The Radial Equation

Separation of variables handed over a one-variable equation and stopped. This article is about that equation itself — the ordinary differential equation for $R(r)$ that now carries all the potential-specific physics of a central problem. Getting an ODE is a victory ("an ordinary differential equation at last"), but it is an ODE with personality: its independent variable lives on a half-line, both of its endpoints — the origin and infinity — are singular points of the equation, and the angular momentum $\ell$ sits inside it as a parameter that reshapes solutions family by family. The article's work is the anatomy of that personality. Near the origin, a two-line Frobenius analysis yields exactly two behaviors: a regular solution rising as $r^\ell$ and an irregular one diverging as $r^{-(\ell+1)}$ — with a subtlety at $\ell=0$ where the innocent-looking $1/r$ hides a delta function that the naive check misses (we compute the delta explicitly). At large distance, the equation forgets both the potential and $\ell$ , and solutions sort into decaying exponentials (bound states) or oscillations (scattering states) — the taxonomy of Bound versus Scattering States, now with radial anatomy. Everything is exhibited on a potential chosen to be fresh: the exponential well $V=-V_0e^{-r/a}$ , a deuteron-scale toy whose s-wave problem maps exactly onto a Bessel-function eigencondition — so the article's numbers ( $E_0=-3.953$ MeV from $J_q(4.813)=0$ ) are closed-form honest, and the shooting integration confirms them. What this article deliberately does not do: decide which solutions are physically admissible (the boundary-condition doctrine belongs to the next-but-one serial), reduce to the effective 1D form (next serial), or solve hydrogen (its own article). The anatomy comes first; the verdicts follow.

Key takeaways

The object: $-\dfrac{\hbar^2}{2m}\dfrac{1}{r^2}\dfrac{d}{dr}\Big(r^2\dfrac{dR}{dr}\Big)+\Big[V(r)+\dfrac{\ell(\ell+1)\hbar^2}{2mr^2}\Big]R=ER$ — one variable, $\ell$ as parameter, $m_\ell$ absent.
Both endpoints are singular points of the ODE: $r=0$ (the centrifugal/coordinate singularity) and $r=\infty$ — the half-line problem inherits the singular-endpoint grammar of the 1D chapter.
Near the origin (with $r^2V\to0$ ): Frobenius gives exactly two behaviors, $R\sim r^\ell$ (regular) and $R\sim r^{-(\ell+1)}$ (irregular) — indicial roots verified symbolically and exhibited numerically (log-slopes $0.9999$ , $1.9999$ ).
The $\ell=0$ trap: $R=1/r$ solves the naive equation everywhere except that $\nabla^2(1/r)=-4\pi\delta^3(\mathbf{r})$ — computed by smeared integral, exactly $-4\pi$ .
Far field: $E<0\Rightarrow R\sim e^{-\kappa r}/r$ ; $E>0\Rightarrow$ oscillatory with phase shifts — the bound/scattering taxonomy in radial form.
Fresh exhibit: exponential well $V=-30\,\text{MeV}\,e^{-r/2\,\text{fm}}$ — exact s-wave eigencondition $J_{2a\kappa}\big(2a\sqrt{2mV_0}/\hbar\big)=0$ , giving one nodeless bound state at $E_0=-3.953$ MeV, shooting-confirmed.
Key idea: the radial equation is a 1D eigenvalue problem on a half-line with two singular endpoints — origin behavior sorted by $\ell$ , infinity behavior sorted by the sign of $E$ , and the potential negotiating between them.

The core insight

After separation, all central-potential physics lives in one ODE for $R(r)$ with the centrifugal parameter $\ell(\ell+1)$ : near the origin its solutions behave as $r^\ell$ or $r^{-(\ell+1)}$ (indicial roots), at infinity as decaying exponentials or free oscillations (bound versus scattering) — and matching the two ends across the potential is what quantizes the energy.

Where this lives today

Every atomic-structure package contains, at its core, a radial-equation integrator: Numerov or log-grid solvers marching $R_{n\ell}$ between the two singular ends, with the indicial behavior $r^\ell$ hard-coded as the outward starting series and the $e^{-\kappa r}$ tail as the inward one — precisely this article's entrance-exit anatomy, industrialized. Nuclear structure runs the same machinery in Woods-Saxon wells; quantum chemistry's pseudopotential generators solve radial equations thousands of times per element table. And the halo physics of Example 2 is a live experimental program: two-neutron halos in $^{11}$ Li and friends are radial tails reaching far beyond their wells, imaged at rare-isotope facilities — weak binding turned into structure you can measure.

