Stationary Phase, Coalescing Saddles, and Stokes Geometry

For an oscillatory integral

$$I(\Lambda,\alpha) = \int_\Gamma a(z,\alpha)e^{i\Lambda\Phi(z,\alpha)}\,\mathrm dz, \qquad \Lambda\to+\infty,$$

rapid oscillations suppress regions where the phase has no stationary point. An isolated nondegenerate stationary point instead contributes a Fresnel Gaussian: the Hessian determinant fixes its magnitude and the Hessian signature fixes its phase. If two saddles merge, their separate Gaussian approximations cease to be uniform and a generic fold is described by one Airy approximation. Stokes and equal-magnitude curves then organize different changes: saddle coefficients can switch on a phase-alignment curve, whereas dominance can exchange on an equal-magnitude curve. With the contour and analytic continuation held fixed, the exact integral does not jump when its asymptotic description changes.

Every one of these statements depends on the limiting direction of $\Lambda$, the oriented contour or boundary-value prescription, endpoint behavior, phase and root branches, and a stated parameter region. This page derives the reusable finite-dimensional method; it does not infer a continuum path integral from a formal saddle sum.

Required background. Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation supplies the fixed-order meaning of an asymptotic series and the distinction between a pointwise saddle expansion and an approximation uniform in a control parameter.

Helpful background. Laplace Method and Steepest Descent develops global contour accessibility, orientation, and branch tracking for isolated saddles. The local facts needed here are recalled below, but a critical point still contributes only when it belongs to a legal deformation of the original contour.

Stationary-phase prescription and saddle data

Unless stated otherwise, $\Lambda>0$, the phase $\Phi$ is real on a real contour, and $e^{i\Lambda\Phi}$ is the oscillatory convention. Fresnel and Airy integrals over the full real line are Abel limits: a factor $e^{-\epsilon x^2}$ is inserted and $\epsilon\downarrow0$ is taken before the large-$\Lambda$ limit. Replacing $e^{i\Lambda\Phi}$ by $e^{-i\Lambda\Phi}$ conjugates the signature phases. For comparison with a decay integral $e^{-\Lambda S}$, use

$$S=-i\Phi.$$

The input to the method consists of the phase and amplitude, the oriented contour and its endpoints, all singularities and branch cuts, the parameter domain, and the requested fixed asymptotic order. A reproducible calculation proceeds as follows:

1. Fix the convergence or Abel prescription and the branches of every multivalued quantity.
2. Locate stationary points, endpoints, singularities, and possible pinches, then determine which stationary points are accessible from the original contour.
3. Partition the contour into stationary neighborhoods and a nonstationary remainder.
4. Use integration by parts on the remainder and a quadratic normal form near each isolated nondegenerate point.
5. Test the Hessian gap and saddle separation. If they are not uniform, identify the local degeneracy and replace the separated expansions by the appropriate canonical integral.
6. Sum contributions before judging their size, and report an absolute remainder wherever interference can make the leading sum vanish.
7. Check the result against an exact special-function representation, a regulated Gaussian, a differential identity, or numerical quadrature in the declared sector.

The local series is usually easy to generate. The difficult work is proving contour accessibility, tracking every relevant saddle, and constructing a stable normal form as parameters vary. Those global tasks also determine when a local Airy formula is the wrong model.

Cancellation away from stationary points

On a finite real interval with $\Phi'(x)\neq0$, one integration by parts gives

$$\begin{aligned} \int_A^B a(x)e^{i\Lambda\Phi(x)}\,\mathrm dx &= \left[ \frac{a(x)e^{i\Lambda\Phi(x)}} {i\Lambda\Phi'(x)} \right]_A^B\\ &\quad -\frac{1}{i\Lambda} \int_A^B e^{i\Lambda\Phi(x)} \frac{\mathrm d}{\mathrm dx} \left( \frac{a(x)}{\Phi'(x)} \right)\mathrm dx. \end{aligned}$$

If $a$ is smooth and compactly supported in a region where $|\Phi'|\geq c>0$, this operation can be repeated: for every fixed $M$, that region contributes $O(\!\left(\Lambda^{-M}\right))$. The estimate is not true merely because there is no stationary point. A nonzero amplitude at a finite endpoint leaves the displayed $O(\!\left(\Lambda^{-1}\right))$ boundary term; uncontrolled behavior at infinity can also defeat the argument.

