Asymptotic Series and Poincaré Expansions

A vast number of the functions physics cares about — Bessel and gamma functions, error-function tails, WKB wavefunctions, Feynman perturbation series — are never handed to us in closed form. What we can almost always get instead is their behaviour near a limit: how they grow as $x\to\infty$ , or collapse as $x\to0^+$ . An asymptotic series is the tool that captures that behaviour with a controllable error, and its defining feature is startling: the most useful such series are usually divergent. Henri Poincaré, in an 1886 paper on the irregular integrals of linear differential equations, cut the knot by redefining what a "good" expansion means. A convergent series fixes the argument and lets the number of terms run to infinity. An asymptotic series does the opposite: it fixes the number of terms and lets the argument run to its limit, asking only that the truncation error be smaller than the last term you kept. That single change of quantifier is the whole subject. It explains why summing Stirling's series for the factorial to five terms is brilliant and to fifty terms is a disaster; why $e^{-x}$ is invisible to every inverse-power expansion, so a series can fingerprint a function without determining it; and why factorial coefficient growth, the exponentially small floor on accuracy, the Stokes phenomenon, and the machinery of Borel summation are all facets of one gem. This article builds the Poincaré definition from the ground up, proves the coefficients unique, derives the canonical divergent series by hand, and shows exactly where to stop.

Key takeaways

An asymptotic series $f(x)\sim\sum_n a_n\phi_n(x)$ means that for each fixed number of terms $N$ , the truncation error vanishes faster than the last retained term as $x$ approaches the limit point.
This inverts convergence: convergence fixes $x$ and sends $N\to\infty$ ; asymptoticity fixes $N$ and sends $x\to x_0$ . The two limits are independent, and asymptotic series are usually divergent.
Given a gauge sequence the coefficients are unique and computable one at a time — yet the function is not uniquely fixed, because a beyond-all-orders term like $e^{-x}$ has every inverse-power coefficient equal to zero.
The coefficients typically grow like $n!$ , so terms shrink then grow: partial sums improve to a best accuracy near $N\approx x$ and deteriorate thereafter. Truncate at the smallest term.
That best accuracy is a floor of order $e^{-x}$ — exponentially small in the expansion parameter, the realm of superasymptotics.
Key idea: Hold the number of terms fixed and let the argument approach its limit, $f(x)-\sum_{n=0}^{N}a_n\phi_n(x)=o(\phi_N(x))$ . The error beats the last kept term — even when the full series diverges.

The core insight

Hold the number of terms fixed and let the argument approach its limit: $f(x)-\sum_{n=0}^{N}a_n\phi_n(x)=o(\phi_N(x))$ . The truncation error beats the last kept term — even when the full series diverges, because usefulness and convergence are different properties.

Where this lives today

The Poincaré definition is not a historical curiosity — it is load-bearing across modern theoretical physics. Perturbation theory in quantum field theory is asymptotic, not convergent: Dyson's 1952 argument shows the QED series in the fine-structure constant must diverge, its factorial growth realised by the proliferation of Feynman diagrams. In QCD the "renormalon" singularities in the Borel plane quantify how the perturbative series senses nonperturbative condensates, and the entire resurgence programme — trans-series that stitch perturbative and instanton sectors together — is the current language for extracting exponentially small effects in gauge theory, string theory, and matrix models. Closer to the numerics, every high-precision special-function library (the DLMF, GSL, the gamma and Bessel routines in every scientific stack) leans on Stirling-type and Watson's-lemma asymptotics, with optimal truncation and hyperasymptotic refinements pushing below the $e^{-x}$ floor when digits matter. And the Stokes phenomenon, once an obscurity about Airy integrals, now governs how WKB switches on subdominant waves at caustics — the mathematics behind rainbow-scattering intensities and semiclassical tunnelling rates. Wherever a physical answer is an expansion in a large parameter — $\hbar$ , $1/N$ , a coupling, a distance — Poincaré's bookkeeping is what makes it rigorous.