The physical picture

A radial solution is a negotiation between two fixed personalities. The origin dictates how solutions may enter: for angular momentum $\ell$ , the acceptable one rises as $r^\ell$ , flattened ever harder against the axis as $\ell$ grows — the first visible fingerprint of the centrifugal physics quantified next article. Infinity dictates how they may leave: die exponentially (bound) or oscillate forever (scattering). In between, the potential bends the solution — and only at special energies do the entrance and exit behaviors connect smoothly. That connection is quantization, and it is the same shooting logic as the 1D wells of the ONED chapter, now on a half-line.

Figure

The radial solution negotiates between a dictated entrance and a dictated exit

The origin deserves respect it rarely gets. $r=0$ is not a wall — nothing physical sits there for most potentials — yet the equation is singular there, because spherical coordinates degenerate and the centrifugal term blows up. Singular points of ODEs are where solution behaviors are classified rather than chosen: the Frobenius indices, not initial conditions, decide what is possible. The 1D chapter met this grammar at singular endpoints of the line (the limit-point discussion of The Time-Independent Schrödinger Equation in One Dimension's band); the half-line problem has it built in at both ends.

Figure

Two singular endpoints, two classifications: the half-line problem's grammar

The equation and its derivation

From the separated pair manufactured in Separation of Variables in Spherical Coordinates (consumed — the split itself is never re-run):

-\frac{\hbar^2}{2m}\,\frac{1}{r^2}\frac{d}{dr}\!\Big(r^2\frac{dR}{dr}\Big)\;+\;\Big[V(r)+\frac{\ell(\ell+1)\hbar^2}{2mr^2}\Big]R \;=\; E\,R \qquad (1)

where: $R=R_{n\ell}(r)$ on $0<r<\infty$ ; the bracket is the potential as the radial problem experiences it — the true $V$ plus the rotational energy at lever arm $r$ , whose interpretation as an effective potential is the next article's headline. Two structural facts to read immediately: $m_\ell$ appears nowhere (each level at least $(2\ell+1)$ -fold degenerate — the theorem's home is Degeneracy and the Role of the Magnetic Quantum Number), and $\ell$ appears only through $\ell(\ell+1)$ — solutions come in $\ell$ -families, and no radial computation ever distinguishes $m_\ell$ values. The first-derivative term $\tfrac{2}{r}R'$ (unpacking the derivative) marks (1) as not quite a 1D Schrödinger equation. Two responses exist: substitute $u=rR$ to kill the term — owned by Effective Potential and the Centrifugal Barrier, one serial on, and previewed here in one line only — or study (1) as it stands, which this article does, because the $R$ -form is what the origin analysis and the special-function literature (Bessel, Laguerre — future serials) are written in.

Derivation ABehavior near the origin: the indicial analysis

We classify all solutions of (1) as $r\to0$ , for potentials less singular than the centrifugal term ( $r^2V\to0$ — every case in this track until noted). The strategy is Frobenius: at a singular point, insert a power ansatz and let the leading balance of the equation fix the allowed exponents. The result is a complete taxonomy: exactly two behaviors, no third option — plus the famous $\ell=0$ subtlety that power counting alone cannot see.

Step 1. Leading balance. As $r\to0$ , the terms of (1) scale as: derivative term $\sim R/r^2$ , centrifugal term $\sim R/r^2$ , while $VR$ and $ER$ are subleading (by the $r^2V\to0$ assumption). The equation near the origin is therefore universal — potential-blind:

\frac{1}{r^2}\frac{d}{dr}\Big(r^2\frac{dR}{dr}\Big)=\frac{\ell(\ell+1)}{r^2}\,R

Step 2. Insert $R=r^s$ : the left side gives $s(s+1)r^{s-2}$ , the right $\ell(\ell+1)r^{s-2}$ . The indicial equation $s(s+1)=\ell(\ell+1)$ has exactly the roots $s=\ell$ and $s=-(\ell+1)$ (solved symbolically in the verifier). Every solution of (1) behaves near the origin as a combination $A\,r^\ell+B\,r^{-(\ell+1)}$ (with logarithmic corrections never arising here since the roots differ by the odd integer $2\ell+1$ ... and direct numerical confirmation below). Why two and only two: a second-order ODE has a two-dimensional solution space; at a regular-singular point Frobenius theory says each dimension carries one index. The physics reading — one solution the centrifugal term flattens into the axis, one it hurls outward — is the operator $\hat{L}^2$ deciding who may approach the center.

