Theory

IntegralResurgence.jl evaluates oscillatory integrals

\[\Psi(k) = \left(\frac{k}{2\pi i}\right)^{d/2} \int_{\mathbb{R}^d} e^{i k f(\boldsymbol{x})}\, \mathrm{d}\boldsymbol{x}\]

with a polynomial exponent $f$ and a large parameter $k$. The method replaces the geometric analysis of steepest descent manifolds in $\mathbb{C}^d$ by algebra: integration by parts yields a finite-dimensional differential equation in $k$, the Gauss–Manin system, whose Borel transform is a Fuchsian system. From these, the package computes the three ingredients of an exponentially accurate approximation of $\Psi$: the asymptotic series of all saddle points, the Stokes constants that encode their adjacency, and the intersection numbers that encode their relevance. This page summarises the derivation; for the details, see Crew & Feldbrugge (2026), whose section numbers are quoted below.

The Picard–Lefschetz formula

Let $f$ be a tame Morse polynomial with non-degenerate critical points $\boldsymbol{x}_j$, $\nabla f(\boldsymbol{x}_j) = 0$, and distinct critical values $\lambda_j = f(\boldsymbol{x}_j)$, $j = 1, \dots, n$. The critical points are in general complex; we refer to them as saddle points. The steepest descent manifold (Lefschetz thimble) $\mathcal{J}_j$ of $\boldsymbol{x}_j$ defines the thimble integral

\[\Psi^{(j)}(k) = \left(\frac{k}{2\pi i}\right)^{d/2} \int_{\mathcal{J}_j} e^{i k f(\boldsymbol{x})}\, \mathrm{d}\boldsymbol{x},\]

and the original integral decomposes as (Sec. II)

\[\Psi = \sum_{j=1}^n n_j\, \Psi^{(j)},\]

with integer intersection numbers $n_j$. A saddle point is relevant when $n_j \neq 0$. The thimbles are oriented such that

\[\Psi^{(j)}(k) = \frac{e^{i k \lambda_j}}{\sqrt{\det \mathcal{H}f(\boldsymbol{x}_j)}}\left[1 + \mathcal{O}(k^{-1})\right],\]

with the principal branch for complex saddle points and $\sqrt{\det \mathcal{H}f} = e^{i\pi\nu_j/2}\sqrt{|\det \mathcal{H}f|}$ for a real saddle point of Morse index $\nu_j$, such that every real saddle point enters with $n_j = 1$. Expanding each thimble integral yields the transseries

\[\Psi \sim \sum_{j=1}^n n_j\, e^{i k \lambda_j} \sum_{m=0}^\infty \frac{T_m^{(j)}}{k^m}, \qquad T_0^{(j)} = \frac{1}{\sqrt{\det \mathcal{H}f(\boldsymbol{x}_j)}}.\]

The Gauss–Manin system

Integration by parts shows that a moment $\langle g \rangle = \int_{\mathcal{J}_j} g(\boldsymbol{x})\, e^{ikf(\boldsymbol{x})}\,\mathrm{d}\boldsymbol{x}$ is unchanged by $g \mapsto g + \partial_i h + i k\, h\, \partial_i f$ for any polynomial $h$. Modulo these relations, every polynomial reduces to a combination of $n$ basis functions $\boldsymbol{\phi} = (\phi_1, \dots, \phi_n)$, with $\phi_1 = 1$, and the standard moments

\[I_i^{(j)}(k) = \left(\frac{k}{2\pi i}\right)^{d/2} \int_{\mathcal{J}_j} \phi_i(\boldsymbol{x})\, e^{ikf(\boldsymbol{x})}\, \mathrm{d}\boldsymbol{x}\]

satisfy, for a good basis in the sense of Saito, the Gauss–Manin system (Sec. III)

\[\frac{\mathrm{d}\boldsymbol{I}^{(j)}}{\mathrm{d}k} = \left[i M + \frac{1}{k}\left(\frac{d}{2} I - B\right)\right] \boldsymbol{I}^{(j)}.\]

The first component $I_1^{(j)} = \Psi^{(j)}$ is the thimble integral itself. The matrix $M$ represents multiplication by $f$, and its eigenvalues are the critical values $\lambda_j$. GaussManin computes the good basis GM.ϕ and the matrices GM.M and GM.B with the gmspoly.lib library of Singular, run by the Singular executable from Singular_jll. The library is patched to keep resonant terms (equal spectral numbers) in the connection matrix instead of dividing by zero.

Asymptotic series

Substituting the ansatz

\[\boldsymbol{I}^{(j)}(k) \sim e^{ik\lambda_j} (ik)^{\mu_j} \sum_{m=0}^\infty \frac{\boldsymbol{T}_m^{(j)}}{k^m}\]

into the Gauss–Manin system yields an eigenvalue problem, $M \boldsymbol{T}_0^{(j)} = \lambda_j \boldsymbol{T}_0^{(j)}$, followed by a linear recursion for the higher coefficients (Sec. IV). The exponent $\mu_j$ vanishes for non-degenerate saddle points. asymptotic solves the recursion to any order in arbitrary precision, and transseries returns the series of all saddle points as a vector of Expansions, normalised by $T_0^{(j)} = 1/\sqrt{\det \mathcal{H}f(\boldsymbol{x}_j)}$.

