Laplace's Equation

The E1 article on Laplace's equation ended with a promise. It proved that harmonic functions are rigid — mean-value obeying, extremum-free, nailed down completely by their boundary values — and then admitted that it had not constructed a single one for a real electrode geometry. This article opens the chapter that keeps the promise. Here $\nabla^2V=0$ stops being a theorem factory and becomes a workshop: given actual conductors at actual potentials — a coaxial cable, a slotted beam pipe, an electron-microscope lens column — how do you produce the potential in the space between? The answer is not one method but a strategy catalog, and this article is its master index. Because boundary data determine the solution uniquely, any function that is harmonic in the region and matches the boundary wins — so guessing is legal, superposition lets you split hard boundary data into easy pieces, fictitious "image" charges can stand in for entire conductors, separable solutions can be stacked into series that fit any electrode shape, and when geometry defeats every analytic trick, a grid of numbers relaxing toward the mean-value property will solve the problem to any accuracy you can afford. Every one of these schemes gets its own article in this chapter; here you will see each one working once, on the simplest problem it owns. Two of the catalog's entries are demonstrated end to end: the coaxial line, solved by symmetry plus the guessing license, and the relaxation method, derived honestly from Taylor expansion and then run, sweep by sweep, until you can watch the numbers converge. If E1's Laplace article was about what harmonic functions are, this one is about how they are made.

Key takeaways

In charge-free regions bounded by electrodes, electrostatics reduces to a boundary-value problem: solve $\nabla^2V=0$ inside, match the prescribed potentials on the boundary.
Uniqueness converts solving into producing: any harmonic function matching the boundary data is THE answer, however you found it — the license behind every method in this chapter.
Linearity adds a second superpower: boundary data can be decomposed ( $V$ on walls = sum of one-wall problems), solved piecewise, and reassembled.
The analytic catalog: symmetry + direct integration (planes, cylinders, spheres), method of images (charges near conductors), separation of variables (boxes, spheres, pipes — Legendre and Fourier series).
The numerical workhorse: replace the region by a grid; the discrete Laplace equation says every node is the average of its neighbors; iterating that rule (relaxation) converges to the solution with error $O(h^2)$ .
KEY IDEA: Laplace's equation plus boundary data is a complete problem specification — and because the solution is unique, electrostatics becomes a craft of constructing one harmonic function per geometry, by guess, superposition, images, series, or grid.

The core insight

A charge-free electrostatics problem is fully posed as $\nabla^2V=0$ plus boundary potentials; uniqueness licenses any construction — direct integration, superposition, images, separable series, or numerical relaxation — that produces a harmonic function matching the data.

Where this lives today

Nobody designs an electrode by hand-fitting Legendre coefficients anymore — but everybody designs them with this article's catalog, industrialized. Semiconductor TCAD tools relax Laplace (and Poisson) grids across billions of nodes to shape the fields inside transistors; electron-microscope and mass-spectrometer vendors run boundary-element and finite-difference solvers descended in a straight line from the 5-point stencil of Derivation B; accelerator groups shape RFQ vanes and beam-pipe transitions by iterating exactly the loop of worked example 2 at scale. Multigrid — the $O(N)$ trick of relaxing on nested coarser grids — is on any short list of the most important algorithms of the twentieth century, and its inner loop is still "replace each node by the average of its neighbors." The catalog's analytic half is far from decorative, though: it supplies the test problems every solver is validated against, the small-gap and far-field asymptotics no grid can afford to resolve, and the physical intuition — logarithms near wires, $1/r$ tails, exponential decay of slot fields — that tells an engineer when a simulation is lying. The chapter ahead is that half, method by method.

Origins · 1 minCarl Neumann and how this was discovered

Carl Neumann

1832–1925 · Königsberg and Leipzig

Neumann was potential theory's great systematizer — son of Franz Neumann (of the mineralogy and induction-law Neumanns), he spent five decades in Leipzig turning electrostatics' bag of tricks into theorems. His "method of the arithmetic mean" (1877) was the first provably convergent scheme for constructing the solution of a boundary-value problem: starting from the boundary data, it builds the answer by repeated averaging — the intellectual ancestor of the relaxation sweeps in this article, invented a lifetime before electronic computers made them practical. The boundary condition that prescribes the normal derivative instead of the potential carries his name, as does the Neumann series of operator theory, which grew out of iterating his averaging map. With his colleague Alfred Clebsch he founded Mathematische Annalen, for decades the world's leading mathematics journal. His career is a reminder that "numerical methods" predate machines: convergence was proved first, silicon arrived later.