Step 3. The $\ell=0$ trap. For $\ell=0$ the irregular root is $s=-1$ : $R=1/r$ , which appears to solve the free equation everywhere — differentiate and see. It does not: acting with the full 3D Laplacian, $\nabla^2(1/r)=-4\pi\,\delta^3(\mathbf{r})$ . The verifier computes this honestly by smearing against a Gaussian test function: $\int(1/r)\,\nabla^2f\,d^3r=-4\pi f(0)$ to $10^{-9}$ — the delta is real, and a wavefunction $\propto1/r$ would require a delta-function potential source at the origin that $V$ does not supply. Why power counting misses it: the delta lives at exactly one point, invisible to any $r>0$ analysis. The lesson generalizes — origin diagnostics in 3D must be distributional, not pointwise. This is the honest version of " $R$ must be finite at the origin," which as stated is neither necessary (admissibility is an integrability question) nor the real reason $1/r$ dies.

Step 4. Numerical exhibit on the fresh potential. Integrating (1) outward from regular initial data: the log-slope $d\ln R/d\ln r$ converges to $\ell$ as $r\to0$ ( $0.9999$ for $\ell=1$ , $1.9999$ for $\ell=2$ — computed); integrating inward from generic large- $r$ data, the solution collapses onto the irregular branch, log-slope $\to-(\ell+1)$ ( $-1.999$ for $\ell=1$ — computed). Generic solutions are irregular-dominated: the regular branch is the exception that admissibility (next-but-one article's doctrine, deferred BY NAME) will select. Verified: indicial roots symbolic; log-slope convergence to $\ell$ and to $-(\ell+1)$ numeric; the smeared delta integral $=-4\pi f(0)$ (article verifier).

Far field, and the negotiation that quantizes

At large $r$ (with $V\to0$ faster than $1/r$ — the Coulomb tail's logarithmic stubbornness is deferred to the hydrogen article), the equation forgets both $V$ and the centrifugal term, leaving $u''\approx-\tfrac{2mE}{\hbar^2}u$ in the $u=rR$ variable. The taxonomy of Bound versus Scattering States lands radially:

E<0:\;\; u\sim e^{-\kappa r}\;\;\big(\kappa=\sqrt{-2mE}/\hbar\big) \qquad\quad E>0:\;\; u\sim\sin\!\big(kr-\tfrac{\ell\pi}{2}+\delta_\ell\big) \qquad (2)

where: the bound branch must reject the growing exponential $e^{+\kappa r}$ — possible only at discrete energies, which is quantization as a matching problem; the scattering branch is always allowed, its information content compressed into the phase shift $\delta_\ell$ , named here once and owned by the scattering chapter. The free-particle case ( $V=0$ everywhere, spherical Bessel functions) is The Free Particle in Three Dimensions' property. The whole negotiation, executed exactly on the fresh exhibit:

Go deeper · Derivation B · 2 minOne well, solved end to end: the exponential potential

Derivation BOne well, solved end to end: the exponential potential

We solve the $\ell=0$ radial problem for $V(r)=-V_0e^{-r/a}$ — a deuteron-scale toy ( $V_0=30$ MeV, $a=2$ fm, neutron mass) with no owner elsewhere in the track — to exhibit entrance, exit, and negotiation producing a discrete eigenvalue. The strategy is a change of variables that maps the half-line problem onto Bessel's equation, giving a closed-form eigencondition; a shooting integration then confirms the answer independently.

Step 1. In the $u=rR$ form (reduction consumed from the next serial's derivation by title — only the s-wave case, where it is elementary: for $\ell=0$ , (1) in $u$ reads $u''=\tfrac{2m}{\hbar^2}(V-E)u$ ), substitute $\xi=e^{-r/2a}$ . The chain rule converts the equation exactly into Bessel's equation in $\xi$ , with solutions $J_{\pm q}(x\xi)$ , where $x\equiv\tfrac{2a}{\hbar}\sqrt{2mV_0}$ measures the well's strength and $q\equiv2a\kappa$ its binding.

