This post is dedicated to a proof of the following result.
Christmas Determinant Integral. Let be given constants such that
. Then, the determinant integral
where
evaluates to
Proof. Let be a 2×2 matrix
that is diagonalisable, so that there exists an inveritable matrix
and a diagonal matrix
satisfying (where of course
). As such, we have
With the above decomposition, it can be shown that the exponential of reads
or
Assuming that are constants,
is given, and that
and
are integrable with respect to Lebesgue measure over
, we have
Here, I’ve let denote the integration of the matrix
entry-wise. This operation is discussed on my Determinant Integrals page.
Now, assume , and suppose
. Then,
which implies
or rather
(1)…….
With the choice of made above, it can be shown that
Setting and
where is an arbitrary constant, it follows from (1) that
Here, I have used the fact that
holds for all . This follows by using integration by parts and a trigonometric substitution to reduce the integral to the problem of evaluating
. Wolfram Alpha indicates that this latter integral is equal to
We now have that takes the form
for , with
Setting to be the real number given by
the proposed result follows by setting
,
and
, where
are arbitrary constants such that
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