This is the localization principle behind stationary phase. A partition of unity isolates the critical points, while all remaining compactly supported pieces are removed to arbitrary algebraic order. NIST DLMF 2026, §2.3(iv) states the corresponding endpoint and stationary-point expansions.

Isolated real stationary points

Let $a\in C_c^\infty((A,B))$ and let $\Phi\in C^\infty((A,B);\mathbb R)$. Suppose the critical points $x_j$ in $\operatorname{supp}a$ are finite in number, separated, and nondegenerate:

$$\Phi'(x_j)=0, \qquad \Phi''(x_j)\neq0.$$

Then, for every fixed $N$,

$$\begin{aligned} I(\Lambda) ={}& \sum_j e^{i\Lambda\Phi(x_j)} e^{i\pi\sigma_j/4} \sqrt{\frac{2\pi} {\Lambda|\Phi''(x_j)|}}\\ &\times \left[ \sum_{n=0}^{N-1} \frac{C_{j,n}}{\Lambda^n} \right] +O\!\left(\Lambda^{-N-1/2}\right), \end{aligned}$$

where

$$\sigma_j=\operatorname{sgn}\Phi''(x_j), \qquad C_{j,0}=a(x_j).$$

The remainder constant may depend on $N$, $a$, $\Phi$, and the fixed parameter region. Uniformity in an auxiliary parameter additionally requires common bounds on derivatives, $|\Phi'|$ away from the critical neighborhoods, critical-point separation, and $|\Phi''(x_j)|$. If $a(x_j)=0$, the leading term at that point is not a relative approximation; continue to the first nonzero additive coefficient.

Where the signature phase comes from

Near $x_j$, the one-dimensional Morse coordinate puts the phase into the exact local form

$$\Phi(x)=\Phi(x_j)+\frac{\sigma_j y^2}{2}.$$

The normalization is anchored by the regulated Fresnel integral

$$\begin{aligned} &\lim_{\epsilon\downarrow0} \int_{\mathbb R} \exp\!\left[ -\epsilon y^2+\frac{i\Lambda h y^2}{2} \right]\mathrm dy\\ &\qquad= e^{i\pi\operatorname{sgn}(h)/4} \sqrt{\frac{2\pi}{\Lambda|h|}}, \qquad h\in\mathbb R\setminus\{0\}. \end{aligned}$$

The square root is obtained by continuation from $\epsilon>0$; it is not chosen afterward. Taylor-expanding the transformed amplitude and integrating its even terms produces the full series. For one stationary point, write $\Phi_k=\Phi^{(k)}(x_0)$ and $a_k=a^{(k)}(x_0)$. The first correction is

$$\begin{aligned} I_{x_0}(\Lambda) ={}& e^{i\Lambda\Phi_0+i\pi\operatorname{sgn}(\Phi_2)/4} \sqrt{\frac{2\pi}{\Lambda|\Phi_2|}}\\ &\times \left[ a_0+\frac{iD_1}{\Lambda} +O\!\left(\Lambda^{-2}\right) \right], \end{aligned}$$

with

$$\begin{aligned} D_1={}& \frac{a_2}{2\Phi_2} -\frac{a_1\Phi_3}{2\Phi_2^2} -\frac{a_0\Phi_4}{8\Phi_2^2}\\ &+\frac{5a_0\Phi_3^2}{24\Phi_2^3}. \end{aligned}$$

This formula retains the sign of $\Phi_2$. It is a useful check on both the Fresnel phase and the factors of $i$.

