This post is dedicated to a proof of the following result.

Christmas Determinant Integral.Letbe 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 . \\\\