In this post we discuss example solutions of the following integral determinant equation that was derived on my Determinant Integrals page:
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
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.