Several variables

For a nondegenerate real critical point $q_\star\in\mathbb R^d$, let $H$ be the real symmetric Hessian and define

$$\operatorname{sig}H=n_+-n_-,$$

where $n_+$ and $n_-$ count its positive and negative eigenvalues. With the same support and separation qualifications,

$$\begin{aligned} \int_{\mathbb R^d} a(q)e^{i\Lambda\Phi(q)}\,\mathrm d^dq \sim{}& e^{i\Lambda\Phi(q_\star)} \left(\frac{2\pi}{\Lambda}\right)^{d/2}\\ &\times \frac{e^{i\pi\operatorname{sig}H/4}} {\sqrt{|\det H|}} a(q_\star). \end{aligned}$$

The absolute determinant fixes the magnitude; the signature factor carries the oscillatory phase. A zero eigenvalue makes both parts of this formula inapplicable. For several accessible stationary points, calculate each oriented contribution and add them before taking an absolute value. Diagonalizing the real symmetric Hessian reduces the local coefficient to a product of the regulated one-dimensional Fresnel factors. Hunter 2004, §§3.3–3.4, pp. 35–40, PDF gives the Fresnel, nonstationary, one-dimensional nondegenerate, and cubic degenerate stationary-phase constructions.

Why two separate Gaussians fail

Consider the canonical fold phase

$$\Phi(u;\zeta) = \frac{u^3}{3}-\zeta u.$$

For real $\zeta>0$, its stationary points and phase values are

$$\begin{gathered} u_\pm=\pm\sqrt\zeta,\\ \Phi(u_+;\zeta)=-\frac23\zeta^{3/2}, \qquad \Phi(u_-;\zeta)=+\frac23\zeta^{3/2}. \end{gathered}$$

Their phase separation is $4\zeta^{3/2}/3$. Each Gaussian neighborhood has width $O((\Lambda\sqrt\zeta)^{-1/2})$, whereas the saddle separation is $O(\!\left(\sqrt\zeta\right))$. They are genuinely separate only when

$$\Lambda\zeta^{3/2}\gg1.$$

Thus the separated approximation loses uniformity when

$$\zeta=O\!\left(\Lambda^{-2/3}\right), \qquad u=O\!\left(\Lambda^{-1/3}\right).$$

The divergence of each Gaussian coefficient as $\zeta^{-1/4}$ is not a divergence of the integral. It says that the quadratic neighborhoods overlap and must be replaced by a single cubic neighborhood.

The exact Airy fold

For real $\zeta$, define the Abel-prescribed canonical integral

$$\mathcal A(\Lambda,\zeta) = \lim_{\epsilon\downarrow0} \int_{\mathbb R} e^{-\epsilon u^2} \exp\!\left[ i\Lambda\left( \frac{u^3}{3}-\zeta u \right) \right]\mathrm du.$$

The change of variable $t=\Lambda^{1/3}u$ and the standard real Airy integral give the exact identity

$$\mathcal A(\Lambda,\zeta) = 2\pi\Lambda^{-1/3} \operatorname{Ai}\!\left(-\Lambda^{2/3}\zeta\right).$$

This one function covers all three real regimes. For fixed $\zeta>0$,

$$\begin{aligned} \mathcal A(\Lambda,\zeta) = 2\sqrt{\frac{\pi}{\Lambda\sqrt\zeta}} \biggl[ &\cos\!\left( \frac{2\Lambda\zeta^{3/2}}{3}-\frac{\pi}{4} \right)\\ &+O\!\left( \frac{1}{\Lambda\zeta^{3/2}} \right) \biggr]. \end{aligned}$$

The two oscillatory saddle contributions have combined into a cosine. The error inside the brackets is additive, so the statement remains meaningful at zeros of the leading cosine. For $\zeta=-q<0$ with fixed $q>0$,

$$\mathcal A(\Lambda,-q) = \sqrt{\frac{\pi}{\Lambda\sqrt q}} e^{-2\Lambda q^{3/2}/3} \left[ 1+O\!\left( \frac{1}{\Lambda q^{3/2}} \right) \right].$$

At coalescence,

$$\mathcal A(\Lambda,0) = \frac{2\pi\Lambda^{-1/3}} {3^{2/3}\Gamma(2/3)}.$$

The finite $\Lambda^{-1/3}$ answer confirms that the divergent separate $\Lambda^{-1/2}\zeta^{-1/4}$ terms were the wrong local representation. The normalization and sectorial expansions follow from NIST DLMF 2026, §9.5(i) and NIST DLMF 2026, §9.7(ii).

