Line Style
Calligraphy penParameters
Static Circle:Mobile Circle:
Tracing Stick:
This is for illustration only and comes with no guarantees concerning accuracy etc.
Please email me any suggestions for improvements though
(boalch at imj-prg.fr).
Background:
Click on "draw", and the program will draw the Stokes diagram of the
exponential factor $\< x^{3/2} >$.
Here $x$ is a coordinate on the complex plane and this means
we consider the growth/decay of the
two branches
of the function $\exp(x^{3/2})$ as $x$ tends to infinity along any ray.
This picture appears in
Stokes' 1857 paper [S1857], and was reproduced on the title page of [BY2015].
Thus the dashed line is a (small!) circle around the point
$x=\infty$ in the Riemann sphere, and the solid line encodes the growth/decay of
$\exp(\pm x^{3/2})$. For example along the positive real axis, there are two real branches and the right-most curve indicates the branch
$\exp(+x^{3/2})$ has maximal growth there (one of the directions along which the solid curve is furthest from the dashed line is along the real axis).
The other branch $\exp(-x^{3/2})$ lies inside the dashed line, and has maximal decay along the positive real axis.
In contrast along the ray $\arg(x)=\pi/3$ the dominance of the two branches
changes; this is an "oscillating" or "Stokes" direction (in the terminology of [W1976]).
Away from the Stokes directions the two branches have a well-defined (dominance) order.
The Stokes diagram arose in Stokes' study of the linear
differential equation $y''=xy$ for the Airy functions, since
formal solutions to this equation at $x=\infty$ involve the
exponential functions $\exp((2/3)x^{3/2})$,
and the Stokes diagram of
these functions looks the same
(we can ignore the constant $2/3$ here).
Of course the Stokes diagram is not intrinsically defined,
but it is representing something that can be defined intrinsically.
Let $\d$ denote the circle of real directions at $\infty$
and let $\I=\< x^{3/2} > \to \d$
denote the degree two covering map given by the germs at
$\infty$
of the functions $\pm x^{3/2}$ along various directions.
Thus $\I=\< x^{3/2}>$ denotes a circle
(basically the germ of the Riemann surface of these functions).
A point $p\in \I$ lies over some direction
$d\in \d$, and $p$ "is"
a choice of one of the two branches of the function
$x^{3/2}$
(on a germ of an open sector spanning the direction $d$).
The Stokes directions $\IS\subset \d$
are well-defined
and for any direction $d$ that is not a Stokes direction
the set $\I_d$
(the two points in the fibre of $\I$ over $d\in \d$)
has a well-defined dominance ordering, given by the dominance ordering of the two functions $\exp(\pm x^{3/2})$.
All of this is intrinsic, and the Stokes diagram is a
(non-canonical) projection of the Stokes
circle $\I$ to the plane near $\infty$,
indicating these dominance orderings,
and the directions along which the dominance changes.
This intrinsic formalism works in general:
for example
one can list all the possible exponential factors that occur at infinity
for any algebraic linear differential equation on the complex plane.
They make up a huge collection of circles $\cI$, the
exponential local system,
equipped with a covering map $\cI\to \d$.
Each component of $\cI$ is a circle of the form
$\< q >$ for some expression
$$ q = a_1 x^{k_1}+\cdots + a_m x^{k_m}$$
for rational numbers $k_i>0$ and complex numbers $a_i$.
Any such Stokes circle $\< q >$
has three numbers attached to it: