Stokes phenomenon

Across a Stokes curve, an exponentially small term can switch on or off another exponentially small term.

Across an anti-Stokes curve, a subdominant exponentially small term can switch to a dominant exponentially large term or vice versa.

This change in behaviour across the Stokes and anti-Stokes curves is directly related to the divergence of the asymptotic expansion. The usual type of divergence seen in an asymptotic series that exhibits Stokes phenomenon is known as factorial-over-power divergence and has the typical form

${\displaystyle \sum _{j}^{\infty }\epsilon ^{j}{\frac {A\Gamma (j+\alpha )}{\chi ^{j+\alpha }}},}$

Where ${\displaystyle A}$ is a function known as the prefactor, ${\displaystyle \chi }$ is a function known as the Singulant and ${\displaystyle \Gamma }$ is the gamma function.

Stokes curves are determined using the condition ${\displaystyle \Im \{\chi \}=0}$ and ${\displaystyle \Re \{\chi \}>0}$. Anti Stokes curve are determined by the condition ${\displaystyle \Re \{\chi \}=0}$.

The Airy function Ai(x) is one of two solutions to a simple differential equation

${\displaystyle y''-xy=0,\,}$

which it is often useful to approximate for many values of x – including complex values. For large x of given argument the solution can be approximated by a linear combination of the functions

${\displaystyle {\frac {e^{\pm {\frac {2}{3}}x^{3/2}}}{x^{1/4}}}.}$

However, the linear combination has to change as the argument of x passes certain values (when x crosses a branch cut) because these approximations contain multi-valued functions. In contrast, the Airy function is single valued and indeed entire and therefore, in order to make sense of the approximation, one has to choose a single value out of the multiple possible values (this imposes a branch cut for the approximation, by implication). For example, if we regard the limit of x as large and real, and would like to approximate the Airy function for both positive and negative values, we would find that

Plot of Ai(x) in red and Bi(x) in blue

${\displaystyle {\begin{aligned}\mathrm {Ai} (x)&{}\sim {\frac {e^{-{\frac {2}{3}}x^{3/2}}}{2{\sqrt {\pi }}\,x^{1/4}}}\\\mathrm {Ai} (-x)&{}\sim {\frac {\sin({\frac {2}{3}}x^{3/2}+{\frac {1}{4}}\pi )}{{\sqrt {\pi }}\,x^{1/4}}}\\\end{aligned}}}$

Stokes lines and anti-Stokes lines for the Airy function

which are two very different expressions. What has happened is that as we have increased the argument of x from 0 to pi (rotating it around through the upper half complex plane) we have crossed an anti-Stokes line, which in this case is at ${\displaystyle \operatorname {arg} \,x=\pi /3}$. At this anti-Stokes line, the coefficient of ${\displaystyle {\frac {e^{-{\frac {2}{3}}x^{3/2}}}{x^{1/4}}}}$ is forced to jump. The coefficient of ${\displaystyle {\frac {e^{+{\frac {2}{3}}x^{3/2}}}{x^{1/4}}}}$ can jump at this line but is not forced to; it can change gradually as arg x varies from π/3 to π because it is not determined in this region.

There are three anti-Stokes lines with arguments π/3, π. –π/3, and three Stokes lines with arguments 2π/3, 0. –2π/3.

The Airy function example can be generalized to a broad class of second order linear differential equations as follows. By standard changes of variables, a second order equation can often be changed to one of the form

${\displaystyle {\frac {d^{2}w}{dz^{2}}}=f(z)w}$

where f is holomorphic in a simply-connected region and w is a solution of the differential equation. Then in some cases the WKB method gives an asymptotic approximation for w as a linear combination of functions of the form

${\displaystyle {\frac {e^{\pm \int _{a}^{z}{\sqrt {f(z')}}\,dz'}}{f(z)^{1/4}}}}$

for some constant a. (Choosing different values of a is equivalent to choosing different coefficients in the linear combination.) The anti-Stokes lines and Stokes lines are then the zeros of the real and imaginary parts, respectively, of

${\displaystyle \int _{a}^{z}{\sqrt {f(z^{'})}}\,dz^{'}.}$

If a is a simple zero of f then locally f looks like ${\displaystyle f'(a)(z-a)}$. Solutions will locally behave like the Airy functions; they will have three Stokes lines and three anti-Stokes lines meeting at a.

• Borel summation

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