2 Asymptotic ApproximationsAreas2.6 Distributional Methods2.8 Differential Equations with a Parameter

- §2.7(i) Regular Singularities: Fuchs–Frobenius Theory
- §2.7(ii) Irregular Singularities of Rank 1
- §2.7(iii) Liouville–Green (WKBJ) Approximation
- §2.7(iv) Numerically Satisfactory Solutions

An *ordinary point* of the differential equation

2.7.1 | $$\frac{{d}^{2}w}{{dz}^{2}}+f(z)\frac{dw}{dz}+g(z)w=0$$ | ||

is one at which the coefficients $f(z)$ and $g(z)$ are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.

Other points ${z}_{0}$ are *singularities* of the differential equation. If both
$(z-{z}_{0})f(z)$ and ${(z-{z}_{0})}^{2}g(z)$ are analytic at ${z}_{0}$, then ${z}_{0}$ is a
*regular singularity* (or *singularity of the first kind*). All other
singularities are classified as *irregular*.

In a punctured neighborhood $\mathbf{N}$ of a regular singularity ${z}_{0}$

2.7.2 | $f(z)$ | $={\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{f}_{s}{(z-{z}_{0})}^{s-1},$ | ||

$g(z)$ | $={\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{g}_{s}{(z-{z}_{0})}^{s-2},$ | |||

with at least one of the coefficients ${f}_{0}$, ${g}_{0}$, ${g}_{1}$ nonzero. Let
${\alpha}_{1}$, ${\alpha}_{2}$ denote the *indices* or *exponents*, that is,
the roots of the *indicial equation*

2.7.3 | $$Q(\alpha )\equiv \alpha (\alpha -1)+{f}_{0}\alpha +{g}_{0}=0.$$ | ||

Provided that ${\alpha}_{1}-{\alpha}_{2}$ is not zero or an integer, equation (2.7.1) has independent solutions ${w}_{j}(z)$, $j=1,2$, such that

2.7.4 | $${w}_{j}(z)={(z-{z}_{0})}^{{\alpha}_{j}}\sum _{s=0}^{\mathrm{\infty}}{a}_{s,j}{(z-{z}_{0})}^{s},$$ | ||

$z\in \mathbf{N}$, | |||

with ${a}_{0,j}=1$, and

2.7.5 | $$Q({\alpha}_{j}+s){a}_{s,j}=-\sum _{r=0}^{s-1}\left(({\alpha}_{j}+r){f}_{s-r}+{g}_{s-r}\right){a}_{r,j},$$ | ||

when $s=1,2,3,\mathrm{\dots}$.

If ${\alpha}_{1}-{\alpha}_{2}=0,1,2,\mathrm{\dots}$, then (2.7.4) applies only in the case $j=1$. But there is an independent solution

2.7.6 | $${w}_{2}(z)={(z-{z}_{0})}^{{\alpha}_{2}}\sum _{\begin{array}{c}s=0\\ s\ne {\alpha}_{1}-{\alpha}_{2}\end{array}}^{\mathrm{\infty}}{b}_{s}{(z-{z}_{0})}^{s}+c{w}_{1}(z)\mathrm{ln}\left(z-{z}_{0}\right),$$ | ||

$z\in \mathbf{N}$. | |||

The coefficients ${b}_{s}$ and constant $c$ are again determined by equating coefficients in the differential equation, beginning with $c=1$ when ${\alpha}_{1}-{\alpha}_{2}=0$, or with ${b}_{0}=1$ when ${\alpha}_{1}-{\alpha}_{2}=1,2,3,\mathrm{\dots}$.

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 ${z}_{0}$.

To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing $z$ in (2.7.1) with $1/z$; see Olver (1997b, pp. 153–154). For corresponding definitions, together with examples, for linear differential equations of arbitrary order see §§16.8(i)–16.8(ii).

If the singularities of $f(z)$ and $g(z)$ at ${z}_{0}$ are no worse than poles,
then ${z}_{0}$ has *rank* $\mathrm{\ell}-1$, where $\mathrm{\ell}$ is the least integer such
that ${(z-{z}_{0})}^{\mathrm{\ell}}f(z)$ and ${(z-{z}_{0})}^{2\mathrm{\ell}}g(z)$ are analytic at
${z}_{0}$. 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

2.7.7 | $f(z)$ | $={\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{\displaystyle \frac{{f}_{s}}{{z}^{s}}},$ | ||

$g(z)$ | $={\displaystyle \sum _{s=0}^{\mathrm{\infty}}}{\displaystyle \frac{{g}_{s}}{{z}^{s}}},$ | |||

these series converging in an annulus $|z|>a$, with at least one of ${f}_{0}$, ${g}_{0}$, ${g}_{1}$ nonzero.