The physical picture

Picture an asymptotic series as a ruler that sharpens the farther away you stand. Each additional term refines the description in the limit $x\to\infty$ at fixed term count — not if you pile on terms at a fixed $x$ . For a typical physical series the coefficients grow like $n!$ , so a term $n!\,x^{-n}$ shrinks while $n<x$ and swells once $n>x$ . The partial sums therefore trace a valley: they descend toward the true value, bottom out at a best accuracy near $N\approx x$ , and then climb away as the divergence takes over. The first figure contrasts this with an ordinary convergent series, whose error simply falls forever.

Figure

Two summation philosophies contrasted. A convergent series (green) has its error fall monotonically as terms are added. A divergent asymptotic series (blue) improves only until N ≈ x, reaches an accuracy floor of order e⁻ˣ, then gets worse — past optimal truncation, adding terms hurts.

Why does the valley have a floor, and how deep? The depth is the size of the smallest term, and for the canonical factorial series that term is exponentially small — roughly $e^{-x}$ . Everything below that floor is invisible to the power series: a quantity like $e^{-x}$ has all its inverse-power coefficients equal to zero, which is why two functions differing by such a term carry an identical asymptotic fingerprint. The second figure plots the actual term magnitudes for the prototype divergent series at $x=10$ .

Figure

Magnitude of successive terms n!∕xⁿ⁺¹ of the exponential-integral series at x = 10, on a log scale. Terms shrink until n ≈ x = 10, where the smallest term ≈ 3.6×10⁻⁵ ≈ √(2π∕x)·e⁻ˣ sets the best achievable accuracy, then grow without bound — the series diverges.

The core equations

An asymptotic expansion is written with the symbol $\sim$ , which is weaker than equality: it records only the leading behaviour term by term against a chosen ladder of comparison functions. The ladder — the gauge or asymptotic sequence — must decrease strictly, $\phi_{n+1}=o(\phi_n)$ , and for large argument the standard choice is the inverse powers $\phi_n=x^{-n}$ .

f(x)\sim\sum_{n=0}^{\infty}a_n\,\phi_n(x)\quad(x\to x_0) \qquad (1)

where: $f(x)$ — the function being approximated; $a_n$ — constant coefficients; $\phi_n(x)$ — the gauge sequence with $\phi_{n+1}=o(\phi_n)$ ; $x_0$ — the limit point, usually $\infty$ or $0$ . Reading it: the right side is not claimed to converge to $f$ , only to reproduce it order by order as $x\to x_0$ . The precise content of $\sim$ is the truncation-error condition: fix the number of terms $N$ , and the remainder must vanish faster than the last term you kept as $x\to x_0$ . This is the definition to memorise — everything else follows from it.

f(x)-\sum_{n=0}^{N}a_n\,\phi_n(x)=o\!\left(\phi_N(x)\right)\quad(x\to x_0) \qquad (2)

where: $N$ — a fixed retained-term count; $o(\phi_N)$ — a remainder negligible against the last kept term. An equivalent and often handier phrasing is that the error is $O(\phi_{N+1})$ , of the order of the first omitted term; since $\phi_{N+1}=o(\phi_N)$ , the two statements coincide. Because the definition pins each order in turn, the coefficients can be extracted recursively by a sequence of limits — subtract everything already known, then divide by the next gauge function. This makes the expansion unique once the gauge is chosen.

a_N=\lim_{x\to x_0}\frac{f(x)-\sum_{n=0}^{N-1}a_n\phi_n(x)}{\phi_N(x)} \qquad (3)

where: $a_N$ — the $N$ -th coefficient, determined one term at a time. Reading it: there is nothing to solve simultaneously; each limit hands you exactly one number, so no two valid expansions in the same gauge can disagree. Uniqueness of the series does not mean uniqueness of the function. A quantity that is smaller than every gauge function contributes nothing to any coefficient. The exponential is the canonical culprit: it is subdominant to all inverse powers.

