Complex Linear ODEs and Singularities

The first task in a complex ODE problem is not to search for a named function. It is to determine which local structures the equation carries and which transformations preserve the global question. This chapter builds that diagnostic discipline for scalar second-order equations and their first-order systems.

The recurring scalar form is

$$y''(z)+p(z)y'(z)+q(z)y(z)=0,$$

with meromorphic coefficients on a declared Riemann surface or domain. From it we will extract fundamental matrices, Wronskians, local exponents, formal exponential factors, sectorial bases, monodromy, and Stokes matrices. Each object requires different hypotheses; no single “singularity type” captures all of them.

Chapter question

What local data does a complex linear ODE carry, and which transformations preserve the problem being solved?

The chapter assumes the just-in-time toolkit and uses the global notation and continuation conventions. Readers who already know Frobenius theory may still want to check the latter, because every connection and Stokes matrix below uses its right-action convention.

The diagnostic workflow

For a new equation, work through the following sequence before invoking a special function or correspondence.

1. Declare the domain and parameters. Include punctures, cuts, asymptotic ends, and the parameter values excluded from the calculation.
2. Choose a working form. Record the scalar equation, a first-order system, or normal form—and keep the transformations connecting them.
3. Locate every singular point. Include infinity and points introduced or removed by a gauge or coordinate change.
4. Apply the local growth test. Decide whether the point is ordinary, regular singular, or irregular in the chosen equation.
5. Compute the local invariant data. At a regular singularity, find indicial roots and resonance. At an irregular singularity, determine the formal exponential factors and any ramification.
6. Choose normalized bases. State leading coefficients, branches, sectors, and the order of basis vectors.
7. Separate formal from analytic claims. A formal normal form need not converge; an asymptotic series needs a sector and an error statement.
8. Check the global compatibility. Verify Wronskian evolution, determinant constraints, monodromy products, and controlled limiting behavior.

The output is a local data sheet, not merely a function name:

Entry	Minimum information
Equation	Coefficients, independent variable, domain, and parameters
Singularities	Location, type, and rank or formal slope where defined
Local bases	Ordering, leading normalization, logarithms, and branches
Irregular data	Formal exponential factors, sectors, and Stokes convention
Transformations	Gauge, coordinate, and parameter maps with inverses
Global target	Connection coefficient, monodromy datum, boundary function, or spectrum

The scalar growth test

Let $x=z-z_0$. The point $z_0$ is ordinary when $p$ and $q$ are holomorphic there. It is a regular singular point when

$$x\,p(z) \quad\text{and}\quad x^2q(z)$$

extend holomorphically to $x=0$. Equivalently, $p$ has at most a simple pole and $q$ at most a double pole. When the coefficients are meromorphic but this test fails, the point is irregular.

At a regular singular point, write

$$p(z)=\frac{p_{-1}}{x}+O(1), \qquad q(z)=\frac{q_{-2}}{x^2}+O(x^{-1}).$$

The indicial polynomial is

$$I(\rho) =\rho(\rho-1)+p_{-1}\rho+q_{-2}.$$

Its roots give the candidate Frobenius exponents. Their difference detects resonance, but an integer difference does not by itself decide whether a logarithm occurs. The recurrence calculation on the regular-singularity page supplies that missing information.

Meromorphic is a hypothesis

If a coefficient has an essential singularity, the point is singular but the finite-Poincaré-rank meromorphic classification used for formal normal forms does not apply without further analysis.

Infinity is a point, not a limit symbol

Classify infinity by the coordinate

$$x=\frac1z, \qquad u(x)=y(1/x).$$

Using

$$\frac{\dd y}{\dd z}=-x^2\frac{\dd u}{\dd x}, \qquad \frac{\dd^2y}{\dd z^2} =x^4\frac{\dd^2u}{\dd x^2} +2x^3\frac{\dd u}{\dd x},$$

the scalar equation becomes

$$\begin{aligned} u''(x) &+\left[ \frac2x-\frac{p(1/x)}{x^2} \right]u'(x)\\ &+\frac{q(1/x)}{x^4}u(x)=0. \end{aligned}$$

The ordinary or regular-singular test is then applied at $x=0$. This coordinate calculation is safer than reasoning only from the large-$z$ appearance of the original coefficients.