Uniformizing a generic fold

Suppose two analytic saddles coalesce at $(z_0,\alpha_0)$. Isolate a contour neighborhood containing this pair, call its contribution $I_{\mathrm{fold}}$, and assume the mapped local contour and amplitude obey uniform bounds. In suitable local coordinates, require

$$\Phi_z=\Phi_{zz}=0, \qquad \Phi_{zzz}\neq0, \qquad \partial_\alpha\Phi_z\neq0.$$

The last condition says that the parameter unfolds the degeneracy transversely. Locally there is a branch-consistent change of variable such that

$$\Phi(z,\alpha) = \Phi_0(\alpha) +\frac{u^3}{3} -\zeta(\alpha)u.$$

If $z_\pm$ map to $u_\pm=\pm\sqrt\zeta$, branches can be fixed continuously by

$$\begin{aligned} \Phi_0 &= \frac{\Phi(z_+)+\Phi(z_-)}{2},\\ \frac43\zeta^{3/2} &= \Phi(z_-)-\Phi(z_+). \end{aligned}$$

Let the transformed amplitude be

$$G(u,\alpha) = a(z(u,\alpha),\alpha) \frac{\mathrm dz}{\mathrm du}.$$

Its values at the two saddles determine a smooth interpolant,

$$G(u)=c_0+c_1u+(u^2-\zeta)H_0(u),$$

where

$$\begin{aligned} c_0 &= \frac{G(\sqrt\zeta)+G(-\sqrt\zeta)}{2},\\ c_1 &= \frac{G(\sqrt\zeta)-G(-\sqrt\zeta)} {2\sqrt\zeta}. \end{aligned}$$

The apparent singularity in $c_1$ has a finite limit as $\zeta\to0$. For a mapped contour in the standard real Airy class, put $\eta=\Lambda^{2/3}\zeta$. Uniformly while $\eta$ remains in a fixed compact set,

$$\begin{aligned} I_{\mathrm{fold}}(\Lambda,\alpha) = 2\pi e^{i\Lambda\Phi_0} \biggl[ &\frac{c_0}{\Lambda^{1/3}} \operatorname{Ai}(-\eta)\\ &-\frac{ic_1}{\Lambda^{2/3}} \operatorname{Ai}'(-\eta) +O\!\left(\Lambda^{-4/3}\right) \biggr]. \end{aligned}$$

The sign of the derivative term follows by differentiating the exact Airy integral with respect to $\zeta$. The remainder begins one recursive integration by parts later because

$$(u^2-\zeta) e^{i\Lambda(u^3/3-\zeta u)} = \frac{1}{i\Lambda} \frac{\mathrm d}{\mathrm du} e^{i\Lambda(u^3/3-\zeta u)}.$$

Further recursion gives a uniform series in $\operatorname{Ai}$ and $\operatorname{Ai}'$ with smooth coefficient functions. Away from the transition region, its Airy asymptotics recover the separate saddle contributions. NIST DLMF 2026, §2.4(v) gives the two-coalescing-saddle reduction, and NIST DLMF 2026, §36.12(i) places it in the wider theory of uniform canonical-integral approximations.

This local cubic form does not select the global solution. A different mapped contour can select a rotated Airy function or a linear combination of Airy solutions. The original oriented contour, its decay sectors, and its singularity obstructions determine that choice.

Stokes geometry without a naming ambiguity

Write two saddle contributions in decay form as $e^{-\Lambda S_\sigma}$ and $e^{-\Lambda S_\tau}$, and set $\Delta S=S_\sigma-S_\tau$. This page uses the following definitions:

• A phase-alignment or Stokes curve satisfies

  $$\operatorname{Im}(\Lambda\Delta S)=0.$$

  A descent connection can become possible there, and a subdominant saddle coefficient can change in a sectorial asymptotic representation.

• An equal-magnitude curve, often called an anti-Stokes curve, satisfies

  $$\operatorname{Re}(\Lambda\Delta S)=0.$$

  The two exponentials have equal magnitude, so dominance can exchange.

Some references reverse the two names. The equations are therefore part of the definitions here. Neither equation by itself proves that both saddles occur in the original contour: contour accessibility remains a separate global question. Crossing an equal-magnitude curve does not by itself change a saddle coefficient, and meeting the phase-alignment condition does not prove that a Stokes multiplier is nonzero.

