Irregular Singularities and Formal Classification

At a regular singular point, powers and logarithms suffice. At an irregular singular point, finite exponential factors enter the formal solutions. Their rational slopes, the regular-singular blocks attached to them, and the action of ramification are intrinsic formal data.

The gauge that exposes these data is generally a divergent formal series. Formal classification therefore answers what asymptotic types are possible, but not yet which analytic solution realizes a type in a given sector. That second question produces Stokes matrices and belongs to analytic sectorial normalization.

Formal theorem versus sectorial theorem

Naming varies. The decomposition below is often called the Levelt–Turrittin or Hukuhara–Levelt–Turrittin formal theorem. Its analytic realization on sufficiently small sectors is also called the Hukuhara–Turrittin theorem. This page uses only the formal statement.

An invariant diagnosis of irregularity

Let $x=z-z_0$ and consider a meromorphic system

$$Y'=A(x)Y, \qquad A(x)\in M_n\!\left(\mathbb C(\{x\})\right).$$

The point $x=0$ is regular singular if a formal meromorphic gauge

$$G\in GL_n\!\left(\mathbb C((x))\right)$$

puts the system into a Fuchsian presentation

$$\widetilde A(x)=\frac{R}{x}+O(1).$$

It is irregular otherwise. Equivalently, all formal slopes vanish in the regular case, whereas an irregular differential module has at least one positive slope.

This formulation is invariant under formal meromorphic gauge. It also records an assumption that is easy to overlook: the coefficient matrix must be meromorphic. An essential singularity of $A$ lies outside the finite-rank formal classification used here.

Why a displayed pole order is not enough

Suppose a chosen frame has

$$A(x) =\frac{A_{-r-1}}{x^{r+1}} +\frac{A_{-r}}{x^r} +\cdots, \qquad A_{-r-1}\neq0, \qquad r\geq0.$$

Then $r$ is the Poincaré rank of this presentation. More generally, the raw rank is

$$r_{\mathrm{raw}} =\max\left\{ \operatorname{poleord}_0(A)-1,\ 0 \right\}.$$

A formal meromorphic gauge, often involving shears, can lower it, so the raw pole order is not an invariant.

Datum	Meaning	Invariance
Raw Poincaré rank	Maximum of pole order minus one and zero	Frame-dependent
Minimal Poincaré rank	Minimum raw rank over formal meromorphic gauges	Invariant integer
Formal slopes	Rational numbers in the Levelt–Turrittin decomposition	Invariant multiset
Katz rank $\kappa$	Maximum formal slope	Invariant rational number
Irregularity	Sum of slopes with multiplicity	Invariant integer
Stokes levels	Positive pole orders in $x$ of nonzero differences $q_\alpha-q_\beta$; on $x=t^p$, divide the $t^{-1}$-degree by $p$	Pair-dependent formal data

The minimal Poincaré rank and Katz rank have the same zero test, but the integer rank loses fractional information. A module of Katz rank $3/2$, for example, needs an integer Poincaré rank at least $2$ in an unramified presentation.

A removable double pole

Let

$$N= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}, \qquad Y'=\frac{N}{x^2}Y.$$

The displayed matrix has raw Poincaré rank $1$. Since $N^2=0$, however,

$$G(x) =\exp\left(\frac{N}{x}\right) =I+\frac{N}{x}$$

is meromorphic. With the book’s left-gauge convention,

$$\widetilde A =GAG^{-1}+G'G^{-1} =0.$$

Thus the differential module is formally ordinary. A double pole with nilpotent leading coefficient is not, by itself, evidence of invariant irregularity.