For coefficients meromorphic near infinity, the formula gives the useful corollary

$$\infty\text{ is regular singular} \quad\Longleftrightarrow\quad \begin{cases} p(z)=O(z^{-1}),\\ q(z)=O(z^{-2}). \end{cases}$$

With the chosen dependent variable $u(x)=y(1/x)$, infinity is ordinary only when the leading derivative term also cancels:

$$p(z)=\frac2z+O(z^{-2}), \qquad q(z)=O(z^{-4}).$$

For a system

$$\frac{\dd Y}{\dd z}=A(z)Y,$$

the transformed matrix is

$$\frac{\dd Y}{\dd x} =-\frac1{x^2}A(1/x)Y.$$

A presentation

$$A(x)=\frac{A_{-1}}x+O(1)$$

is Fuchsian and hence regular singular. Invariantly, a meromorphic system is regular singular when a single-valued formal meromorphic gauge $G\in GL_n(\mathbb C((x)))$ puts it in such a presentation.

The raw scalar companion matrix can contain a double pole in $q$ even at a regular singular point. The meromorphic gauge

$$\widetilde Y= \begin{pmatrix} y\\ xy' \end{pmatrix}$$

gives

$$\widetilde Y' = \begin{pmatrix} 0 & x^{-1}\\ -xq & x^{-1}-p \end{pmatrix} \widetilde Y.$$

When $p$ has at most a simple pole and $q$ at most a double pole, every entry has at most a simple pole, so this presentation is Fuchsian.

Three forms and three kinds of change

The same local solution space can be presented in several forms, but the transformations do not all act in the same way.

Operation	Formula	What changes
Scalar-to-system conversion	$Y=(y,y')^{\mathsf T}$	Presentation; the scalar cyclic vector is remembered
Scalar gauge	$y=f(z)\psi$	Coefficients, exponents, and possibly global branches
System gauge	$\widetilde Y=G(z)Y$	Connection matrix by $\widetilde A=GAG^{-1}+G'G^{-1}$
Coordinate change	$z=z(w)$	Coefficients, singular locations, and normal-form Schwarzian term
Constant basis change	$\widetilde\Phi=\Phi H$	Matrix representatives of connection and monodromy data
Singular parameter scaling	$z=z(\zeta,\varepsilon)$	Potentially the singularity type and normalization

The first-order companion system is

$$\frac{\dd}{\dd z} \begin{pmatrix} y\\ y' \end{pmatrix} = \begin{pmatrix} 0 & 1\\ -q & -p \end{pmatrix} \begin{pmatrix} y\\ y' \end{pmatrix}.$$

Its trace is $-p$. Removing the first derivative from the scalar equation uses

$$y =\exp\left(-\frac12\int^z p(\zeta)\,\dd\zeta\right)\psi$$

and gives

$$\psi''+T\psi=0, \qquad T=q-\frac12p'-\frac14p^2.$$

Locally this produces a traceless companion system. Globally, the exponential factor may be multivalued: a trace-removing $GL(2)$ gauge requires a trivialization of the determinant local system, and the scalar half-density gauge also requires a compatible square root. These are global lift choices, not consequences of the local algebra. The system formulation and oper page treat the two issues separately.

What normal form reveals

At a finite point, normal form makes the regular-singular threshold especially transparent:

$$\psi''+T(z)\psi=0$$

is regular singular when $T$ has a pole of order at most two. If

$$T(z) =\frac{t_{-2}}{x^2}+O(x^{-1}),$$

then its exponents satisfy

$$\sigma(\sigma-1)+t_{-2}=0.$$

The scalar gauge shifts exponents. Since

$$y\sim x^{-p_{-1}/2}\psi,$$

one has

$$\sigma=\rho+\frac{p_{-1}}2.$$

The two normal-form exponents sum to $1$, and their difference equals $\rho_+-\rho_-$. Analytic continuation of the scalar gauge multiplies the monodromy matrix by a common scalar, so its $PGL(2,\mathbb C)$ class is unchanged. At resonance, however, the exponent difference alone does not distinguish scalar from nontrivially unipotent projective monodromy.

If $T$ has a pole of order greater than two, the point is irregular. The leading formal behavior is suggested by

$$\psi \sim \exp\left( \pm\int^z\sqrt{-T(\zeta)}\,\dd\zeta \right),$$

but this expression is only diagnostic. Repeated leading eigenvalues, ramification, lower-order terms, and formal gauge transformations determine the actual formal classification.