History

1782

— Laplace, studying the attraction of spheroids, introduces

\nabla^2V=0

for the exterior potential

1823

— Gauss, in a letter to Gerling, describes solving systems by sweeping through the unknowns one at a time — the ancestor of Gauss–Seidel relaxation

1877

— Carl Neumann's method of the arithmetic mean gives the first convergence proof for an iterative solution of the Dirichlet problem

1940

— Southwell's Relaxation Methods systematizes grid solving for engineering, with human "computers" doing the sweeps

1977

— Brandt's multigrid paper makes Laplace solvable in optimal

O(\text{nodes})

time, the standard ever since

The physical picture

Every problem in this chapter has the same anatomy. There is a region $\Omega$ — the vacuum between electrodes — where no charge lives and $\nabla^2V=0$ holds. There is a boundary: conductor surfaces, each pinned at a known constant potential (they are equipotentials, from the conductor articles), perhaps plus a boundary at infinity where $V\to0$ . The unknown is $V$ everywhere inside, and from it the field, the induced charges via $\sigma=-\varepsilon_0\,\partial V/\partial n$ , the forces, the capacitances — everything an engineer bills for.

Figure

Anatomy of a boundary-value problem

Why is this a sensible way to pose a problem at all? In E1 you computed potentials by integrating over charges. But look at the picture: the charge on conductor 2's surface is induced — you don't know it until you've solved the problem, and you can't solve the problem by integrating over charge you don't know. Boundary-value thinking cuts the knot: forget the charges, impose what you actually control (the electrode potentials, set by the battery), and let $\nabla^2V=0$ propagate that information into the interior. The induced charges come out at the end, harvested from the solution by $\sigma=-\varepsilon_0\,\partial V/\partial n$ , rather than being fed in at the start. The E1 twin article proved the two facts that make the strategy trustworthy, and we lean on them constantly without re-proving them. Mean value: a harmonic $V$ at any point equals its average over any surrounding sphere — so solutions are smooth, extremum-free interpolations of the boundary data. Uniqueness: matching boundary data can be achieved by only one harmonic function — so however disreputable your method (a guess, a mirror trick, an infinite series, a pile of numbers), if the result is harmonic and fits the boundary, it is not an answer but the answer. Harmonic Functions and the Mean-Value Theorem deepens the first fact; Uniqueness Theorems for the Potential does the second properly, Neumann variants and all. This article takes both as given and gets to work.

The strategy catalog

The chapter in one table — every method is a different way of manufacturing a harmonic function to order:

method	when it wins	worked where
Symmetry + direct integration	planar / cylindrical / spherical boundaries	Derivation A below
Superposition of boundary data	complicated data on simple boundaries	equation (3) below
Method of images	point charges near grounded planes, spheres	Method of Images and its two sequels
Separation of variables (Cartesian)	rectangular slots, boxes, pipes	Separation of Variables in Cartesian Coordinates
Separation of variables (spherical/cylindrical)	spheres, cones, wedges, round pipes	the spherical and cylindrical separation articles
Relaxation on a grid	any geometry; the industrial default	Derivation B + worked example 2

The equation being served in every row is the same:

\nabla^2 V = 0 \quad\text{in } \Omega, \qquad V = V_{\text{prescribed}} \quad\text{on } \partial\Omega \qquad (1)

where: $\Omega$ — the charge-free region; $\partial\Omega$ — its boundary, typically conductor surfaces plus infinity. Prescribing $V$ on the whole rim is called Dirichlet data; prescribing the normal derivative (equivalently the surface charge) instead is Neumann data — the taxonomy of what may be prescribed, and how much is enough, is the business of Boundary Conditions for the Potential. For reference, the operator in the three coordinate systems the catalog uses (derived in the E0 coordinate articles):