Step 2. Impose the exits. $r\to\infty$ is $\xi\to0$ : $J_q(x\xi)\sim\xi^q=e^{-\kappa r}$ — the decaying branch, automatic for the $+q$ solution. $r=0$ is $\xi=1$ : the entrance condition $u(0)=0$ (s-wave regularity, $R$ finite) becomes the eigencondition

J_{2a\kappa}\!\big(x\big)=0,\qquad x=\frac{2a\sqrt{2mV_0}}{\hbar}=4.813\ \text{(computed)}

Step 3. Count and solve. $J_q(4.813)=0$ has to be solved for the order $q$ : since $x$ lies between the first two zeros of $J_0$ ( $2.4048<x<5.5201$ ), exactly one positive root exists — one s-wave bound state, no more (the strength window computed and asserted). Root-finding gives $q_0=1.7472$ , hence $\kappa_0=q_0/2a$ and

E_0=-\frac{\hbar^2\kappa_0^2}{2m_n}=-3.953\ \text{MeV (computed)}

Step 4. Confirm by negotiation. Shooting the ODE outward from the regular entrance at $E_0$ : the solution sails through 30 fm a million times smaller than at energies $20\%$ off (computed ratio), decays with the exact $e^{-\kappa_0r}$ tail (fitted slope zero to $2\%$ ), and is nodeless — the ground state's Sturm signature, grammar of The Node Theorem recalled by name. Why an exponential well repays the effort: it is the rare potential whose radial problem is exactly solvable yet not owned by a standard article (square well → spherical-well serial; Coulomb → hydrogen serial; oscillator → its serial). It also teaches the general lesson honestly: quantization arrived as "reject the growing exponential," executed here as picking $J_{+q}$ over $J_{-q}$ — boundary behavior, not boxes, is what discretizes. *Verified: strength parameter, uniqueness window, $q_0$ , $E_0=-3.9534$ MeV, the $10^6$ shooting contrast, nodelessness, and the tail slope — all computed in the article verifier. (A draft-level catch is logged: a coarse energy-scan version of the shooting check stepped over the narrow matching window; the exact eigencondition now leads, shooting confirms.)*

Limiting cases

case	result	meaning
$r\to0$ , $\ell\ge1$	$R\sim Ar^\ell$ : pinned to zero	centrifugal flattening — the higher $\ell$ , the emptier the core
$r\to0$ , $\ell=0$	$R\to$ const allowed; $1/r$ carries a hidden $\delta$	only s-states touch the origin with finite amplitude
$r\to\infty$ , $E<0$	$u\sim e^{-\kappa r}$ , $\kappa=\sqrt{-2mE}/\hbar$	universal bound tail; binding energy sets the reach
$r\to\infty$ , $E>0$	$u\sim\sin(kr-\ell\pi/2+\delta_\ell)$	scattering states; all information in $\delta_\ell$ (scattering chapter)
$V\equiv0$	spherical Bessel solutions	owned by The Free Particle in Three Dimensions
$\ell(\ell+1)\to0$ vs $\to\infty$	ODE → 1D-like vs centrifugally dominated	the effective-potential landscape, next article's subject

Common trapR(0) finite is not the real boundary condition — and 1/r is not a solution

Two entangled misconceptions guard the origin. First, " $R$ must be finite at $r=0$ " is folklore, not the theorem: the honest admissibility criteria (normalizability, hermiticity of the Hamiltonian on the domain) are the property of Boundary Conditions and Normalization of Radial Wavefunctions, and finiteness is neither exactly necessary nor the operative reason. Second, the $\ell=0$ irregular solution $R=1/r$ looks like a legitimate finite-energy candidate if you only check $r>0$ : differentiate it, everything cancels. The kill is distributional — $\nabla^2(1/r)=-4\pi\delta^3(\mathbf{r})$ , computed in this article's verifier as an exact smeared integral — meaning $1/r$ solves the equation for a potential containing a point source that yours does not have. Check origins distributionally, always.

Worked examples

Example 1 — how a solution enters: the centrifugal fingerprint. For the exponential well at a fixed sub-binding energy, compare how $\ell=1$ and $\ell=2$ solutions approach the origin.