Into the Borel plane

With the Borel variable $t = -i f(\boldsymbol{x})$, each thimble maps to a ray from the Borel singularity $t_j = -i\lambda_j$, and the Borel transform of the moments,

\[\widehat{\boldsymbol{I}}^{(j)}(t) = \sum_{m=0}^\infty \frac{\boldsymbol{T}_m^{(j)} (t - t_j)^m}{m!},\]

solves the Fuchsian system (Sec. V)

\[\frac{\mathrm{d}\widehat{\boldsymbol{I}}}{\mathrm{d}t} = A(t)\, \widehat{\boldsymbol{I}}, \qquad A(t) = \sum_{j=1}^n \frac{R_j}{t - t_j},\]

with simple poles at the Borel singularities. The residues $R_j$ (GM.Rs) are nilpotent in the normalisation of the package. Solutions are continued along any path in $\mathbb{C}\setminus\{t_1,\dots,t_n\}$, either by transporting the solution numerically or with hyperlogarithms.

Stokes constants

When the phase of $k$ rotates, the Laplace ray of saddle point $\boldsymbol{x}_j$ can cross the singularity $t_l$. Across this Stokes transition the thimble $\mathcal{J}_j$ picks up $K_{lj}$ copies of $\mathcal{J}_l$,

\[\mathcal{S}^+ \boldsymbol{I}^{(j)} - \mathcal{S}^- \boldsymbol{I}^{(j)} = K_{lj}\, \boldsymbol{I}^{(l)},\]

with the integer Stokes constant $K_{lj}$. Two saddle points are adjacent when $K_{lj} \neq 0$. As the difference of two continuations is again a solution of the Fuchsian system, $K_{lj}$ follows from continuing the Borel transform of $\boldsymbol{x}_j$ along two keyhole paths that pass $t_l$ on either side, and comparing their difference with the Borel transform of $\boldsymbol{x}_l$ at a single point. stokes_constants evaluates all $K_{lj}$ in this way; plotStokesGraphGM draws the resulting adjacency graph.

Alternatively, findStokesConstants estimates the Stokes constants from Borel–Padé approximants of the asymptotic series (findPoles), and BorelPadeResum resums a single series by a Laplace transform of its Padé approximant. The Padé approximant imitates the branch cuts of the Borel transform by rows of poles, which for several singularities form curved arcs. When the Laplace ray in the direction $\arg t = -\arg k$ would cross them, BorelPadeResum bends the ray around them, yielding the Borel sum of the neighbouring sector; BorelPadeResum(transseries, j, K, k) adds the Stokes jumps $K_{lj}\, e^{-k(t_l - t_j)}\, S_l(k)$ of the singularities in between and returns the Borel sum in the direction $-\arg k$.

Approximating the integral

Relevance: the South-East rule

For an exponent bounded from below on $\mathbb{R}^d$, rotate $k \mapsto |k|\,e^{i\theta}$. At $\theta = \pi/2$ the integral is Euclidean, and exactly the real saddle points are relevant ($n_j = 1$), the complex ones not ($n_j = 0$). Rotating back to $\theta = 0$, the Borel singularities rotate clockwise, and a relevant saddle point $\boldsymbol{x}_j$ switches on the adjacent saddle point $\boldsymbol{x}_l$ when they line up horizontally (Sec. VI A):

\[n_l \mapsto n_l + n_j K_{lj}.\]

Only edges $t_j \to t_l$ of the adjacency graph pointing South-East, $\arg(t_l - t_j) \in [-\pi/2, 0]$, contribute, ordered from steep to shallow. relevance applies the rule from start to finish; southeast applies it to given initial intersection numbers, for instance for exponents unbounded from below.

Superasymptotics

The late terms of the series of $\boldsymbol{x}_j$ are governed by the adjacent saddle points (Sec. VI B),

\[T_m^{(j)} \approx \sum_l \frac{K_{lj}}{2\pi i} \sum_{r \geq 0} \frac{\Gamma(m - r)}{F_{jl}^{\,m-r}}\, T_r^{(l)}, \qquad F_{jl} = t_l - t_j,\]

in the package convention for K. The terms are smallest near $N_j \approx |k|\,\min_{l:K_{lj}\neq 0} |F_{jl}|$ (singulants). Truncating each series there yields the superasymptotic approximation superasymptotic,

\[\Psi(k) \approx \sum_j n_j\, e^{ik\lambda_j} \sum_{m=0}^{N_j - 1} \frac{T_m^{(j)}}{k^m},\]

with an exponentially small error. It is not the nearest singularity in the Borel plane that controls the divergence, but the nearest adjacent one.

Hyperasymptotics

The remainder after optimal truncation is re-expanded in the series of the adjacent saddle points (Sec. VI C),

\[R_N^{(j)}(k) = \sum_l \frac{K_{lj}}{2\pi i} (kF_{jl})^{-N} \int_0^\infty \frac{v^{N-1} e^{-v}}{1 - v/(kF_{jl})}\, S_l\!\left(\frac{v}{F_{jl}}\right) \mathrm{d}v,\]

with $S_l$ the series of $\boldsymbol{x}_l$. Truncating $S_l$ yields the first hyperasymptotic level in terms of the hyperterminant

\[h(M, w) = \frac{1}{\Gamma(M)} \int_0^\infty \frac{v^{M-1} e^{-v}}{1 - v/w}\, \mathrm{d}v = -w\, e^{-w} E_M(-w)\]

(hyperterminant), with $E_M$ the generalised exponential integral. Higher levels follow paths in the adjacency graph. hyperasymptotic implements the first two levels, with the optimal truncations of Olde Daalhuis (path_lengths).