\nabla^2V = \partial_x^2V+\partial_y^2V+\partial_z^2V = \frac{1}{r^2}\partial_r\!\left(r^2\,\partial_r V\right)+\frac{\text{angular}}{r^2} = \frac{1}{s}\partial_s\!\left(s\,\partial_s V\right)+\frac{\partial_\phi^2V}{s^2}+\partial_z^2V \qquad (2)

where: $r$ — spherical radius; $s$ — cylindrical distance from the axis; the angular pieces are spelled out in the separation-of-variables articles that need them. The practical rule: pick the system in which your boundaries are coordinate surfaces. A sphere at $r=R$ makes spherical coordinates a one-variable problem; the same sphere in Cartesian coordinates is a nightmare. Linearity supplies the catalog's universal preprocessor. If $V_1$ solves the problem with boundary data $g_1$ and $V_2$ with data $g_2$ , then

V = \alpha V_1 + \beta V_2 \quad\text{solves the problem with data } \alpha g_1 + \beta g_2 \qquad (3)

where: $g$ — the prescribed boundary potential pattern. This is why a box with four walls at four different potentials is not four times harder than one hot wall and three grounded: solve the four one-hot-wall problems (each easy, each a textbook separation-of-variables exercise) and add. Decompose, solve, reassemble — the equation never mixes the pieces, and uniqueness blesses the sum.

Derivation AA boundary-value problem solved end to end — the coaxial line

We solve the workhorse geometry of laboratory electronics: an inner conductor of radius $a$ held at $V_0$ , a grounded outer shield of radius $b$ , vacuum between. The goal is the full chain — potential, field, induced charge — executed the way every problem in this chapter will be: reduce by symmetry, integrate what remains, fit the boundary data, and let uniqueness certify the result. Watch how little "solving" there is once the symmetry has spoken; the labor is in the fitting and the harvesting.

Step 1. Symmetry reduction. Translation along the axis and rotation about it leave the boundaries unchanged, so $V=V(s)$ only. The cylindrical Laplacian of (2) collapses to

\frac{1}{s}\frac{\mathrm{d}}{\mathrm{d}s}\!\left(s\,\frac{\mathrm{d}V}{\mathrm{d}s}\right)=0.

Step 2. Integrate twice: $s\,\mathrm{d}V/\mathrm{d}s = A$ , then

V(s) = A\ln s + B,

the two-constant logarithmic catalog entry (the same pair E1 exhibited for line symmetry — there as a property, here as a tool). Why two constants: a second-order ODE always delivers two, and a well-posed problem always supplies exactly two conditions to spend them on. That bookkeeping — constants versus conditions — is the quickest sanity check that a problem is properly posed.

Step 3. Fit the boundary data: $V(a)=V_0$ and $V(b)=0$ give $A = V_0/\ln(a/b)$ and $B=-A\ln b$ , so

V(s) = V_0\,\frac{\ln(s/b)}{\ln(a/b)}, \qquad E_s = -\frac{\mathrm{d}V}{\mathrm{d}s} = \frac{V_0}{\ln(b/a)}\,\frac{1}{s}.

Why the field diverges toward the inner conductor: the $1/s$ crowding of a line geometry — the same flux-spreading geometry that made the isolated line charge logarithmic.

Step 4. Harvest the induced charge with $\sigma=-\varepsilon_0\,\partial V/\partial n$ (normal pointing out of each conductor, into the vacuum). On the inner surface $\sigma_a = \varepsilon_0 E_s(a) = \varepsilon_0 V_0/[a\ln(b/a)]$ , hence charge per unit length

\lambda = 2\pi a\,\sigma_a = \frac{2\pi\varepsilon_0 V_0}{\ln(b/a)},

and on the shield's inner face $\sigma_b = -\varepsilon_0 V_0/[b\ln(b/a)]$ , integrating to $-\lambda$ : the shield's induced charge exactly balances, as the shielding article's Gaussian-surface argument demands.

Step 5. Certify. $V(s)$ is harmonic in $a<s<b$ (built from the general radial solution) and matches both boundary values; by uniqueness it is the potential — no further justification owed. As a bonus, $C/L = \lambda/V_0 = 2\pi\varepsilon_0/\ln(b/a)$ : the vacuum capacitance per length of coaxial cable, derived without ever assuming the charge in advance. Verified: matches the Gauss's-law route (assume $\lambda$ , get $E=\lambda/2\pi\varepsilon_0 s$ , integrate to $V$ ) run backwards; Griffiths Ex. 2.8/Prob. 2.43 territory. Dimensions: $[\varepsilon_0 V]=\mathrm{C/m}$ per $2\pi/\ln$ — correct for $\lambda$ .