Formal solutions are

2.7.8 | $${\mathrm{e}}^{{\lambda}_{j}z}{z}^{{\mu}_{j}}\sum _{s=0}^{\mathrm{\infty}}\frac{{a}_{s,j}}{{z}^{s}},$$ | ||

$j=1,2$, | |||

where ${\lambda}_{1}$, ${\lambda}_{2}$ are the roots of the *characteristic
equation*

2.7.9 | $${\lambda}^{2}+{f}_{0}\lambda +{g}_{0}=0,$$ | ||

2.7.10 | $${\mu}_{j}=-({f}_{1}{\lambda}_{j}+{g}_{1})/({f}_{0}+2{\lambda}_{j}),$$ | ||

${a}_{0,j}=1$, and

2.7.11 | $$({f}_{0}+2{\lambda}_{j})s{a}_{s,j}=\begin{array}{l}(s-{\mu}_{j})(s-1-{\mu}_{j}){a}_{s-1,j}\\ \phantom{\rule{2em}{0ex}}+\sum _{r=1}^{s}\left({\lambda}_{j}{f}_{r+1}+{g}_{r+1}-(s-r-{\mu}_{j}){f}_{r}\right){a}_{s-r,j},\end{array}$$ | ||

when $s=1,2,\mathrm{\dots}$. The construction fails iff ${\lambda}_{1}={\lambda}_{2}$, that is, when ${f}_{0}^{2}=4{g}_{0}$: this case is treated below.

For large $s$,

2.7.12 | ${a}_{s,1}$ | $\sim {\displaystyle \frac{{\mathrm{\Lambda}}_{1}}{{({\lambda}_{1}-{\lambda}_{2})}^{s}}}{\displaystyle \sum _{j=0}^{\mathrm{\infty}}}{a}_{j,2}{({\lambda}_{1}-{\lambda}_{2})}^{j}\mathrm{\Gamma}\left(s+{\mu}_{2}-{\mu}_{1}-j\right),$ | ||

2.7.13 | ${a}_{s,2}$ | $\sim {\displaystyle \frac{{\mathrm{\Lambda}}_{2}}{{({\lambda}_{2}-{\lambda}_{1})}^{s}}}{\displaystyle \sum _{j=0}^{\mathrm{\infty}}}{a}_{j,1}{({\lambda}_{2}-{\lambda}_{1})}^{j}\mathrm{\Gamma}\left(s+{\mu}_{1}-{\mu}_{2}-j\right),$ | ||

where ${\mathrm{\Lambda}}_{1}$ and ${\mathrm{\Lambda}}_{2}$ are constants, and the $J$th remainder terms in the sums are $O\left(\mathrm{\Gamma}\left(s+{\mu}_{2}-{\mu}_{1}-J\right)\right)$ and $O\left(\mathrm{\Gamma}\left(s+{\mu}_{1}-{\mu}_{2}-J\right)\right)$, respectively (Olver (1994a)). Hence unless the series (2.7.8) terminate (in which case the corresponding ${\mathrm{\Lambda}}_{j}$ is zero) they diverge. However, there are unique and linearly independent solutions ${w}_{j}(z)$, $j=1,2$, such that

2.7.14 | $${w}_{j}(z)\sim {\mathrm{e}}^{{\lambda}_{j}z}{(({\lambda}_{2}-{\lambda}_{1})z)}^{{\mu}_{j}}\sum _{s=0}^{\mathrm{\infty}}\frac{{a}_{s,j}}{{z}^{s}}$$ | ||

as $z\to \mathrm{\infty}$ in the sectors

2.7.15 | $$-\frac{3}{2}\pi +\delta \le \mathrm{ph}\left(({\lambda}_{2}-{\lambda}_{1})z\right)\le \frac{3}{2}\pi -\delta ,$$ | ||

$j=1$, | |||

2.7.16 | $$-\frac{1}{2}\pi +\delta \le \mathrm{ph}\left(({\lambda}_{2}-{\lambda}_{1})z\right)\le \frac{5}{2}\pi -\delta ,$$ | ||

$j=2$, | |||

$\delta $ 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 ${w}_{j}(z)$ can be continued analytically into any other sector. Typical connection formulas are