The unramified formal normal form

First suppose no fractional power of $x$ is required. After formal meromorphic gauge, the system decomposes into blocks

$$\frac{\dd Z_\alpha}{\dd x} = \left[ q_\alpha'(x)I_{n_\alpha} +\frac{L_\alpha}{x} \right]Z_\alpha,$$

where

$$q_\alpha(x)\in x^{-1}\mathbb C[x^{-1}]$$

is a polar polynomial with no constant term, and $L_\alpha$ is constant. A formal fundamental matrix has the shape

$$\widehat\Phi(x) =\widehat H(x) \operatorname{diag}_{\alpha} \left( \ee^{q_\alpha(x)}x^{L_\alpha} \right),$$

with

$$x^{L_\alpha} =\exp\left( L_\alpha\operatorname{Log}x \right), \qquad \widehat H\in GL_n\!\left(\mathbb C((x))\right).$$

After extracting integral shears, one can normalize the remaining factor as a formal power-series unit. An equivalent compact notation is

$$\widehat\Phi =\widehat H\,\ee^Qx^\Lambda, \qquad [Q,\Lambda]=0,$$

where $Q$ is block diagonal and scalar on each exponential block.

Several qualifications prevent this formula from being misused:

• $L_\alpha$ need not be diagonal. It carries the regular-singular Jordan data within its exponential block.
• The polar polynomial represents the intrinsic class of $q_\alpha$ modulo holomorphic formal series.
• The exponential factors, their multiplicities, and the formal classes of their attached regular blocks are invariant up to permutation.
• Integral shears shift eigenvalues of $L_\alpha$ by integers without changing their monodromy eigenvalues.
• A block with $q_\alpha=0$ is regular singular.

Define

$$d_\alpha = \begin{cases} \deg_{x^{-1}}q_\alpha,&q_\alpha\neq0,\\ 0,&q_\alpha=0. \end{cases}$$

The block then has slope $d_\alpha$. Unramified slopes are integers.

Ramification and rational slopes

A general meromorphic connection may require a finite cover

$$x=t^p.$$

The pulled-back system is

$$\frac{\dd Y}{\dd t} =p\,t^{p-1}A(t^p)Y.$$

The formal Levelt–Turrittin theorem gives, over $\mathbb C((t))$, a decomposition

$$\widehat M_t \simeq \bigoplus_\alpha \left( \mathcal E^{q_\alpha} \otimes R_\alpha \right).$$

Here $\mathcal E^{q_\alpha}$ is the rank-one exponential module generated by $\ee^{q_\alpha(t)}$,

$$q_\alpha(t)\in t^{-1}\mathbb C[t^{-1}],$$

and $R_\alpha$ is regular singular. In matrices,

$$\widehat\Phi(t) =\widehat H(t) \operatorname{diag}_\alpha \left( \ee^{q_\alpha(t)}t^{L_\alpha} \right).$$

Define

$$d_\alpha = \begin{cases} \deg_{t^{-1}}q_\alpha,&q_\alpha\neq0,\\ 0,&q_\alpha=0. \end{cases}$$

The corresponding slope in the original $x$-coordinate is

$$s_\alpha=\frac{d_\alpha}{p}, \qquad \kappa=\max_\alpha s_\alpha.$$

The smallest cover on which this decomposition exists is part of the formal type. “Unramified” means that its minimal degree is $p=1$, not merely that the leading matrix happens to be diagonalizable in the original frame.

The deck transformation

$$t\longmapsto\zeta t, \qquad \zeta=\ee^{2\pi\ii/p},$$

preserves the multiset of exponential factors but can permute its elements:

$$q_\alpha(t) \longmapsto q_\alpha(\zeta t).$$

This permutation is essential in the formal monodromy.

Formal monodromy is not the full monodromy

In the unramified case, positive continuation leaves each polar polynomial and the formal Laurent gauge unchanged. It acts through

$$x^{L_\alpha} \longmapsto x^{L_\alpha}\ee^{2\pi\ii L_\alpha}.$$

Thus, in an ordered formal basis,

$$M_{\mathrm f} =\operatorname{diag}_\alpha \left( \ee^{2\pi\ii L_\alpha} \right).$$

In the ramified case, one positive circuit in the $x$-plane lifts only to $t\mapsto\zeta t$, not to a full circuit in the $t$-plane. It may permute exponential blocks. The generalized formal monodromy combines that permutation with the identifications and power factors on the regular blocks. It is generally wrong to represent one $x$-loop simply by $\exp(2\pi\ii L/p)$.