For $e^{i\Lambda\Phi}$, the translation $S=-i\Phi$ gives

$$\begin{array}{ll} \text{phase alignment:} & \operatorname{Re}(\Lambda\Delta\Phi)=0,\\[2mm] \text{equal magnitude:} & \operatorname{Im}(\Lambda\Delta\Phi)=0. \end{array}$$

For the cubic fold on a fixed branch, $\Delta\Phi=4\zeta^{3/2}/3$. Equal-magnitude rays are therefore $\arg\zeta=0,\pm2\pi/3$, while phase-alignment rays are $\arg\zeta=\pm\pi/3,\pi$. The exact Airy function is entire across these rays; what changes is its useful sectorial decomposition into saddle exponentials. Near a Stokes curve, an exponentially improved description replaces a sharp coefficient jump by a smooth transition. NIST DLMF 2026, §36.5(i) defines Stokes sets geometrically, while NIST DLMF 2026, §2.11(iv) explains the smooth switching in exponentially improved asymptotics.

Coalescence and Stokes switching should not be conflated. Coalescence is a local degeneration of the Hessian and requires a new canonical scale. Stokes switching is a global reorganization of well-defined saddle contributions under analytic continuation. They can interact, as the Airy model shows, but they answer different diagnostic questions.

A regulated cubic source integral

For real $J$, consider the dimensionless zero-dimensional analogue of a Lorentzian source integral

$$\begin{aligned} \mathcal Z(J,\hbar) = \lim_{\epsilon\downarrow0} \int_{\mathbb R} \exp\!\biggl[ &-\epsilon\phi^2\\ &+\frac{i}{\hbar} \left( \frac{\phi^3}{3}-J\phi \right) \biggr]\mathrm d\phi, \qquad \hbar>0. \end{aligned}$$

The Abel limit is part of the definition and is taken before $\hbar\downarrow0$. In the notation of the canonical fold,

$$\mathcal Z(J,\hbar) = \mathcal A(\hbar^{-1},J).$$

Thus the example transfers the preceding mathematics into source-and-action notation; it is not a second derivation. Rescaling $\phi=\hbar^{1/3}t$ gives the exact answer

$$\mathcal Z(J,\hbar) = 2\pi\hbar^{1/3} \operatorname{Ai}\!\left( -\frac{J}{\hbar^{2/3}} \right).$$

For $J>0$, two real stationary points contribute:

$$\phi_\pm=\pm\sqrt J, \qquad \Phi(\phi_\pm;J)=\mp\frac23J^{3/2}.$$

Their Hessians have opposite signs. Adding the two Fresnel terms gives

$$\mathcal Z(J,\hbar) \sim 2\sqrt{\pi\hbar}\,J^{-1/4} \cos\!\left( \frac{2J^{3/2}}{3\hbar}-\frac{\pi}{4} \right),$$

provided $J^{3/2}/\hbar\gg1$. Neither saddle alone reproduces the real interference pattern.

For $J=-q<0$, the saddles are $\phi_\pm=\pm i\sqrt q$. The Abel-selected continuation contains the decaying saddle because

$$\Phi(i\sqrt q;-q) = \frac{2i}{3}q^{3/2},$$

and therefore

$$\mathcal Z(-q,\hbar) \sim \sqrt{\pi\hbar}\,q^{-1/4} e^{-2q^{3/2}/(3\hbar)}.$$

The other algebraic solution of the saddle equation would grow exponentially and is not added without a contour coefficient. At $J=0$,

$$\mathcal Z(0,\hbar) = \frac{2\pi\hbar^{1/3}} {3^{2/3}\Gamma(2/3)},$$

and the transition window is $J=O(\hbar^{2/3})$. The exact Airy expression is therefore an independent check of the two-saddle phase, the coalescence scale, and the exponentially small continuation.

This model is finite-dimensional. It demonstrates the local mathematics of a soft cubic mode but does not define a Lorentzian QFT functional integral or fix its physical integration cycle. Mariño 2015, §1.3, pp. 12–16 uses an ordinary zero-dimensional integral to make the same controlled bridge to semiclassical path-integral reasoning. Gauge fixing, collective coordinates, functional determinants, renormalization, and physical saddle sectors require the later field-theory treatment.