2.7.17 | ${w}_{1}(z)$ | $={\mathrm{e}}^{2\pi \mathrm{i}{\mu}_{1}}{w}_{1}(z{\mathrm{e}}^{-2\pi \mathrm{i}})+{C}_{1}{w}_{2}(z),$ | ||

${w}_{2}(z)$ | $={\mathrm{e}}^{-2\pi \mathrm{i}{\mu}_{2}}{w}_{2}(z{\mathrm{e}}^{2\pi \mathrm{i}})+{C}_{2}{w}_{1}(z),$ | |||

in which ${C}_{1}$, ${C}_{2}$ are constants, the so-called *Stokes multipliers*.
In combination with (2.7.14) these formulas yield asymptotic
expansions for ${w}_{1}(z)$ in $\frac{1}{2}\pi +\delta \le \mathrm{ph}\left(({\lambda}_{2}-{\lambda}_{1})z\right)\le \frac{5}{2}\pi -\delta $, and ${w}_{2}(z)$ in
$-\frac{3}{2}\pi +\delta \le \mathrm{ph}\left(({\lambda}_{2}-{\lambda}_{1})z\right)\le \frac{1}{2}\pi -\delta $. Furthermore,

2.7.18 | ${\mathrm{\Lambda}}_{1}$ | $=-\mathrm{i}{\mathrm{e}}^{({\mu}_{2}-{\mu}_{1})\pi \mathrm{i}}{C}_{1}/(2\pi ),$ | ||

${\mathrm{\Lambda}}_{2}$ | $=\mathrm{i}{C}_{2}/(2\pi ).$ | |||

Note that the coefficients in the expansions (2.7.12),
(2.7.13) for the “late” coefficients, that is, ${a}_{s,1}$, ${a}_{s,2}$
with $s$ large, are the “early” coefficients ${a}_{j,2}$, ${a}_{j,1}$ with $j$
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 ${f}_{0}^{2}=4{g}_{0}$ is handled by *Fabry’s transformation*:

2.7.19 | $w$ | $={\mathrm{e}}^{-{f}_{0}z/2}W,$ | ||

$t$ | $={z}^{1/2}.$ | |||

The transformed differential equation either has a regular singularity at $t=\mathrm{\infty}$, 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 $({a}_{1},{a}_{2})$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous. Then in $({a}_{1},{a}_{2})$ the differential equation

2.7.20 | $$\frac{{d}^{2}w}{{dx}^{2}}=(f(x)+g(x))w$$ | ||

has twice-continuously differentiable solutions

2.7.21 | ${w}_{1}(x)$ | $={f}^{-1/4}(x)\mathrm{exp}\left({\displaystyle \int {f}^{1/2}(x)dx}\right)\left(1+{\u03f5}_{1}(x)\right),$ | ||

2.7.22 | ${w}_{2}(x)$ | $={f}^{-1/4}(x)\mathrm{exp}\left(-{\displaystyle \int {f}^{1/2}(x)dx}\right)\left(1+{\u03f5}_{2}(x)\right),$ | ||

such that

2.7.23 | $$|{\u03f5}_{j}(x)|,\frac{1}{2}{f}^{-1/2}(x)|{\u03f5}_{j}^{\prime}(x)|\le \mathrm{exp}\left(\frac{1}{2}{\mathcal{V}}_{{a}_{j},x}\left(F\right)\right)-1,$$ | ||

$j=1,2$, | |||

provided that $$. Here $F(x)$ is the
*error-control function*

2.7.24 | $$F(x)=\int \left(\frac{1}{{f}^{1/4}}\frac{{d}^{2}}{{dx}^{2}}\left(\frac{1}{{f}^{1/4}}\right)-\frac{g}{{f}^{1/2}}\right)dx,$$ | ||

and $\mathcal{V}$ denotes the variational operator (§2.3(i)). Thus

2.7.25 | $${\mathcal{V}}_{{a}_{j},x}\left(F\right)=\left|{\int}_{{a}_{j}}^{x}\left|\frac{1}{{f}^{1/4}(t)}\frac{{d}^{2}}{{dt}^{2}}\left(\frac{1}{{f}^{1/4}(t)}\right)-\frac{g(t)}{{f}^{1/2}(t)}\right|dt\right|.$$ | ||

Assuming also $$, we have

2.7.26 | $${w}_{1}(x)\sim {f}^{-1/4}(x)\mathrm{exp}\left(\int {f}^{1/2}(x)dx\right),$$ | ||

