhood of . At irregular singular points the guarantee disappears entirely, and the series solution method may fail completely.
Related
Stated in
- Theorem 8.4 (Fuch's Theorem)§8.2 Method of Frobenius