e^{-x}\sim 0+0\cdot x^{-1}+0\cdot x^{-2}+\cdots\quad(x\to+\infty) \qquad (4)

where: $e^{-x}$ — a function whose entire inverse-power asymptotic series vanishes. Consequently $f(x)$ and $f(x)+Ce^{-x}$ share an identical series; the series is a faithful but incomplete fingerprint, and recovering the missing $e^{-x}$ is the business of exponential asymptotics. The prototype divergent asymptotic series is the exponential integral. Repeated integration by parts generates inverse powers with factorially growing coefficients, so the series has zero radius of convergence yet reproduces the function beautifully for large $x$ .

e^{x}\!\int_x^\infty\frac{e^{-t}}{t}\,\mathrm{d}t=\int_x^\infty\frac{e^{x-t}}{t}\,\mathrm{d}t\sim\sum_{n=0}^{\infty}\frac{(-1)^n\,n!}{x^{n+1}} \qquad (5)

where: $n!$ — factorial coefficient growth, forcing a zero radius of convergence; $x$ — a large positive argument. The very growth that kills convergence is what makes the early terms shrink so fast that a handful are superbly accurate. The factorial growth dictates where to stop. The term ratio is $(n+1)/x$ , so terms shrink while $n<x$ and grow afterward; the smallest term sits at $N\approx x$ and its size is the best accuracy you can extract.

|R_N(x)|\le\frac{(N+1)!}{x^{N+2}},\qquad N_{\mathrm{opt}}\approx x,\qquad \text{floor}\sim\sqrt{\tfrac{2\pi}{x}}\,e^{-x} \qquad (6)

where: $R_N$ — the remainder after $N$ terms; $N_{\mathrm{opt}}$ — the error-minimising term index. Optimal truncation buys an exponentially small error $\sim e^{-x}$ — the superasymptotic floor — with no resummation. The most famous asymptotic series in physics is Stirling's. It starts from the integral definition of the factorial and delivers the leading Gaussian estimate; the full logarithmic series carries Bernoulli numbers that themselves grow like $(2k)!$ .

n!=\int_0^\infty t^{n}e^{-t}\,\mathrm{d}t,\qquad n!\sim\sqrt{2\pi n}\,\left(\tfrac{n}{e}\right)^{n} \qquad (7)

where: $n$ — a large argument; $\sqrt{2\pi n}$ — the Gaussian prefactor from Laplace's method applied to the integrand's peak at $t=n$ .

\ln\Gamma(z)\sim\left(z-\tfrac12\right)\ln z - z+\tfrac12\ln(2\pi)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)\,z^{2k-1}} \qquad (8)

where: $B_{2k}$ — Bernoulli numbers ( $B_2=\tfrac16$ , $B_4=-\tfrac1{30}$ ), growing like $(2k)!$ ; $z$ — large, with $|\arg z|<\pi$ . Reading it: this is the backbone of every numerical gamma-function routine, accurate to many digits for $z\gtrsim 10$ with only two or three terms.

Derivation AWhy the Poincaré coefficients exist and are unique

The definition of an asymptotic expansion would be empty if a function could carry two different ones in the same gauge, or if the coefficients were not computable. We settle both at once. The strategy is to build the coefficients constructively — peel off the leading behaviour, divide by the next gauge function, take a limit, repeat — showing each limit exists whenever the expansion does; then show any second expansion in the same gauge must reproduce the identical numbers term by term. Throughout, fix a gauge sequence $\{\phi_n\}$ with $\phi_{n+1}=o(\phi_n)$ as $x\to x_0$ , and suppose $f\sim\sum a_n\phi_n$ in the Poincaré sense.

Step 1. The leading coefficient is $a_0=\lim_{x\to x_0} f(x)/\phi_0(x)$ . This limit exists precisely because the expansion assumes $f\sim a_0\phi_0$ at leading order; if it failed to exist, $f$ would simply have no asymptotic expansion in this gauge. Why: the whole construction bootstraps from the fact that "leading behaviour" is a well-defined limit, not a fitted guess.

