Exponentially small terms and Stokes phenomena

For a fixed positive constant $c$, the exponential $e^{-c/\varepsilon}$ is little-o of every fixed power as the positive parameter tends to zero. Set the reciprocal parameter equal to a growing real variable and compare exponential decay with polynomial growth. Consequently adding any fixed multiple of this exponential leaves every coefficient of an existing power expansion unchanged. Boundary data, an integral representation, or a summation prescription can determine that multiple, but the formal algebraic coefficients alone cannot.

In a complex domain, the same expression may decay, grow, or oscillate depending on the real part of $c/\varepsilon$. A beyond-all-orders claim therefore belongs to a declared direction or sector where exponential decay persists, not automatically to an entire punctured complex neighborhood. Bounds uniform on closed subsectors can deteriorate near rays where the exponential loses its separation. This is a continuation of the domain issue seen in boundary layers: apparently negligible terms become relevant when the observation regime changes.

Return to the factorial integral from the first lesson. Its term magnitudes have successive ratio $(n+1)\varepsilon/2$. They decrease until the index is near twice the reciprocal parameter and then increase. For a fixed numerical parameter, retaining indefinitely more terms is therefore counterproductive. Choosing a truncation near the least term gives a useful remainder bound because this particular integral has a proved next-term estimate. A generic divergent expansion does not inherit that estimate merely from a similar ratio.

Stirling's formula shows that the least factorial term has an exponential factor of size $e^{-2/\varepsilon}$, multiplied by an algebraic prefactor. This diagnoses an exponential scale, but does not determine every exponentially small coefficient or prove a full transseries representation. A fixed-index Poincaré expansion and a parameter-dependent least-term truncation are different assertions; the latter needs estimates valid as the index grows. Here the finite geometric remainder bound holds for every index, so that step is justified directly.

Suppose two asymptotic contributions have relative exponential factor $e^{-\chi(z)/\varepsilon}$ with positive real parameter and a chosen analytic branch of the singulant $\chi$. Use the convention that a candidate Stokes curve satisfies zero imaginary part of $\chi$ with positive real part, so the second contribution is phase aligned and exponentially subdominant. Use anti-Stokes for equal-magnitude curves where the real part is zero. Some authors interchange these names; the actual equations and branch choices should therefore accompany the terminology. These loci alone do not guarantee a nonzero switching coefficient.

Across an accessible Stokes curve, a sectorwise asymptotic representation can acquire a subdominant term determined by analytic continuation and global data. A connection formula specifies its coefficient for a particular equation and normalization. The exact analytic solution does not jump. Exponentially improved descriptions can resolve smooth switching in a parameter-dependent angular neighborhood, but their smoothing scale and profile need additional hypotheses and are not universal consequences of drawing a line. Airy functions provide a canonical example with documented sector restrictions; importing a multiplier from another branch or normalization would change the answer.