$x\to {a}_{1}+$, | |||

2.7.27 | $${w}_{2}(x)\sim {f}^{-1/4}(x)\mathrm{exp}\left(-\int {f}^{1/2}(x)dx\right),$$ | ||

$x\to {a}_{2}-$. | |||

Suppose in addition $|\int {f}^{1/2}(x)dx|$ is unbounded as $x\to {a}_{1}+$ and $x\to {a}_{2}-$. Then there are solutions ${w}_{3}(x)$, ${w}_{4}(x)$, such that

2.7.28 | $${w}_{3}(x)\sim {f}^{-1/4}(x)\mathrm{exp}\left(\int {f}^{1/2}(x)dx\right),$$ | ||

$x\to {a}_{2}-$, | |||

2.7.29 | $${w}_{4}(x)\sim {f}^{-1/4}(x)\mathrm{exp}\left(-\int {f}^{1/2}(x)dx\right),$$ | ||

$x\to {a}_{1}+$. | |||

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

2.7.30 | $${w}_{1}(x)/{w}_{4}(x)\to 0,$$ | ||

$x\to {a}_{1}+$, | |||

${w}_{1}(x)$ is a *recessive* (or *subdominant*) solution as
$x\to {a}_{1}+$, and ${w}_{4}(x)$ is a *dominant* solution as $x\to {a}_{1}+$.
Similarly for ${w}_{2}(x)$ and ${w}_{3}(x)$ as $x\to {a}_{2}-$.

2.7.31 | $$\frac{{d}^{2}w}{{dx}^{2}}=(x+\mathrm{ln}x)w,$$ | ||

$$. | |||

We cannot take $f=x$ and $g=\mathrm{ln}x$ because $\int g{f}^{-1/2}dx$ would diverge as $x\to +\mathrm{\infty}$. Instead set $f=x+\mathrm{ln}x$, $g=0$. By approximating

2.7.32 | $${f}^{1/2}={x}^{1/2}+\frac{1}{2}{x}^{-1/2}\mathrm{ln}x+O\left({x}^{-3/2}{(\mathrm{ln}x)}^{2}\right),$$ | ||

we arrive at

2.7.33 | ${w}_{2}(x)$ | $\sim {x}^{-(1/4)-\sqrt{x}}\mathrm{exp}\left(2{x}^{1/2}-\frac{2}{3}{x}^{3/2}\right),$ | ||

2.7.34 | ${w}_{3}(x)$ | $\sim {x}^{-(1/4)+\sqrt{x}}\mathrm{exp}\left(\frac{2}{3}{x}^{3/2}-2{x}^{1/2}\right),$ | ||

as $x\to +\mathrm{\infty}$, ${w}_{2}(x)$ being recessive and ${w}_{3}(x)$ dominant.

One pair of independent solutions of the equation

2.7.35 | $${d}^{2}w/{dz}^{2}=w$$ | ||

is ${w}_{1}(z)={\mathrm{e}}^{z}$, ${w}_{2}(z)={\mathrm{e}}^{-z}$. Another is ${w}_{3}(z)=\mathrm{cosh}z$, ${w}_{4}(z)=\mathrm{sinh}z$. In theory either pair may be used to construct any other solution

2.7.36 | $$w(z)=A{w}_{1}(z)+B{w}_{2}(z),$$ | ||

or

2.7.37 | $$w(z)=C{w}_{3}(z)+D{w}_{4}(z),$$ | ||

where $A,B,C,D$ are constants. From the numerical standpoint, however, the pair
${w}_{3}(z)$ and ${w}_{4}(z)$ has the drawback that severe numerical cancellation can
occur with certain combinations of $C$ and $D$, for example if $C$ and $D$ are
equal, or nearly equal, and $z$, or $\mathrm{\Re}z$, is large and negative. This
kind of cancellation cannot take place with ${w}_{1}(z)$ and ${w}_{2}(z)$, and for this
reason, and following Miller (1950), we call ${w}_{1}(z)$ and ${w}_{2}(z)$ a
*numerically satisfactory pair*
of solutions.

The solutions ${w}_{1}(z)$ and ${w}_{2}(z)$ are respectively recessive and dominant as
$\mathrm{\Re}z\to -\mathrm{\infty}$, and *vice versa* as
$\mathrm{\Re}z\to +\mathrm{\infty}$. 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 $\frac{1}{2}\pi $ out of phase.