Step 2. Subtract the leading piece to define $g_1(x):=f(x)-a_0\phi_0(x)$ . By the definition of $a_0$ this remainder satisfies $g_1=o(\phi_0)$ , so it is genuinely smaller than the term just removed. Why: peeling one term must leave a strictly smaller remainder, or the definition of $a_0$ was inconsistent — this is the induction's engine.

Step 3. The next coefficient is $a_1=\lim_{x\to x_0} g_1(x)/\phi_1(x)$ , which exists by the order-one hypothesis of the expansion, and one continues inductively: at each stage subtract the known partial sum and divide by the next gauge function, recovering equation (3). Why: each coefficient is fixed by a single limit involving only lower-order data, so the recursion never stalls and never branches.

Step 4. For uniqueness, suppose $f\sim\sum a_n\phi_n$ and also $f\sim\sum b_n\phi_n$ . Subtracting, $\sum(a_n-b_n)\phi_n\sim 0$ ; dividing by $\phi_0$ and taking the limit forces $a_0=b_0$ , and inductively $a_n=b_n$ for all $n$ . Hence the expansion is unique — even though, by equation (4), the function is not. Why: this is the sharp asymmetry of the subject: the map function → series is many-to-one, but for a fixed gauge the series is a genuine invariant. *Verified: standard result in Bender & Orszag §3.4 and Stone & Goldbart; every limit uses only $\phi_{n+1}=o(\phi_n)$ , and the $e^{-x}$ example shows uniqueness of the series coexisting with non-uniqueness of the function.*

Go deeper · Derivation B · 2 minThe asymptotic series for the complementary error function

Derivation BThe asymptotic series for the complementary error function

The complementary error function $\mathrm{erfc}(x)=\tfrac{2}{\sqrt\pi}\int_x^\infty e^{-t^2}\,\mathrm{d}t$ governs Gaussian tails throughout statistical physics and diffusion, and for large $x$ its Taylor series is useless while its asymptotic series is superb. We derive it by repeated integration by parts, exhibit the double-factorial coefficients that make it diverge, and bound the remainder to confirm the Poincaré property — seeing every hallmark, from factorial growth to the exponential prefactor, arise from one elementary manipulation.

Step 1. Write the integrand as a total derivative: $e^{-t^2}=-\dfrac{1}{2t}\dfrac{\mathrm{d}}{\mathrm{d}t}e^{-t^2}$ . This is the standard device for Gaussian tails, isolating a piece that integrates exactly. Why: forcing the exponential into $\mathrm{d}v$ form is what makes each integration by parts produce a clean boundary term plus a smaller integral.

Step 2. One integration by parts gives $\displaystyle\int_x^\infty e^{-t^2}\mathrm{d}t=\frac{e^{-x^2}}{2x}-\frac12\int_x^\infty\frac{e^{-t^2}}{t^2}\,\mathrm{d}t$ . The boundary term at $t=x$ supplies the leading factor $e^{-x^2}/2x$ ; the new integral carries an extra $1/t^2$ and is smaller by $1/x^2$ . Why: the boundary term is the next series coefficient and the leftover integral is the remainder — the two roles that make the process asymptotic.

Step 3. Iterating $N$ times produces $\displaystyle\int_x^\infty e^{-t^2}\mathrm{d}t=\frac{e^{-x^2}}{2x}\Big[1-\frac{1}{2x^2}+\frac{3}{(2x^2)^2}-\cdots\Big]+R_N$ , with coefficients $(-1)^n(2n-1)!!$ . The double factorial grows factorially, so the bracketed series diverges for every fixed $x$ . Why: divergence is not a defect here — the same growth makes the leading terms plummet, which is why two terms already nail several digits at moderate $x$ .

Step 4. Bound the remainder using $t\ge x$ on the domain: $|R_N|\le\dfrac{(2N+1)!!}{2^{N+1}}\displaystyle\int_x^\infty\frac{e^{-t^2}}{t^{2N+2}}\mathrm{d}t=O\!\Big(\dfrac{e^{-x^2}}{x^{2N+3}}\Big)$ , which is smaller than the last retained term. The Poincaré criterion (2) holds despite the divergence. Why: this is the crux — a rigorous bound, not an act of faith, guarantees the truncated series is trustworthy.