After $p$ positive $x$-circuits, the lift completes one $t$-circuit and the power part acts by $\exp(2\pi\ii L)$. Even this remains only formal monodromy. Actual analytic continuation also crosses singular directions whose sector boundaries carry Stokes jumps, and accumulates an ordered product of Stokes matrices.

A formal gauge can diverge

The exponential factors are finite polar polynomials, but the normalizing matrix $\widehat H$ generally diverges. Consider

$$\frac{\dd}{\dd x} \begin{pmatrix} y\\ c \end{pmatrix} = \begin{pmatrix} -x^{-2}&x^{-1}\\ 0&0 \end{pmatrix} \begin{pmatrix} y\\ c \end{pmatrix}.$$

For $c=1$, the first row is

$$x^2y'+y=x.$$

Writing $\widehat y=\sum_{n\geq1}a_nx^n$ gives

$$a_1=1, \qquad a_n=-(n-1)a_{n-1},$$

and therefore

$$\widehat y(x) =\sum_{n=1}^{\infty} (-1)^{n-1}(n-1)!x^n.$$

The formal fundamental matrix

$$\widehat\Phi(x) = \begin{pmatrix} 1&\widehat y(x)\\ 0&1 \end{pmatrix} \begin{pmatrix} \ee^{1/x}&0\\ 0&1 \end{pmatrix}$$

has a factorially divergent normalizer.

A series $\widehat f(x)=\sum_{n\geq0}f_nx^n$ is Gevrey of order $s$ if there are constants $C,A>0$ such that

$$\lVert f_n\rVert \leq CA^n\Gamma(1+sn).$$

The displayed series is Gevrey-$1$. On suitable sectors it has analytic realizations with the same asymptotic expansion, but on overlaps where $\operatorname{Re}(1/x)<0$ they can differ by a multiple of the flat homogeneous solution $\ee^{1/x}$. That invisible difference is the seed of a Stokes jump.

In the coordinate in which the formal decomposition is written—$t$ after ramification—a single integer level $k$ typically gives Gevrey order $1/k$ and $k$-summability away from singular directions. On $x=t^p$, the corresponding $x$-level is $k/p$. With several degrees among the differences $q_\alpha-q_\beta$, the normalizer is generally multisummable. The relevant levels come from differences of exponential factors, not merely from their individual degrees.

Newton polygons for scalar equations

The Newton polygon extracts slopes without first solving the full gauge problem. Fix the Euler derivation

$$\vartheta=x\frac{\dd}{\dd x}$$

and write a scalar operator as

$$P =\sum_{j=0}^{n}b_j(x)\vartheta^j, \qquad b_j\in\mathbb C((x)).$$

Let $\nu_j=\operatorname{ord}_x b_j$. This page uses

$$N(P) =\operatorname{conv} \bigcup_{b_j\neq0} \left\{ (u,v):u\leq j,\ v\geq\nu_j \right\}.$$

The positive slopes of its finite lower boundary are the formal slopes; the horizontal length of an edge is its multiplicity. Multiplying $P$ by a Laurent monomial translates the polygon vertically and changes none of these slopes.

Suppose a positive-slope edge $s$ has supporting line

$$\nu_j=sj+c.$$

If

$$b_j(x) =b_{j,0}x^{\nu_j}+\cdots,$$

define its edge polynomial by

$$\chi_s(\lambda) = \sum_{(j,\nu_j)\text{ on the edge}} b_{j,0}\lambda^j.$$

For a nonzero root $\lambda$,

$$\vartheta\ee^q \sim \lambda x^{-s}\ee^q.$$

Hence

$$xq'(x)=\lambda x^{-s}, \qquad q(x) =-\frac{\lambda}{s}x^{-s}.$$

Further Newton steps determine lower exponential terms, formal powers, and series corrections.