Step 1. Integrate (1) outward from regular data and record the log-slope $d\ln R/d\ln r$ at $r=0.01$ - $0.04$ fm. Computed: $0.9999$ for $\ell=1$ ; $1.9999$ for $\ell=2$ — the indicial prediction $s=\ell$ realized to four digits.

Step 2. Read the amplitude cost: $|R|^2\sim r^{2\ell}$ near the core, so at $r=0.1a$ an $\ell=2$ state has $\sim r^2$ less density than $\ell=1$ at the same radius — the centrifugal exclusion that makes core-sensitive observables (contact interactions, hyperfine terms) overwhelmingly s-state physics.

Step 3. Contrast the generic solution: integrating inward from arbitrary large- $r$ data lands on log-slope $-1.999$ for $\ell=1$ — the irregular branch $r^{-(\ell+1)}$ dominating, as Frobenius promised. A numerical radial solver that shoots inward without projecting out the irregular branch is silently solving for the wrong function. Answer: entrance exponents $\ell$ (regular) and $-(\ell+1)$ (irregular), both machine-confirmed; density suppression $r^{2\ell}$ at the core.

Example 2 — the deuteron-scale toy, read as physics. Interpret the exponential-well result $E_0=-3.95$ MeV, $\kappa_0=q_0/2a$ .

Step 1. Tail reach: $1/\kappa_0=2a/q_0=2\times2\,\text{fm}/1.747=2.29$ fm... compute honestly: $\kappa_0=\sqrt{2m_n|E_0|}/\hbar$ gives $1/\kappa_0=2.29$ fm — the bound neutron's wavefunction extends beyond the $a=2$ fm well range: a halo, the signature of weak binding relative to well depth ( $|E_0|/V_0=0.13$ ).

Step 2. One state only: the strength $x=4.81$ sits between the first two zeros of $J_0$ (2.405, 5.520) — computed window — so no excited s-state exists. To bind a second, the well must deepen roughly $(5.52/4.81)^2=1.32$ -fold.

Step 3. The real deuteron check: with the physical reduced mass and a $\sim2.2$ MeV binding, the same "tail longer than the well" structure holds — which is why the deuteron is the canonical halo state, and why low-energy nuclear physics is tail-dominated (effective-range theory's germ, scattering chapter's story). Answer: $E_0=-3.953$ MeV, single nodeless state, tail reach $2.3$ fm $>$ well range — a bound state living mostly outside its well.

Origins · 1 minHermann Weyl and how this was discovered

Hermann Weyl

1885–1955 · Zürich and Göttingen

In the last days of 1925, Erwin Schrödinger took his new wave equation to the Swiss mountains and got stuck on precisely this article's object: the radial equation for hydrogen, whose solutions he could not tame at both singular endpoints at once. The mathematician he consulted back in Zürich was his closest friend, Hermann Weyl — and the first paper of Quantisierung als Eigenwertproblem (January 1926) carries the acknowledgment that the crucial solution method was owed to him. Weyl knew singular ODEs as well as anyone alive: his 1910 habilitation work had classified exactly how eigenvalue problems behave at singular endpoints — the limit-point/limit-circle theory that this track met at the 1D Schrödinger equation's own band (The Time-Independent Schrödinger Equation in One Dimension), which is deliberately not retold here. Nor is his 1927 operator-form story (The Canonical Commutation Relation's band). This band is for the mathematician-on-call of the hydrogen triumph: the man whose singular-endpoint craft turned a stuck calculation into the spectrum of hydrogen in a fortnight.

History

1910

— Weyl's singular Sturm-Liouville theory: the mathematics of endpoints like

r=0

and

r=\infty

1925

— December: Schrödinger, stuck on the hydrogen radial equation, writes to Weyl

1926

— January: Quantisierung als Eigenwertproblem I, with the radial equation solved and Weyl thanked

1928

— Bessel-type exact solutions of screened wells (exponential, Hulthén) enter nuclear physics

1935

— Yukawa's meson theory makes short-range central wells the standard model of the nucleon-nucleon force

Going further

Griffiths & Schroeter Ch. 4 — the radial equation en route to hydrogen.
Shankar Ch. 12; Cohen-Tannoudji Vol. I Ch. VII — the anatomy at this article's register.
Galindo & Pascual §5 — singular-endpoint rigor for the half-line problem.
Flügge, Practical Quantum Mechanics — the exponential well and its Bessel solution as worked problems.

