An ordinary point of the differential equation
is one at which the coefficients and are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.
Other points are singularities of the differential equation. If both and are analytic at , then is a regular singularity (or singularity of the first kind). All other singularities are classified as irregular.
In a punctured neighborhood of a regular singularity
with at least one of the coefficients , , nonzero. Let , denote the indices or exponents, that is, the roots of the indicial equation
Provided that is not zero or an integer, equation (2.7.1) has independent solutions , , such that
with , and
If , then (2.7.4) applies only in the case . But there is an independent solution
The coefficients and constant are again determined by equating coefficients in the differential equation, beginning with when , or with when .
The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from .
If the singularities of and at are no worse than poles, then has rank , where is the least integer such that and are analytic at . Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. Then
these series converging in an annulus , with at least one of , , nonzero.
Formal solutions are
where , are the roots of the characteristic equation
when . The construction fails iff , that is, when : this case is treated below.
For large ,
where and are constants, and the th remainder terms in the sums are and , respectively (Olver (1994a)). Hence unless the series (2.7.8) terminate (in which case the corresponding is zero) they diverge. However, there are unique and linearly independent solutions , , such that
as in the sectors
being an arbitrary small positive constant.
Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution can be continued analytically into any other sector. Typical connection formulas are
in which , are constants, the so-called Stokes multipliers. In combination with (2.7.14) these formulas yield asymptotic expansions for in , and in . Furthermore,
Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, , with large, are the “early” coefficients , with small. This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b). See §2.11(v) for other examples.
The exceptional case is handled by Fabry’s transformation:
The transformed differential equation either has a regular singularity at , or its characteristic equation has unequal roots.
For error bounds for (2.7.14) see Olver (1997b, Chapter 7). For the calculation of Stokes multipliers see Olde Daalhuis and Olver (1995b). For extensions to singularities of higher rank see Olver and Stenger (1965). For extensions to higher-order differential equations see Stenger (1966a, b), Olver (1997a, 1999), and Olde Daalhuis and Olver (1998).
For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
In a finite or infinite interval let be real, positive, and twice-continuously differentiable, and be continuous. Then in the differential equation
has twice-continuously differentiable solutions
provided that . Here is the error-control function
and denotes the variational operator (§2.3(i)). Thus
Assuming also , we have
Suppose in addition is unbounded as and . Then there are solutions , , such that
The solutions with the properties (2.7.26), (2.7.27) are unique, but not those with the properties (2.7.28), (2.7.29). In fact, since
is a recessive (or subdominant) solution as , and is a dominant solution as . Similarly for and as .
We cannot take and because would diverge as . Instead set , . By approximating
we arrive at
as , being recessive and dominant.
One pair of independent solutions of the equation
is , . Another is , . In theory either pair may be used to construct any other solution
where are constants. From the numerical standpoint, however, the pair and has the drawback that severe numerical cancellation can occur with certain combinations of and , for example if and are equal, or nearly equal, and , or , is large and negative. This kind of cancellation cannot take place with and , and for this reason, and following Miller (1950), we call and a numerically satisfactory pair of solutions.
The solutions and are respectively recessive and dominant as , and vice versa as . This is characteristic of numerically satisfactory pairs. In a neighborhood, or sectorial neighborhood of a singularity, one member has to be recessive. In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere.
In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are out of phase.