Rank-two normal form

For

$$\psi''+T(x)\psi=0, \qquad T(x)=t_{-m}x^{-m}+\cdots, \qquad t_{-m}\neq0,$$

the Euler form is

$$\left[ \vartheta^2-\vartheta+x^2T(x) \right]\psi=0.$$

When $m>2$, the relevant Newton points are

$$(0,2-m), \qquad (1,0), \qquad (2,0).$$

The positive edge has slope

$$s=\frac{m-2}{2}$$

with multiplicity two, and its edge polynomial is

$$\chi_s(\lambda) =\lambda^2+t_{-m}.$$

Thus

$$\lambda_\pm^2=-t_{-m}, \qquad q_\pm(x) =-\frac{\lambda_\pm}{s}x^{-s}+\cdots.$$

This agrees with the leading balance

$$q_\pm \sim \pm\int\sqrt{-T(x)}\,\dd x.$$

If $m$ is odd, $s$ is half-integral and a quadratic cover is required. If $m\leq2$, there is no positive slope and the point is regular singular.

An exact commuting model

Consider

$$Y' = \left( \frac{B}{x^2} +\frac{C}{x} \right)Y, \qquad [B,C]=0.$$

On a chosen logarithm branch,

$$\Phi(x)=\ee^{-B/x}x^C$$

is an exact fundamental matrix, because

$$\Phi' = \left( \frac{B}{x^2} +\frac{C}{x} \right)\Phi.$$

Positive continuation gives

$$\Phi^\gamma =\Phi\,\ee^{2\pi\ii C}, \qquad M_0=\ee^{2\pi\ii C}.$$

There is no divergent normalizer and no Stokes jump in this already-normal model.

To read its invariant formal type, use the commuting Jordan decomposition

$$B=B_{\mathrm s}+B_{\mathrm n}.$$

Since $B_{\mathrm n}$ is nilpotent, $G=\ee^{B_{\mathrm n}/x}$ is a finite Laurent polynomial, and the left gauge by $G$ replaces $B$ with $B_{\mathrm s}$. Equivalently, the factor $\ee^{-B_{\mathrm n}/x}$ in $\Phi$ is meromorphic and carries no intrinsic exponential type. The genuine exponential factors are

$$q_\beta(x)=-\frac{\beta}{x},$$

where $\beta$ ranges over the eigenvalues of $B_{\mathrm s}$. The matrix $C$ restricts to the corresponding generalized eigenspaces. Thus a purely nilpotent $B$ produces no positive slope despite the double pole.

A common nonzero scalar $B=\beta I$ is still irregular as a $GL_n$ connection, although its common exponential disappears after projectivization.

Airy at infinity

The Airy equation

$$y''-zy=0$$

has no finite singularity. Set

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

Then

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

or, in Euler form,

$$\left( \vartheta^2+\vartheta-x^{-3} \right)u=0.$$

The Newton points are $(0,-3)$, $(1,0)$, and $(2,0)$. The positive slope is

$$\kappa=\frac32$$

with multiplicity two. Since the edge polynomial is

$$\chi_{3/2}(\lambda) =\lambda^2-1,$$

the exponential factors are

$$q_\pm(x) =\pm\frac23x^{-3/2}.$$

On the quadratic cover $x=t^2$,

$$q_\pm(t) =\pm\frac{2}{3t^3},$$

and formal solutions have the shape

$$\widehat u_\pm(x) =x^{1/4} \exp\left( \pm\frac23x^{-3/2} \right) \widehat h_\pm(x^{3/2}).$$

One positive $x$-circuit lifts to $t\mapsto-t$ and exchanges the two exponential factors. With the natural normalization

$$\widehat h_\pm(0)=1, \qquad \widehat h_-(w)=\widehat h_+(-w),$$

continuation also sends $x^{1/4}$ to $\ii x^{1/4}$. Hence

$$\widehat u_+^\gamma=\ii\widehat u_-, \qquad \widehat u_-^\gamma=\ii\widehat u_+.$$

In the ordered basis $(\widehat u_+,\widehat u_-)$,

$$M_{\mathrm f} =\ii \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad M_{\mathrm f}^2=-I.$$

Rescaling or reordering the formal basis conjugates this matrix. The invariant content is the deck permutation together with the nontrivial power-factor phase.