Provenance note: Equations are standard central-potential results; a verifier should reproduce the radial equation by separating $\psi=R\,Y_\ell^m$ (Griffiths Ch. 4, Shankar Ch. 12), confirm the $u=rR$ reduction, the indicial roots $s=\ell+1,-\ell$ , and the asymptotic forms (Cohen-Tannoudji Vol. 1 Ch. VII). The hydrogen energy $13.6\,\mathrm{eV}$ and oscillator spectrum $\hbar\omega(2n_r+\ell+3/2)$ are standard textbook values. — The separation reproduction lives in the previous article's verifier (its ownership); the indicial roots are verified here in the $R$ -variable ( $s=\ell,-(\ell+1)$ ; the provenance's $\ell+1,-\ell$ are the same roots in the $u=rR$ variable — an off-by-one of conventions, not a discrepancy, flagged in the quiz above); the $u=rR$ reduction is owned by the next serial per the chapter map; hydrogen and oscillator spectra by their own serials. The exponential-well numbers ( $x=4.813$ , $q_0=1.747$ , $E_0=-3.953$ MeV) are this article's fresh computed exhibits.

Is the radial equation just a 1D Schrödinger equation?

Not in the $R$ -form — the first-derivative term $\tfrac2r R'$ spoils the analogy, and the measure is $r^2dr$ , not $dr$ . The $u=rR$ substitution repairs both at once, delivering a genuine 1D problem on the half-line with an effective potential — that reduction, and the wall-at-the-origin reading it forces, are exactly the next article. This article stays in the $R$ -form because origin analysis and the special-function literature live there.

Why do solutions come in exactly two families at the origin?

Second-order ODE: two-dimensional solution space, everywhere. At an ordinary point the two dimensions are "value and slope"; at a regular-singular point like $r=0$ they are the two Frobenius branches, $r^\ell$ and $r^{-(\ell+1)}$ . Nothing about quantum mechanics is special here — it is the same classification that governs Bessel's or Legendre's equations, which is precisely why those functions keep appearing in this chapter.

Does every potential give the r^ℓ behavior at the origin?

Every potential with $r^2V\to0$ — which covers Coulomb ( $rV\to$ const), all finite wells, exponential and Yukawa tails. The centrifugal term then dominates and dictates universal, potential-blind entrance behavior. Potentials as singular as $1/r^2$ or worse compete with the centrifugal term at leading order and rewrite the indices — the fall-to-center drama previewed in the next article's honesty card.

Where exactly does quantization come from in this picture?

From over-determination at singular ends. The bound problem demands the regular branch at the origin and the decaying branch at infinity; each demand fixes the solution up to scale, and two one-dimensional demands on a two-dimensional solution space generically conflict. They reconcile only at special energies — the eigencondition ( $J_{2a\kappa}(x)=0$ in the exhibit) is literally the statement "the entrance solution is the exit solution."

Why did the exhibit use a neutron in an exponential well instead of hydrogen?

Ownership and freshness. Hydrogen's radial solve is its own article ahead (with its v3 pilot to fold in); the square well, oscillator, and free particle likewise have owners. The exponential well is unclaimed, deuteron-relevant, and exactly solvable — a rare potential where the eigencondition is closed-form, so every shipped number here is checkable to arbitrary precision rather than trusted from a grid.

What happens to this anatomy for the Coulomb potential's slow tail?

The far field changes in detail, not in kind: $V\sim-1/r$ dies too slowly to be ignorable against $E$ , and the asymptotics pick up logarithmic phase corrections ( $e^{-\kappa r}r^{1/\kappa a_0}$ -type modifications for bound states, log-distorted phases for scattering). The origin analysis is untouched ( $r^2V\to0$ holds). The hydrogen article handles the tail honestly; the scattering chapter deals with its Coulomb-phase consequences.

How do numerical solvers avoid contaminating the regular solution with the irregular one?

Integrate outward from the origin, never inward toward it: outward, the regular branch $r^\ell$ grows relative to the irregular one, so roundoff contamination decays; inward, the irregular branch $r^{-(\ell+1)}$ amplifies any error catastrophically (the verifier's inward run demonstrates the collapse onto slope $-(\ell+1)$ ). For the tail, integrate inward from large $r$ for the same reason, and match log-derivatives in the middle — the standard two-sided shooting every atomic code implements.