Step 5. Restoring the prefactor gives the boxed result $\mathrm{erfc}(x)\sim\dfrac{e^{-x^2}}{x\sqrt\pi}\sum_{n=0}^{\infty}(-1)^n\dfrac{(2n-1)!!}{(2x^2)^n}$ . Numerically at $x=3$ the leading term is $2.321\times10^{-5}$ , the first correction pulls it to $2.192\times10^{-5}$ , and the exact value is $2.209\times10^{-5}$ — two significant figures from two terms. Why: the payoff is concrete accuracy from almost no work, the reason asymptotics dominates special-function computation. *Verified: matches Arfken & Weber and the DLMF entry for $\mathrm{erfc}$ ; remainder bound from monotonicity of $t^{-(2N+2)}$ on $[x,\infty)$ ; $\mathrm{erfc}(3)=2.2090\times10^{-5}$ reproduced numerically.*

Go deeper · Derivation C · 1 minThe prototype divergent series and its optimal-truncation floor

Derivation CThe prototype divergent series and its optimal-truncation floor

The exponential-integral series (5) is the cleanest laboratory for the whole phenomenon: it is manifestly divergent, its remainder is elementary to bound, and the location and size of its smallest term can be read off exactly. We generate the series by integration by parts, prove the Poincaré remainder bound, then locate the optimal truncation and compute the exponentially small accuracy floor it delivers — the quantitative content of the second figure, closing the loop between "the series diverges" and "you still get $e^{-x}$ accuracy."

Step 1. Let $f(x)=\int_x^\infty t^{-1}e^{x-t}\,\mathrm{d}t$ . Integrating by parts with $u=t^{-1}$ , $\mathrm{d}v=e^{x-t}\mathrm{d}t$ gives $f(x)=\dfrac1x-\int_x^\infty t^{-2}e^{x-t}\,\mathrm{d}t$ ; the boundary term is the leading $1/x$ . Why: each pass converts the current integral into the next inverse-power term plus a smaller integral, exactly as for erfc.

Step 2. After $N$ steps, $f(x)=\sum_{n=0}^{N}\dfrac{(-1)^n n!}{x^{n+1}}+(-1)^{N+1}(N+1)!\displaystyle\int_x^\infty\frac{e^{x-t}}{t^{N+2}}\mathrm{d}t$ . The factorial multiplying the integral is where the divergence lives. Why: making the remainder explicit lets us bound it rather than hand-wave the tail.

Step 3. Since $t\ge x$ throughout, the remainder obeys $|R_N(x)|\le\dfrac{(N+1)!}{x^{N+2}}$ , which is exactly the size of the first omitted term — the Poincaré property, equation (6). Why: the error is controlled by the next term, so the series is safe to truncate anywhere, and best truncated at its smallest term.

Step 4. The consecutive-term ratio is $\dfrac{|a_{n+1}\phi_{n+1}|}{|a_n\phi_n|}=\dfrac{n+1}{x}$ , below one for $n<x-1$ and above one afterward, so the smallest term sits at $N_{\mathrm{opt}}\approx x$ . Why: this single ratio is the practical rule — stop when the terms stop shrinking.

Step 5. At $N\approx x$ , Stirling gives the smallest term $\dfrac{N!}{x^{N+1}}\Big|_{N=x}\sim\sqrt{\tfrac{2\pi}{x}}\,e^{-x}$ . This is the accuracy floor: at $x=10$ it equals $3.6\times10^{-5}$ , matching the valley bottom in the figure, and it can only be beaten by resummation. Why: the floor is exponentially small in the expansion parameter — "superasymptotics" — and its size sets the stage for Borel summation and the Stokes phenomenon. *Verified: remainder bound from $t\ge x$ ; term ratio $(n+1)/x$ ; smallest term at $x=10$ computed as $3.60\times10^{-5}=\sqrt{2\pi/10}\,e^{-10}$ , versus $e^{-10}=4.54\times10^{-5}$ .*