The exact Airy solutions are entire in $z$, so their analytic continuation around any finite loop is trivial. Near $x=0$, the ordered Stokes factors compensate the full formal monodromy in the analytic monodromy product.

Turning point versus singular point

The point $z=0$ is a simple WKB turning point because the two WKB eigenvalues $\pm\sqrt z$ coalesce there. It is nevertheless an ordinary point of the exact Airy equation. Infinity is the irregular singular point.

Formal type in parameter families

For a normal-form family

$$\psi''=Q(z,\lambda)\psi,$$

a singular point is where the exact coefficient is singular. A turning point is a zero of $Q$, where the WKB eigenvalues $\pm\sqrt Q$ coalesce. A turning point can be ordinary for the exact ODE; it is the WKB diagonalization and spectral cover that become singular.

Formal type must be stated on parameter strata. The intrinsic type at a fixed pole can change when a leading polar coefficient vanishes, exponential factors at that pole coalesce, or the minimal ramification degree changes.

Separately, collisions of turning points change the WKB spectral cover and Stokes graph; they affect the local formal type only if a collision with a pole changes its polar jet. Degeneration of a chosen cyclic vector or normalization may spoil that presentation without changing the underlying differential module.

A sharp rank-two model is

$$Y' =\frac1{x^2} \begin{pmatrix} 0&1\\ \lambda&0 \end{pmatrix}Y.$$

For $\lambda\neq0$, the leading eigenvalues are $\pm\sqrt\lambda$ and

$$q_\pm(x) =\mp\frac{\sqrt\lambda}{x}, \qquad \kappa=1.$$

At $\lambda=0$, the leading matrix is nilpotent and the double pole is meromorphically removable, so $\kappa=0$. The coefficient family is holomorphic in $\lambda$, but an ordered formal eigenbasis branches in $\sqrt\lambda$ and cannot extend uniformly through the exceptional value.

A formal-classification workflow

For a local meromorphic system:

1. record the chosen coordinate, frame, and raw pole order;
2. test whether nilpotent leading terms can be lowered by a meromorphic shear;
3. compute the Newton slopes or an equivalent invariant reduction;
4. introduce the minimal ramification $x=t^p$;
5. list the exponential factors with multiplicities and deck action;
6. attach the regular-singular blocks and their formal monodromy;
7. state the parameter stratum on which these data remain valid;
8. postpone analytic sector choices and Stokes matrices until the formal normalizer has been sectorially realized.

Common pitfalls

Calling pole order invariant. It is the Poincaré rank of a presentation, not necessarily of the differential module. Nilpotent leading terms are the first warning that reduction may lower it.

Suppressing the ramified cover. A factor such as $x^{-3/2}$ requires a branch or the declared cover $x=t^2$. One $x$-loop can then permute formal blocks.

Equating formal and analytic monodromy. Analytic continuation also crosses singular directions whose sector boundaries carry Stokes jumps. Its matrix contains an ordered product of Stokes factors whose order depends on the initial sector and loop convention.

Treating the formal normalizer as convergent. The finite exponential polynomials do not imply convergence of $\widehat H$. Factorial divergence already occurs in the elementary triangular example above.

Discarding a common scalar exponential. It is invisible in the projectivized connection but remains part of the $GL_n$ formal type.

Calling a turning point an ODE singularity. Airy’s origin is the standard counterexample: it is an exact ordinary point and a WKB turning point.

Exercises

1. Remove a nilpotent double pole. Classify

$$Y'=\frac{N}{x^2}Y, \qquad N^2=0.$$

Solution

The displayed raw Poincaré rank is $1$. Set

$$G=\ee^{N/x}=I+\frac{N}{x}, \qquad G^{-1}=I-\frac{N}{x}.$$

Then

$$G'=-\frac{N}{x^2}, \qquad GAG^{-1}=\frac{N}{x^2},$$

so

$$\widetilde A =GAG^{-1}+G'G^{-1} =0.$$

All formal slopes vanish; the point is formally ordinary.