Uniformity, conditioning, and stop rules

A separated stationary-phase expansion is uniform on a parameter set only when the support or contour is stable; stationary points stay separated from one another, endpoints, and singularities; nonzero Hessian eigenvalues have a common lower bound; $|\nabla\Phi|$ is bounded below off the stationary neighborhoods; and the required derivatives and branches are controlled uniformly.

For a fold approximation, verify in addition that exactly two nearby saddles are involved, the cubic and transverse-unfolding conditions hold, and the mapped contour remains in one fixed Airy contour class. Useful numerical checks are:

• compare the signature phase with the regulated Fresnel integral;

• recover separated stationary phase from the large-argument Airy expansion on both sides of the transition;

• compare against the exact Airy identity or direct regulated quadrature;

• compute a cancellation indicator

  $$\kappa_{\mathrm{sum}} = \frac{|I_1|+|I_2|}{|I_1+I_2|};$$

  a large value warns that relative error in the summed leading term is ill-conditioned;

• use scaled Airy functions or logarithmic exponential weights in a decay sector instead of subtracting overflowing saddle terms.

Stop the Airy calculation if $\Phi_{zzz}$ also vanishes, three saddles meet, a saddle collides with an endpoint, pole, or branch point, more than one Hessian direction becomes soft, the contour is pinched, or its canonical contour class changes. Higher degeneracy can require a Pearcey-type or another canonical integral; a saddle–endpoint collision needs a one-sided model; and a symmetry zero mode needs collective coordinates. NIST DLMF 2026, §2.4(vi) catalogues these distinct coalescence mechanisms. They are not corrections to a universal Airy formula.

Common pitfalls

Calling every nonstationary contribution negligible. Repeated integration by parts gives superalgebraic decay only after boundary, support, and derivative hypotheses are checked. A finite endpoint often contributes at order $\Lambda^{-1}$.

Dropping the signature phase. The magnitude uses $\sqrt{|\det H|}$, but the phase uses $e^{i\pi\operatorname{sig}H/4}$. Neither factor replaces the other.

Adding every solution of the saddle equation. A stationary point contributes only with the coefficient fixed by the original oriented contour and its legal deformations.

Following separate saddles through coalescence. Their divergent Gaussian coefficients signal a nonuniform representation. Keep the Airy function intact in the $\Lambda^{-2/3}$ transition window.

Using “Stokes line” without an equation. Naming conventions vary. State whether the condition is phase alignment or equal magnitude and write it in terms of the action difference.

Claiming that the exact integral jumps. A sectorial saddle coefficient can change while the analytically continued exact function remains smooth. A genuine discontinuity requires an independently specified change of boundary value, contour, or physical prescription.

Reporting a relative error at an interference zero. When leading saddle terms cancel, an absolute remainder can remain valid while relative error becomes unbounded.

Check your understanding

1. Retrieve the signature factor

Suppose

$$\Phi(x) = \Phi_0-\frac{\kappa}{2}(x-x_0)^2+\cdots, \qquad \kappa>0,$$

and $x_0$ is the only stationary point in the compact support of $a$. What is its leading contribution for the convention $e^{i\Lambda\Phi}$?

Solution

Here $\Phi''(x_0)=-\kappa$, so the signature is $-1$. The contribution is

$$e^{i\Lambda\Phi_0-i\pi/4} \sqrt{\frac{2\pi}{\Lambda\kappa}}\,a(x_0).$$

The statement also assumes the increasing real orientation and no competing endpoint term.

2. Restore the endpoint term

Evaluate

$$\int_0^1 e^{i\Lambda x}\,\mathrm dx$$

and explain why it is not smaller than every power of $\Lambda^{-1}$ despite having no stationary point.

Solution

Direct integration gives

$$\int_0^1 e^{i\Lambda x}\,\mathrm dx = \frac{e^{i\Lambda}-1}{i\Lambda}.$$

The amplitude does not vanish at either endpoint, so the first integration by parts boundary term survives at $O(\!\left(\Lambda^{-1}\right))$. Compact support inside the interval was an essential hypothesis of the superalgebraic estimate.

3. Match the Airy fold to two saddles

For fixed real $\zeta>0$, localize the Abel-prescribed integral $\mathcal A(\Lambda,\zeta)$ around its two saddles and control the complementary region before applying stationary phase. Show how the two local terms combine.