Limiting cases

Case	Result	Meaning
Convergent and asymptotic	$\lim_{N\to\infty}\sum_{n=0}^{N}a_n\phi_n(x)=f(x)$	A series can be both, e.g. a Taylor series inside its disc; but convergence is far stronger and rarer than asymptoticity.
Divergent asymptotic series	$a_n\sim C\,n!\,A^{-n}\Rightarrow$ radius of convergence $=0$	Factorial growth makes the series diverge for every $x\neq x_0$ , yet it stays asymptotic and numerically excellent.
Beyond all orders	$e^{-x}=o(x^{-n})$ for every $n$	Invisible to a Poincaré power series; the origin of non-uniqueness, of the accuracy floor, and of the Stokes phenomenon.
Generalised gauge	$\phi_n=x^{-n/2},\ (\log x)^{-n},\ e^{-nx},\dots$	Any strictly decreasing sequence works; half-integer powers at turning points, logs in boundary layers.
Optimal truncation	error $\sim\sqrt{2\pi/x}\,e^{-x}$ at $N\approx x$	Best a bare series can do; hyperasymptotics and Borel summation push below this floor.

Common trapAdding more terms always makes an asymptotic series better

It does not — and this is the single most common and most costly misconception. Past the smallest term (near $N\approx x$ ) the factorial growth wins and every extra term makes the approximation worse, eventually catastrophically so; optimal accuracy comes from stopping at the smallest term, not from summing everything you can compute. Two further traps ride along with it. First, never assume the series determines the function: any multiple of a beyond-all-orders term such as $e^{-x}$ is asymptotically invisible, so extracting it requires exponential asymptotics, not more terms. Second, do not confuse an asymptotic $\sim$ with an equality — the right-hand side is a controlled approximation, not the function, and treating a divergent series as if it converged (for instance, resumming a QED perturbation series naively) gives nonsense.

Worked examples

1 · Optimal truncation of the exponential integral (fully worked)

Estimate $f(5)=e^{5}E_1(5)=\int_5^\infty e^{5-t}t^{-1}\mathrm{d}t$ using its asymptotic series $\sum_n(-1)^n n!/x^{n+1}$ at $x=5$ , and state the best accuracy attainable.

Step 1. List the term magnitudes $|n!/5^{n+1}|$ : $0.200,\ 0.0400,\ 0.0160,\ 0.00960,\ 0.00768,\ 0.00768,\ 0.00922,\dots$ The terms shrink through $n=4$ , tie at $n=4$ and $n=5$ , and grow thereafter — consistent with $N_{\mathrm{opt}}\approx x=5$ from the ratio rule $(n+1)/x$ . The smallest term is $\approx 7.7\times10^{-3}$ .

Step 2. Sum to the smallest term. The partial sums are $S_3=0.16640$ and $S_4=0.17408$ ; because the series alternates, the true value is trapped between consecutive partial sums. The optimal estimate is $S_4=0.1741$ (or the bracket midpoint $0.1702$ ).

Step 3. Compare with the exact value $e^{5}E_1(5)=0.17042$ . The error of $S_4$ is $|0.17408-0.17042|=3.7\times10^{-3}$ , comfortably below the smallest term and of the order $e^{-5}=6.7\times10^{-3}$ predicted by the floor $\sqrt{2\pi/5}\,e^{-5}=7.6\times10^{-3}$ . Best accuracy $\approx 4\times10^{-3}$ , i.e. order $e^{-x}$ ; pushing to $n=8$ gives error $0.013$ — worse, exactly as warned.

2 · Two terms of the erfc series at x = 3

Use the derived series $\mathrm{erfc}(x)\sim\dfrac{e^{-x^2}}{x\sqrt\pi}\sum_n(-1)^n\dfrac{(2n-1)!!}{(2x^2)^n}$ to evaluate $\mathrm{erfc}(3)$ and compare with the exact $2.2090\times10^{-5}$ .

Step 1. Leading prefactor: $\dfrac{e^{-9}}{3\sqrt\pi}=\dfrac{1.2341\times10^{-4}}{5.3174}=\mathbf{2.321\times10^{-5}}$ . This one term is already correct to about 5%.