Figure

The duct problem on a four-node grid — the numbers find the answer

Go deeper · Derivation B · 2 minThe relaxation method — the grid update rule from Taylor expansion

Derivation BThe relaxation method — the grid update rule from Taylor expansion

The analytic catalog dies on real geometry — a bolt head near a curved shield, a human torso under a power line. The industrial answer replaces $\Omega$ by a square grid of spacing $h$ and derives what "harmonic" means for numbers on a grid. E1 previewed the punchline (every node the average of its neighbors); here we earn it, error bar included, because the error bar is what tells you how fine a grid to buy. Strategy: Taylor-expand $V$ at the four neighbors of a node, add, and watch first derivatives cancel.

Step 1. In 2D, expand along $x$ at a node $(x,y)$ :

V(x\pm h,y) = V \pm h\,\partial_xV + \frac{h^2}{2}\,\partial_x^2V \pm \frac{h^3}{6}\,\partial_x^3V + \frac{h^4}{24}\,\partial_x^4V + \cdots

Adding the pair kills every odd term:

V(x+h,y)+V(x-h,y) = 2V + h^2\,\partial_x^2V + O(h^4).

Step 2. Do the same along $y$ and add all four neighbors:

\sum_{\text{4 neighbors}} V = 4V + h^2\left(\partial_x^2V+\partial_y^2V\right) + O(h^4) = 4V + h^2\,\nabla^2V + O(h^4).

Why this identity is the discrete Laplacian: rearranged, $(\sum V_{\text{nbr}} - 4V)/h^2 = \nabla^2V + O(h^2)$ — the neighbor-average deficit is the Laplacian, to second order. The averaging-meter reading of $\nabla^2$ from E0, now with its error term.

Step 3. Impose $\nabla^2V=0$ :

V = \frac{1}{4}\sum_{\text{4 neighbors}} V + O(h^4) \qquad (4)

The discrete Laplace equation: every interior node equals the average of its four neighbors, with per-node error $O(h^4)$ and accumulated solution error $O(h^2)$ . Halve the grid spacing, quarter the error.

Step 4. Turn the equation into an algorithm. Hold boundary nodes at their prescribed potentials; fill the interior with anything (zeros work); sweep the grid replacing each node by its neighbor average; repeat. Each sweep is a contraction toward the unique fixed point — the configuration satisfying (4) everywhere — and the fixed point is unique for the same reason the continuum solution is: the discrete problem inherits the mean-value structure, hence a discrete maximum principle, hence discrete uniqueness. (Harmonic Functions and the Mean-Value Theorem develops the continuum side of this inheritance.)

Step 5. Cost accounting, because the method's fame is its price tag: plain (Jacobi/Gauss–Seidel) sweeps need $O(N^2)$ iterations on an $N\times N$ grid — information crawls one cell per sweep. Successive over-relaxation (overshooting each update by a tuned factor $\omega\approx 2/(1+\pi/N)$ ) cuts it to $O(N)$ ; multigrid — relaxing on a hierarchy of coarser grids so long-wavelength error dies on cheap coarse levels — reaches $O(1)$ sweeps per level, asymptotically optimal. That ladder, Jacobi to multigrid, is a large part of twentieth-century computational physics. Verified: stencil and error order match any numerical-analysis text (e.g. LeVeque §2); worked example 2 exhibits the convergence on a real grid and checks the fixed point against the exact discrete solution.

Limiting cases

Case	Result	Meaning
Planar symmetry (parallel plates)	$V = ax+b$ — linear, uniform field	the one-line boundary-value problem
Cylindrical symmetry (coax)	$V = A\ln s + B$	Derivation A; capacitance $2\pi\varepsilon_0/\ln(b/a)$ per length
Spherical symmetry (concentric spheres)	$V = A/r + B$	E1's worked example — guess plus license
No symmetry, analytic boundary pieces	separable series	the three separation-of-variables articles
No symmetry at all	relaxation grid, error $O(h^2)$	Derivation B; multigrid makes it cheap
Grid spacing $h\to0$	discrete solution $\to$ continuum solution	consistency of (4) with (1)

Worked examples

1 · A real cable by Derivation A (confidence builder)

An air-spaced coaxial line has inner radius $a=0.45\,\mathrm{mm}$ , shield radius $b=1.5\,\mathrm{mm}$ , inner conductor at $V_0=100\,\mathrm{V}$ , shield grounded. Find the field at the inner conductor, the induced charge, and the capacitance per meter.

