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ZixuanZhang
ZixuanZhang
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Frobenius Method

At a regular singular point, seek with ; matching the lowest power of gives the indicial equation for .

Consider the ODE

We know that is a regular singular point. For convenience, multiply the ODE by to get

We shall try a Frobenius series solution of the form

Since the ODE now looks like an equidimensional equation, we expect to be able to extract a sum when we substitute it in. So we have

  • We shall look at the lowest power of to determine . In this case, it is with .

    Since , we have or .

  • The next lowerest power is with .

    Since , we must have in both cases.

  • For more generality, consider with .

    Now we should consider the two cases for separately.

    • For , we have

      Since , all odd terms are zero. Now, for the even terms, we have

      Therefore, the even terms give one (Taylor series) solution

    • For , we have

      Therefore, for the even terms,

      Again, all odd terms are zero. Therefore, the even terms give another solution (relabelling constant to ):

      Note that this is not a Taylor series but a Frobenius series solution.

Therefore, there are two linearly independent solutions at this regular singular point.

Form

A Frobenius series has the form

where can be real or complex and , so that is unique. The series converges for some neighborhood of . Unlike a power series about an ordinary point, it is not in general a Taylor series.

After multiplying the ODE by a suitable power of , it looks equidimensional, so substituting the series extracts a common factor of . The coefficient of the lowest power , at , is the indicial equation; its roots are the possible values of . Higher powers then give recurrence relations for the coefficients.

Second solutions

Getting one Frobenius-type solution about a regular singular point is guaranteed, but whether there is a second linearly independent one depends on the roots and of the indicial equation:

  1. If is not an integer, two linearly independent solutions are obtained directly:
  1. If is a non-zero integer, one series solution involves the larger root, say . The second solution has the form

where may or may not be zero and is determined in terms of and , so that two arbitrary constants remain.

  1. If , the form is similar to case (2), but the logarithmic term is always present.

For example, in the indicial equation is , so and differ by a non-zero integer: the larger root gives the Taylor-series solution , while for the recurrence demands when , so no series solution exists and the logarithmic form is required instead.

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