Step 2. Apply the bracket $[1-\tfrac{1}{2x^2}+\tfrac{3}{(2x^2)^2}-\cdots]$ with $2x^2=18$ : $1-0.05556+0.00926-\cdots=0.9538$ . Two terms give $2.321\times10^{-5}\times0.9444=\mathbf{2.192\times10^{-5}}$ ; three terms give $\mathbf{2.213\times10^{-5}}$ .

Step 3. The exact value is $2.2090\times10^{-5}$ : two terms agree to two significant figures, three terms to three. Because $2x^2=18$ is large, the smallest term here is of order $e^{-18}$ — an astonishingly deep floor — which is why the erfc series is so accurate even at modest $x$ . Three terms reproduce $\mathrm{erfc}(3)=2.209\times10^{-5}$ to 0.2%.

Physics of the portrait

Asymptotic analysis is where a mathematician's scruple met an astronomer's pragmatism. For two centuries physicists had used divergent series — Stirling's factorial, the perturbative orbits of the planets — with spectacular success and no justification, while mathematicians dismissed them as meaningless. Poincaré reconciled the two.

Origins · 1 minHenri Poincaré and how this was discovered

Henri Poincaré

1854–1912 · the modern definition of asymptotic expansions

Henri Poincaré was the last universalist of mathematics, contributing decisively to topology, celestial mechanics, the theory of differential equations, and the physics that became relativity. His route to asymptotics ran through the three-body problem: the perturbation series astronomers used to predict planetary motion diverged, yet worked, and Poincaré wanted to know why. In an 1886 memoir on the irregular integrals of linear differential equations (Acta Mathematica 8) he gave the definition still used today — that $f\sim\sum a_n x^{-n}$ means the remainder after $N$ terms is smaller than the last kept term as $x\to\infty$ , for each fixed $N$ . The move was to fix $N$ and let $x$ run, divorcing usefulness from convergence, and it retroactively legitimised a century of divergent-series calculation from Stirling and Laplace onward. He understood, too, that such series do not determine their function — an exponentially small ambiguity he could see but not tame. That gap became the twentieth century's subject: Stokes's line-crossing discontinuities, Dingle's terminants, Écalle's resurgence, and Borel summation. Poincaré gave the definition; the corrections he left on the horizon are still active research.

History

1730

— Stirling and De Moivre give the divergent asymptotic series for n!

1857

— Stokes finds exponentially small terms switching on across a Stokes line

1886

— Poincaré publishes the modern definition of an asymptotic expansion

1899

— Borel introduces a summation that revives divergent series

1926

— WKB cast semiclassical quantum mechanics as an asymptotic series in ħ

Going further

Texts. Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers, Ch. 3 (the definitive physics treatment) · Stone & Goldbart, Mathematics for Physics, chapter on asymptotic expansions · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed. · Hassani, Mathematical Physics, on asymptotics and special functions · Byron & Fuller, Mathematics of Classical and Quantum Physics · Dingle, Asymptotic Expansions: Their Derivation and Interpretation (the coefficient-growth and terminant theory).

Provenance. The Poincaré definition, the uniqueness/recursion argument, and the erfc and exponential-integral expansions follow Bender & Orszag (Ch. 3) and Stone & Goldbart. A verifier should reproduce, by repeated integration by parts, the exponential-integral remainder bound $|R_N|\le(N+1)!/x^{N+2}$ and the erfc bound $O(e^{-x^2}/x^{2N+3})$ , and confirm the numerics quoted: $\mathrm{erfc}(3)=2.2090\times10^{-5}$ (leading term $2.321\times10^{-5}$ , two terms $2.192\times10^{-5}$ , three terms $2.213\times10^{-5}$ ); the smallest term of the $x=10$ series $=3.60\times10^{-5}=\sqrt{2\pi/10}\,e^{-10}$ against $e^{-10}=4.54\times10^{-5}$ ; and the $x=5$ example $e^{5}E_1(5)=0.17042$ with optimal error $\approx4\times10^{-3}$ . Bernoulli values $B_2=\tfrac16$ , $B_4=-\tfrac1{30}$ are standard. No fabricated constants.