Solution

Choose disjoint smooth cutoffs around $u_\pm$. On the complement the phase derivative $u^2-\zeta$ stays away from zero; the Abel regulator removes boundary terms, so integration by parts controls that piece in the prescribed limit. At $u_+=\sqrt\zeta$, the phase and Hessian are $-2\zeta^{3/2}/3$ and $+2\sqrt\zeta$. At $u_-=-\sqrt\zeta$, they are $+2\zeta^{3/2}/3$ and $-2\sqrt\zeta$. Hence the two terms are

$$\sqrt{\frac{\pi}{\Lambda\sqrt\zeta}} e^{-2i\Lambda\zeta^{3/2}/3+i\pi/4}$$

and

$$\sqrt{\frac{\pi}{\Lambda\sqrt\zeta}} e^{+2i\Lambda\zeta^{3/2}/3-i\pi/4}.$$

Their sum is

$$2\sqrt{\frac{\pi}{\Lambda\sqrt\zeta}} \cos\!\left( \frac{2\Lambda\zeta^{3/2}}{3}-\frac{\pi}{4} \right),$$

which is the large-negative-argument asymptotic form of the exact Airy result.

4. Transfer the transition scale

For the regulated cubic source integral, when are two separated real saddles valid, when is the Airy form required, and what does this model not establish about QFT?

Solution

For $J>0$, the saddle phase gap is $4J^{3/2}/3$. Separate stationary phase requires

$$\frac{J^{3/2}}{\hbar}\gg1.$$

When $J=O(\!\left(\hbar^{2/3}\right))$, the Gaussian neighborhoods overlap and the uniform expression is

$$2\pi\hbar^{1/3} \operatorname{Ai}\!\left(-J/\hbar^{2/3}\right).$$

The calculation is a regulated ordinary integral. It does not supply a functional measure, gauge fixing, renormalized determinant, physical contour, or Stokes data for a continuum QFT.

What the method establishes

Oscillatory localization is now a controlled sequence rather than a slogan: nonstationary regions cancel subject to boundary hypotheses, each isolated real saddle carries a signed Fresnel phase, and all accessible contributions are summed before their size is assessed. When a Hessian gap closes through a generic two-saddle fold, the Airy normal form replaces the nonuniform Gaussian sum and sets the $\Lambda^{-1/3}$ coordinate and $\Lambda^{-2/3}$ parameter scales. Action-difference equations then distinguish phase alignment from equal magnitude without relying on ambiguous terminology.

Where the method continues

WKB and Eikonal Methods and Turning-Point Matching uses related Airy local models inside differential equations and develops turning-point connection formulas. Special Functions from Equations and Boundary Data develops Airy and other special functions from their differential equations and boundary data.

For physical applications, Saddles, Control Parameters, and Loop Counting develops regulated field-theory stationary points and fluctuation expansions. Complex Saddles, Lefschetz Thimbles, and Integration Cycles treats physical cycle selection, while Stokes Jumps, Saddle Dominance, and Contour Dependence and Resurgence and Transseries treat physical saddle-sector changes and their nonperturbative completion.

References

• John K. Hunter, Asymptotic Analysis and Singular Perturbation Theory, PDF, University of California, Davis (2004), §§2.3 and 3.3–3.4, pp. 27–28 and 35–40. Nonuniformity and Stokes terminology; nonstationary and nondegenerate stationary phase; Fresnel normalization; cubic Airy reduction.

• Marcos Mariño, Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory, Cambridge University Press (2015), §1.3, pp. 12–16. Zero-dimensional contour models, lateral continuations, nontrivial saddles, and the controlled bridge to path-integral reasoning.

• NIST Digital Library of Mathematical Functions, version 1.2.7, released June 15, 2026, National Institute of Standards and Technology, §2.3(iv), “Method of Stationary Phase”, §2.4(v), “Coalescing Saddle Points”, §2.4(vi), “Other Coalescing Critical Points”, §9.5(i), “Real Variable” integral representations, §9.7(ii), “Poincaré-Type Expansions”, §2.11(iv), “Stokes Phenomenon”, §36.5(i), “Stokes Set”, and §36.12(i), “Saddle Points”. Structural formulas, hypotheses, sectors, canonical integrals, and Stokes geometry.