In this post we discuss example solutions of the following integral determinant equation that was derived on my Determinant Integrals page:
(1) ……
Let be a bounded open set. Given an invertible matrix
(see the Determinant Integral page for a description of the set
if needed) we know that its determinant is integrable and
Consequently, if equation (1) holds we have
This leads us to consider examples of bounded open sets and scalar functions
which satisfy
(2) ……
Clearly, equation (2) is equation (1) in the case when is a
matrix, so we’ve massively simplified the problem presented by equation (1). Given an integer
, notice equation (1) on its own does not have a unique
matrix solution
defined a.e in
if there exists a function
satisfying equation (2). With a function
known to satisfy equation (2), we could construct an uncountable number of matrices
for which
a.e in
. For example, consider triangular matrices
whose main diagonals take the form
. We then have full freedom to determine the possibly non-degenerate off-diagonal entries of
, according to whether the matrix is upper or lower triangular.
We now present an example solution pair to equation (2). Denote by
the left-hand side of equation (2):
Let be defined over
with
and
such that
. I’ve allowed simply for the condition
as opposed to the natural condition
because equation (2) holds even if the limits of integration are interchanged, which allows for the case
. On the other hand, equation (2) can’t hold if
or else we have
. Nonetheless, for suitable
there holds
If we let
equation (2) reads . Below we plot this equation implicitly in Python over
for
and
.

In Figure 1 all curves of a given colour constitute the solution locus corresponding to one choice of . We describe this correspondence below:
Light Blue curves
Orange curves
Red curves
Blue curves
Magenta curves
Green curves
In essence, for fixed that generate curves in the plane of positive length, there are infinitely many choices of
that ensure equation (2) holds. In Figure 2 below we extend the plot to
which gives a picturesque pattern that would make for a neat wallpaper!

Between figures 1 and 2 we observe closed curves of decreasing diameter for and
, whereas for
we see small horizontal/vertical spikes which do not cross the red curves although they appear to be touching prior to zooming in. In Figure 2 we see a pattern repeating for any given coloured curve corresponding to a choice of
. This is expected as
is periodic with period
in both its arguments when
is an integer. It would be interesting then to see what family curves we observe when
is not an integer or irrational.