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 〈 x3/2 〉.
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(x3/2) 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=∞ in the Riemann sphere, and the solid line encodes the growth/decay of
exp(± x3/2). For example along the positive real axis, there are two real branches and the right-most curve indicates the branch
exp(x3/2) has maximal growth there (it is furthest from the dashed line). The other branch exp(-x3/2) lies inside the dashed line, and has maximal decay along the positive real axis.
In contrast along the ray arg(x)=π/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=∞ involve the
exponential functions exp((2/3)x3/2), 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 ∂ denote the circle of real directions at ∞
and let I=〈x3/2〉→∂
denote the degree two covering map given by the germs at ∞
of the functions ± x3/2 along various directions.
Thus I=〈x3/2〉 denotes a circle (basically the germ of the Riemann surface of these functions).
A point p ∈I lies over some direction d∈∂, and p "is"
a choice of one of the two branches of the function x3/2
(on a germ of an open sector spanning the direction d).
The Stokes directions 𝕊⊂∂ are well-defined
and for any direction d that is not a Stokes direction the set Id
(the two points in the fibre of I over d∈∂)
has a well-defined dominance ordering.
All of this is intrinsic, and the Stokes diagram is a (non-canonical) projection of the Stokes
circle I to the plane near ∞, indicating these
dominance orderings, and the directions along which they change.
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 ℐ, the
exponential local system,
equipped with a covering map ℐ→∂.
Each component of ℐ is a circle of the form 〈q〉 for
some expression