Four benchmark equations

These models quickly test a classification or transformation convention.

Equation	Finite points	Infinity	Main local feature
Airy: $y''-zy=0$	All ordinary	Irregular	Fractional-power exponential factors
Bessel: $z^2y''+zy'+(z^2-\nu^2)y=0$	$0$ regular singular	Irregular	Resonance when $2\nu\in\mathbb Z$
Gauss hypergeometric	$0$ and $1$ regular singular	Regular singular	Three-point Fuchsian monodromy
Weber: $y''+(\nu+\tfrac12-\tfrac14z^2)y=0$	All ordinary	Irregular	Sector-dependent recessive solutions

The hypergeometric equation is the exact connection benchmark for the first three chapters. Airy and Weber are the local turning-point benchmarks for the exact-WKB branch.

Worked diagnosis: the Bessel equation

Divide the Bessel equation by $z^2$:

$$y''+\frac1z y' +\left(1-\frac{\nu^2}{z^2}\right)y=0.$$

At $z=0$,

$$z\,p(z)=1, \qquad z^2q(z)=z^2-\nu^2,$$

so the origin is regular singular. The indicial equation is

$$\rho(\rho-1)+\rho-\nu^2 =\rho^2-\nu^2=0,$$

with exponents $\rho_\pm=\pm\nu$.

Resonance occurs when $2\nu\in\mathbb Z$. For $\nu\in\mathbb Z$, the standard second solution contains a logarithmic term. For $\nu\in\mathbb Z+\tfrac12$, the integer exponent difference is log-free. This is a concrete demonstration that resonance is a warning, not a proof of logarithmic behavior.

At infinity, the transformed equation is

$$u''+\frac1x u' +\left(\frac1{x^4}-\frac{\nu^2}{x^2}\right)u=0.$$

Because $x^2$ times the coefficient of $u$ still has a pole, $x=0$ is irregular. The normal-form gauge $y=z^{-1/2}\psi$ gives

$$\psi'' +\left[ 1+\frac{\tfrac14-\nu^2}{z^2} \right]\psi=0.$$

The constant leading term at large $z$ produces

$$\psi_\pm(z) \sim \ee^{\pm\ii z} \left( 1+\frac{c_1^{(\pm)}}z +\frac{c_2^{(\pm)}}{z^2} +\cdots \right),$$

and therefore

$$y_\pm(z) \sim z^{-1/2}\ee^{\pm\ii z} \left( 1+\frac{c_1^{(\pm)}}z+\cdots \right).$$

The parameter $\nu$ enters the inverse-power coefficients; both original Bessel branches share the formal power $z^{-1/2}$. A sector and branch are still required before these series become normalized analytic solutions.

Parameters and exceptional loci

Let $D$ and $\Lambda$ be complex domains, choose $z_*\in D$, and suppose

$$A\in\mathcal O(D\times\Lambda), \qquad \Phi_*(\lambda)\in GL(n,\mathbb C)$$

depends holomorphically on $\lambda$. The solution normalized by $\Phi(z_*,\lambda)=\Phi_*(\lambda)$ is jointly holomorphic near $\{z_*\}\times\Lambda$. If $D$ is a common simply connected regular domain for the family, the solution extends jointly holomorphically on $D\times\Lambda$.

Canonical local bases can nevertheless become singular as functions of $\lambda$. Typical exceptional loci include:

• collision of indicial roots;
• integer exponent differences and logarithmic resonance;
• coalescence of formal eigenvalues;
• turning-point collision;
• a parameter value at which the chosen leading normalization vanishes;
• a singular coordinate or gauge transformation.

The remedy is not to declare analytic dependence lost everywhere. Keep a base point and common regular domain valid throughout the parameter neighborhood, use an initial-value fundamental matrix as the regular reference, and treat the canonical basis as a parameter-dependent change of basis whose poles, branching, or Jordan limits must be analyzed.

Chapter route

Page	Question answered
Scalar Equations and First-Order Systems	How do fundamental matrices, Abel’s identity, gauges, and parameters fit together?
Liouville Normal Form and SL(2) Opers	What does removing the first derivative preserve, and why does a Schwarzian term appear?
Regular Singularities: Frobenius, Resonance, and Monodromy	How do Frobenius recurrences, resonance, logarithms, and local monodromy work?
Irregular Singularities and Formal Classification	Which exponential factors and formal invariants replace Frobenius data?
Analytic Classification and Sectorial Normalization	How are divergent formal solutions realized analytically in sectors?
Confluence and Singular Limits	How do regular singularities merge into irregular ones under controlled scaling?
Worked Case Files and Problems	Can the full diagnostic be executed on Bessel, Airy, hypergeometric, and a Poincaré-rank-one model?

By the end of the chapter, a reader should be able to classify a singularity, construct its normalized local or sectorial basis, and translate between scalar and normal forms without discarding branch data.

Common pitfalls

Forgetting infinity. A polynomial coefficient may look entire while creating the equation’s only irregular singularity at infinity. Transform with $x=1/z$ and apply the finite-point test.

Calling every pole Fuchsian. For a scalar second-order equation, $p$ may have at most a simple pole and $q$ at most a double pole. A higher pole in either weighted coefficient is irregular.

Treating a local gauge as globally single-valued. The factor removing $y'$ contains an integral of $p$. Continuation may multiply this factor by a scalar, so the local trace-free frames need not glue as a single-valued gauge of the original bundle. If it is nevertheless used to define new solutions, it defines a scalar-character twist of monodromy; it is not a gauge-equivalent horizontal trivialization of the original determinant local system.

Using a formal exponential as a canonical solution. A formal series may diverge and may represent different analytic solutions in adjacent sectors. Sector, normalization, and summation data come before a Stokes matrix.

Exercises

1. Infinity test. Derive the transformed scalar equation under $x=1/z$. Use it to classify infinity for $y''+z^{-2}y=0$ and for the Airy equation $y''-zy=0$.

Solution

The chain rule gives

$$y'=-x^2u', \qquad y''=x^4u''+2x^3u'.$$

For $y''+z^{-2}y=0$, one obtains

$$u''+\frac2x u'+\frac1{x^2}u=0.$$

Both weighted coefficients are holomorphic, so infinity is regular singular. For Airy,

$$u''+\frac2x u'-\frac1{x^5}u=0,$$

and $x^2(-x^{-5})$ is singular, so infinity is irregular.

2. Gauge-shifted exponents. Prove that the normal-form exponents satisfy $\sigma=\rho+p_{-1}/2$ and $\sigma_++\sigma_-=1$.

Solution

Near the point,

$$\exp\left(-\frac12\int p\,\dd z\right) \sim x^{-p_{-1}/2}.$$

Thus $y\sim x^\rho=x^{-p_{-1}/2}x^\sigma$, which gives $\sigma=\rho+p_{-1}/2$. Vieta’s formula for the original indicial polynomial gives

$$\rho_++\rho_-=1-p_{-1}.$$

Adding $p_{-1}/2$ to each root yields $\sigma_++\sigma_-=1$.

3. Basis change or gauge change? Let $\Phi'=A\Phi$. Compare $\widetilde\Phi=\Phi H$ for constant $H$ with $\widehat\Phi=G(z)\Phi$. Derive the differential equation in each case.

Solution

For constant $H$,

$$(\Phi H)'=A\Phi H,$$

so a solution-basis change leaves $A$ unchanged. For a $z$-dependent left gauge,

$$\widehat\Phi' =G'\Phi+GA\Phi =\left(G'G^{-1}+GAG^{-1}\right)\widehat\Phi.$$

The first operation changes the frame of the solution space; the second changes the connection matrix.

References

• NIST DLMF, Differential equations, for the ordinary, regular-singular, irregular, and infinity tests and classical Fuchs–Frobenius theory.
• NIST DLMF, Differential equations of arbitrary order, for the general Fuchs criterion and elementary examples of confluence.
• G. Teschl, Ordinary Differential Equations and Dynamical Systems, for complex linear systems, analytic dependence, and the Frobenius method.
• W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover, 1987, for singular systems, formal transformations, and asymptotic solutions.
• Y. Sibuya, Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland Mathematics Studies 18, 1975, ISBN 978-0-444-10959-0, for sectorial solutions and global Stokes data of polynomial equations.