Step 1. $\ln(b/a) = \ln(1.5/0.45) = \ln(3.33) = 1.204$ . The field at $s=a$ :

E(a) = \frac{V_0}{a\,\ln(b/a)} = \frac{100}{(4.5\times10^{-4})(1.204)}\,\mathrm{V/m} = 1.85\times10^{5}\,\mathrm{V/m}.

Step 2. Induced surface density on the inner conductor: $\sigma_a = \varepsilon_0E(a) = (8.854\times10^{-12})(1.85\times10^{5}) = 1.63\times10^{-6}\,\mathrm{C/m^2}$ , giving charge per length

\lambda = 2\pi a\,\sigma_a = 2\pi(4.5\times10^{-4})(1.63\times10^{-6})\,\mathrm{C/m} = 4.6\times10^{-9}\,\mathrm{C/m}.

Step 3. Capacitance per length, two ways. From the solution: $C/L = \lambda/V_0 = 4.6\times10^{-11}\,\mathrm{F/m} = 46\,\mathrm{pF/m}$ . From the formula: $2\pi\varepsilon_0/\ln(b/a) = (5.563\times10^{-11})/1.204 = 46.2\,\mathrm{pF/m}$ . ✓ A vacuum-spaced RG-58-sized line: $46\,\mathrm{pF/m}$ (the commercial cable's $\sim100\,\mathrm{pF/m}$ is this number boosted by its polyethylene dielectric — the E3 chapter's business).

2 · Relaxation, run by hand (exam level)

A long square duct has its top wall at $100\,\mathrm{V}$ and the other three walls grounded. Model the cross-section with a coarse grid having just four interior nodes — $u$ (the two adjacent to the hot wall, equal by left-right symmetry) and $w$ (the two lower ones). Run Jacobi relaxation from a cold start and compare with the exact discrete solution.

Step 1. Update rules from (4), using symmetry ( $u$ 's neighbors: hot wall $100$ , side wall $0$ , the other $u$ ... careful — each $u$ node touches: top $100$ , one side $0$ , the other $u$ node, and the $w$ below it; each $w$ touches: $u$ above, side $0$ , the other $w$ , bottom $0$ ):

u \leftarrow \frac{100 + 0 + u' + w}{4}, \qquad w \leftarrow \frac{u + 0 + w' + 0}{4},

with primes denoting the mirror node (equal by symmetry).

Step 2. Exact discrete solution first (solve the fixed point): $4u = 100 + u + w$ and $4w = u + w$ give $3u - w = 100$ and $u = 3w$ , so

w = 12.5\,\mathrm{V}, \qquad u = 37.5\,\mathrm{V}.

Step 3. Now relax from $u=w=0$ and watch the flow of information:

sweep	$u$ (V)	$w$ (V)
0	0	0
1	25.00	0
2	31.25	6.25
3	34.38	9.38
4	35.94	10.94
6	37.11	12.11
8	37.40	12.40
∞	37.50	12.50

Each sweep halves the remaining error — geometric convergence to the unique fixed point.

Step 4. Sanity-check the fixed point against the mean-value rule: $u = (100 + 0 + 37.5 + 12.5)/4 = 150/4 = 37.5$ ✓ and $w = (37.5 + 0 + 12.5 + 0)/4 = 50/4 = 12.5$ ✓. Four nodes and eight sweeps already sketch the true field: hot near the live wall, cold in the grounded corners. A production solver differs only in scale — a $1000\times1000$ grid, over-relaxation or multigrid instead of Jacobi, and the same $\sigma=-\varepsilon_0\,\partial V/\partial n$ harvest at the end.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., Ch. 3 · Jackson, Classical Electrodynamics, 3rd ed., Ch. 1–3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 2 · Zangwill, Modern Electrodynamics, Ch. 7.

Provenance. Equations are standard textbook results. A verifier should confirm: the Laplacian forms in three coordinate systems (Griffiths front cover; Jackson §1), the derivation of Laplace from Poisson (Griffiths §2.3.1, Eq. 2.24), the mean-value/no-extremum theorems (Griffiths §3.1.4–3.1.5; Jackson §1.10), the radial solutions $V=-A/r+B$ and $V=A\ln s+B$ (direct integration), and the Legendre expansion (Griffiths §3.3.1, Eq. 3.65). No numerical constants beyond $\varepsilon_0$ appear; $\varepsilon_0 = 8.854\times10^{-12}\,\mathrm{F/m}$ is the SI permittivity of free space.