Builds on: Order Symbols and Gauge Functions · Regular Perturbation Theory. Leads to: Watson's Lemma · Laplace's Method for Integrals · Optimal Truncation and Superasymptotics · Stokes Phenomenon and Anti-Stokes Lines · Borel Summation and Resurgence · WKB Approximation.

Ask the tutor: Want optimal truncation, the erfc series, or the Stokes phenomenon worked out live for your own integral or perturbation problem? Ask and we'll take it term by term.

If a series diverges, how can it possibly be useful?

Because divergence and usefulness are different properties. A divergent asymptotic series has terms that first shrink rapidly — often the first few terms give several correct digits — before eventually growing. You harvest the accuracy in the shrinking phase and stop at the smallest term. Stirling's series for the factorial diverges for every argument, yet three terms give the gamma function to many digits for $z\gtrsim10$ ; that is why it sits inside every numerical library. The mistake is only to keep summing past the smallest term, where the divergence takes over and wrecks the answer.

How is an asymptotic series different from a Taylor series?

A Taylor series is built to converge: you fix a point inside the radius of convergence and adding terms always improves the estimate. An asymptotic series is built around a limit point (often $x=\infty$ ) and need not converge at all; adding terms improves it only up to the smallest term. The two can coincide — a convergent Taylor series is also asymptotic near its centre — but the reverse is rare. The defining condition is different: Taylor requires the sum to approach $f$ as $N\to\infty$ at fixed $x$ ; asymptotic requires the error to beat the last term as $x\to x_0$ at fixed $N$ .

How do I know when to stop adding terms?

Watch the term sizes and stop at the smallest one — this is optimal truncation. For a factorial series $a_n\sim n!/x^n$ the terms shrink while $n<x$ and grow after, so the smallest term sits near $N\approx x$ and its magnitude is the best accuracy you can expect, typically of order $e^{-x}$ . Practically: compute successive terms, and the moment a term is larger than its predecessor, truncate before it. Adding further terms cannot help and will eventually hurt catastrophically, because the remainder is bounded by the first omitted term.

Why do the coefficients so often grow like n!?

Because the coefficients are usually moments of a fixed profile, and moments of a smooth integrand grow factorially. In Watson's lemma and Laplace's method the $n$ -th coefficient is essentially the $n$ -th derivative or moment of the integrand at the critical point, and repeated differentiation of a function with a finite-distance singularity produces $n!$ -type growth (the singularity sits a finite distance away in the Borel plane). Physically the same growth reflects the number of Feynman diagrams or the combinatorics of the perturbation, as in Dyson's argument that the QED series must diverge. The $n!$ is a feature, not a bug — it encodes where the nearest nonperturbative effect lives.

If two functions share a series, how do I recover the missing piece?

You cannot recover it from the power-series coefficients alone — that information genuinely is not there, since $e^{-x}$ and its multiples contribute zero to every coefficient. Recovering it requires exponential (or "beyond all orders") asymptotics. The practical routes are Borel summation, which resums the divergent series into an analytic function whose ambiguity is exactly the exponentially small term, and resurgence, which relates the large-order growth of the coefficients to the size of the missing exponential. The Stokes phenomenon is the statement that these hidden terms switch on discontinuously as you cross certain rays in the complex plane.

What is the Stokes phenomenon, briefly?

It is the sudden appearance of an exponentially small term in an asymptotic expansion as the argument's phase crosses a special ray, the Stokes line. Consider a function with two competing exponential behaviours, one dominant and one subdominant. Far from the Stokes line the subdominant piece is invisible to the dominant series; as you rotate across the line its coefficient switches on — smoothly, in fact, with an error-function profile (Berry's smoothing). It matters physically because WKB wavefunctions change character across turning points precisely by a Stokes crossing, which is how connection formulae and Bohr–Sommerfeld quantisation emerge from the asymptotics.