2. Read a scalar slope. For

$$\psi''+a x^{-m}\psi=0, \qquad a\neq0, \quad m>2,$$

find the slope, minimal ramification, and leading exponential factors.

Solution

In Euler form,

$$\left( \vartheta^2-\vartheta+a x^{2-m} \right)\psi=0.$$

The positive edge joins $(0,2-m)$ to $(2,0)$, so

$$s=\frac{m-2}{2}.$$

Its edge polynomial is $\lambda^2+a$. For either root $\lambda^2=-a$,

$$q(x) =-\frac{\lambda}{s}x^{-s}.$$

If $m$ is even, the leading type is unramified. If $m$ is odd, $s\in\mathbb Z+\tfrac12$ and the cover $x=t^2$ clears the ramification.

3. Verify the commuting model. Check the fundamental matrix and monodromy of

$$Y' = \left( \frac{B}{x^2} +\frac{C}{x} \right)Y, \qquad [B,C]=0.$$

Solution

Commutativity gives

$$\frac{\dd}{\dd x} \left( \ee^{-B/x}x^C \right) = \left( \frac{B}{x^2} +\frac{C}{x} \right) \ee^{-B/x}x^C.$$

The exponential is single-valued in $x$, while positive continuation sends

$$x^C \longmapsto x^C\ee^{2\pi\ii C}.$$

Thus $M_0=\ee^{2\pi\ii C}$. The nilpotent part of $B$ is meromorphically removable; its semisimple eigenvalues supply the factors $-\beta/x$.

4. Find the divergent normalizer. Solve formally

$$x^2y'+y=x$$

and determine its Gevrey order.

Solution

Put $y=\sum_{n\geq1}a_nx^n$. Equating coefficients gives

$$a_1=1, \qquad a_n+(n-1)a_{n-1}=0.$$

Therefore

$$a_n=(-1)^{n-1}(n-1)!,$$

and

$$\widehat y =x-x^2+2!x^3-3!x^4+\cdots.$$

The coefficients grow like $\Gamma(n)$, so the series is Gevrey-$1$ and divergent. On an overlap where $\operatorname{Re}(1/x)<0$, two sectorial realizations can differ by the flat homogeneous solution $\ee^{1/x}$.

5. Separate an Airy turning point from its singularity. For

$$\psi''=(z-\lambda)\psi,$$

classify the finite points and infinity.

Solution

The coefficient is entire, so every finite point is ordinary. Its zero $z=\lambda$ is a simple turning point. With $x=1/z$ and $u(x)=\psi(1/x)$,

$$u''+\frac2x u' +\left( -x^{-5}+\lambda x^{-4} \right)u=0.$$

The leading Newton term is still $-x^{-3}$ in Euler form, so infinity has slope $3/2$ for every finite $\lambda$. More precisely, on $x=t^2$,

$$q_\pm(t) =\pm \left( \frac{2}{3t^3}-\frac{\lambda}{t} \right) \pmod{\mathbb C[[t]]}.$$

Thus the minimal cover has degree $2$ and $\kappa=3/2$ throughout the finite $\lambda$-plane.

References

• D. G. Babbitt and V. S. Varadarajan, “Formal reduction theory of meromorphic differential equations: a group theoretic view”, for canonical levels, ramified formal gauges, and reduction of nilpotent leading coefficients.
• H. L. Turrittin, “Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point”, for the foundational formal and sectorial reduction theorem.
• A. H. M. Levelt, “Jordan decomposition for a class of singular differential operators”, for canonical decomposition and formal invariants.
• N. M. Katz, “On the calculation of some differential Galois groups”, for slope scaling, integrality of irregularity, and the Airy slope.
• V. S. Varadarajan, “Linear meromorphic differential equations: a modern point of view”, for reduction, asymptotics, Stokes phenomena, and local moduli.
• W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, for formal normalizers, Gevrey classes, and summability.
• NIST DLMF, irregular singularities of rank one and Airy asymptotics, for classical rank-one and Airy benchmarks.