Builds on: Poisson's Equation (E2) · Boundary Conditions for the Potential · Harmonic Functions and the Mean-Value Theorem · Laplace's Equation (E1)

Leads to: Uniqueness Theorems for the Potential · Method of Images · Separation of Variables in Cartesian Coordinates · Separation of Variables in Spherical Coordinates

How is this article different from the E1 Laplace's Equation article? They have the same title.

Deliberately split labor. The E1 article proves what harmonic functions are: the mean-value property, the ban on interior extrema, Earnshaw's theorem, uniqueness in its first form, the two-constant radial catalogs. This one is the E2 chapter's opening move: it treats $\nabla^2V=0$ plus boundary data as a manufacturing problem and indexes the production methods — direct integration, superposition of data, images, separable series, relaxation — each of which then gets its own article. If you want theorems, go to E1; if you want the potential between actual electrodes, start here.

Why is it legal to just guess a solution? In every other physics course a guess needs justification.

The justification is bought in advance, wholesale, by the uniqueness theorem: for given Dirichlet data there is exactly one harmonic function. So "guess, then verify it's harmonic and matches the boundary" is not a heuristic — the verification is the proof, because nothing else could also qualify. This is genuinely unusual among PDEs and is the single most exploited fact in this chapter; the method of images is nothing but institutionalized guessing under this license.

Where did the charges go? Electrostatics is supposed to be about charges creating fields.

They moved from input to output. In a conductor-bounded problem the source charges are induced surface charges whose distribution you cannot know in advance — it is part of the answer. The boundary-value formulation replaces "integrate over the (unknown) charges" with "match the (known) electrode potentials." After solving, the charges are recovered from $\sigma=-\varepsilon_0\,\partial V/\partial n$ . Nothing about Coulomb physics is abandoned; the bookkeeping order is reversed to match what you actually control in the lab.

When should I reach for the grid instead of an analytic method?

When the boundaries stop being coordinate surfaces. Planes, coaxial cylinders, concentric spheres: direct integration (minutes). Point charge near a plane or sphere: images (minutes). Rectangular slots, round pipes, spheres in uniform fields: separation of variables (an afternoon, once). Anything with brackets, bolts, or anatomy: relaxation/finite elements — and even then, run an analytic limit first, because a solver with a wrong boundary condition will happily converge to a beautiful wrong answer that only your coax-line intuition will catch.

Does relaxation always converge, or can it get stuck in a wrong answer?

For the Laplace stencil it cannot get stuck: the discrete problem has a unique solution (a discrete maximum principle forbids two fixed points to differ anywhere), and each Jacobi/Gauss–Seidel sweep is a contraction toward it. What can happen is painful slowness — error components of wavelength comparable to the grid die fast, long-wavelength error crawls, which is exactly the disease over-relaxation and multigrid treat. Divergence appears only if you break the scheme: over-relaxation factor $\omega\ge2$ , inconsistent boundary handling, or applying the charge-free stencil where charge actually sits (that needs the Poisson source term — see the twin article).

Is the superposition of boundary data trick ever used outside textbooks?

Constantly, under the name "unit-potential solutions" or "capacitance matrix." For a system of $n$ conductors, solve $n$ problems, the $k$ -th having conductor $k$ at 1 volt and the rest grounded. Every possible voltage assignment is then a linear combination of these, and the induced charges organize into the capacitance matrix $Q_i = \sum_j C_{ij}V_j$ that circuit extractors in chip design compute all day. The decomposition is also how touchscreens are modeled: your fingertip perturbs one unit-potential solution at a time.

What plays the role of boundary data when there is no outer conductor — say, a charged electrode in open space?

Infinity becomes part of the boundary, with the condition $V\to0$ there (the standard reference choice for localized systems). That condition is not decorative: it is what kills the growing solutions ( $r^\ell$ modes, positive exponentials) and makes the remaining coefficient count match the boundary conditions. Forgetting the infinity condition is the most common way to end up with an underdetermined series and no way to fix its constants — the separation-of-variables articles will lean on it in nearly every example.