Stokes phenomenon

Remarks

Consider the rays where a small term $e^{q_j(z)}$ is maximally subdominant compared with another term $e^{q_i(z)}$. Its coefficient may jump there, and these rays are called Stokes lines in one common convention; other authors call them anti-Stokes lines. Consider a system $Y' = A(z) Y$ where $A$ has a pole at $\infty$. On two overlapping sectors, the actual solutions asymptotic to a fixed formal solution differ by a constant matrix, the Stokes matrix. The phenomenon occurs exactly when some Stokes matrix is not the identity. At a regular singular point it does not occur: the convergent Frobenius series hold in all directions.

Example: the Airy function satisfies $\operatorname{Ai}(z) \sim \frac{e^{-\zeta}}{2 \sqrt{\pi}\, z^{1/4}}$ with $\zeta = \frac{2}{3} z^{3/2}$ in the sector $\abs{\arg z} < \pi$. On the negative real axis, however, $\operatorname{Ai}(-x) \sim \frac{1}{\sqrt{\pi}\, x^{1/4}} \sin\left( \frac{2}{3} x^{3/2} + \frac{\pi}{4} \right)$ for $x \to +\infty$, which is a sum of two exponential terms $e^{\pm i \frac{2}{3} x^{3/2}}$.

Non-example: the function $e^{z}$ solves $y' = y$, which has an irregular singular point at $\infty$, and $e^{z} \sim e^{z} \cdot 1$ holds in every sector; this solution shows no change of expansion.