Double integral of gaussian times mod cos function

[Idoubt]

Homework Statement

I want to evaluate the following definite integral of the form,

$$I = \int\limits_{x = -\infty}^{\infty}\int\limits_{y = -\infty}^{\infty} e^{-ax^2} e^{-by^2} | \cos(c x + d y)| dx dy$$
where a, b, c, and d are constants, as part of a larger problem I am doing,

Homework Equations

$$\cos x = \frac{1}{2} ( e^{ix} + e^{-ix})$$
$$\int\limits_{-\infty}^{\infty} e^{-a(x - b)^2} dx= \sqrt{\frac{\pi}{a}}$$

The Attempt at a Solution

If it wasn't for the abs value on the cos function it would be easy to write it in terms of exponentials and complete squares and perform the integral. As it is I don't know how to approach this, any help would be great. I have given a general form of the integral but for my purpose it's ok to assume a =b.

 

[Ray Vickson]

Homework Statement

I want to evaluate the following definite integral of the form,

$$I = \int\limits_{x = -\infty}^{\infty}\int\limits_{y = -\infty}^{\infty} e^{-ax^2} e^{-by^2} | \cos(c x + d y)| dx dy$$
where a, b, c, and d are constants, as part of a larger problem I am doing,

Homework Equations

$$\cos x = \frac{1}{2} ( e^{ix} + e^{-ix})$$
$$\int\limits_{-\infty}^{\infty} e^{-a(x - b)^2} dx= \sqrt{\frac{\pi}{a}}$$

The Attempt at a Solution

If it wasn't for the abs value on the cos function it would be easy to write it in terms of exponentials and complete squares and perform the integral. As it is I don't know how to approach this, any help would be great. I have given a general form of the integral but for my purpose it's ok to assume a =b.

If you change variables to $cx+dy = z\; \Rightarrow \; y = pz - qx$ your integral becomes
$$I = \int_{x=-\infty}^{\infty} \int_{z = -\infty}^{\infty} p\, e^{-ax^2 - b(pz-qx)^2} |\cos(z)| \, dx \, dz$$
For fixed $z$ you can do the $x$-integral first, and so end up with
$$I = A \int_{-\infty}^{\infty} e^{-r z^2} |\cos(z)| \, dz,$$
where
$$A =p \frac{\sqrt{\pi}}{\sqrt{a+bq^2}} \: \text{and} \; r = \frac{a b p^2}{a+bq^2}$$

You can go a bit farther, by writing
$$\begin{array}{l} I = A \int_{-\infty}^{\infty} e^{-r z^2} \cos(z) \, dz\\
- 2A\left[ \int_{-3\pi/2}^{-\pi/2} + \int_{\pi/2}^{3\pi/2} + \int_{-7\pi/2}^{-5\pi/2} + \int_{5\pi/2}^{7\pi/2} + \int_{-11\pi/2}^{-9\pi/2} +\int_{9\pi/2}^{11\pi/2} + \cdots \right] \,e^{-rz^2} \cos(z) \, dz,
\end{array}$$
That comes from adding to the simple "cos" integral, twice the sum of those "cos" integrals where $\cos(z) < 0$ and noting that these latter cover the intervals $(-3\pi/2,-\pi/2), (\pi/2 , 3\pi/2)$ and the intervals $(\pi/2 + 2 \pi n, 3\pi/2 +2 \pi n), (-3\pi/2 - 2 \pi n, -\pi/2 - 2 \pi n)$ for $n =1,2,\ldots$.

At that point I cannot see how to give a more useful answer, although the integrals can be done in terms or error functions of complex arguments that depend on r and n, but I don't see that as very helpful. The expression might well be amenable to numerical evaluation, since the presence of factors $e^{-4 \pi^2 r n^2}$ means that good accuracy ought to be attainable from a few small values of $n = 0, \pm1, \pm2, \ldots$ and then the individual integrals can be tackled numerically.

 

[Idoubt]

If you change variables to $cx+dy = z\; \Rightarrow \; y = pz - q$ your integral becomes
$$I = \int_{x=-\infty}^{\infty} \int_{z = -\infty}^{\infty} p\, e^{-ax^2 - b(pz-qy)^2} |\cos(z)| \, dx \, dz$$
For fixed $z$ you can do the $x$-integral first, and so end up with
$$I = A \int_{-\infty}^{\infty} e^{-r z^2} |\cos(z)| \, dz,$$
where
$$A =p \frac{\sqrt{\pi}}{\sqrt{a+bq^2}} \: \text{and} \; r = \frac{a b p^2}{a+bq^2}$$

Thank you for the reply. I have a doubt though. If you change variables such that $cx + dy = z \Rightarrow y = \frac{z}{d} - \frac{cx}{d} \equiv pz - q,$ where $q(x) = \frac{cx}{d}$ how can you do the x integral? Do you mean complete squares in the x variable and then inegrate? But then there shouldn't be any q after integration right? But it looks like q is being treated as a constant.

 

[Ray Vickson]

Thank you for the reply. I have a doubt though. If you change variables such that $cx + dy = z \Rightarrow y = \frac{z}{d} - \frac{cx}{d} \equiv pz - q,$ where $q(x) = \frac{cx}{d}$ how can you do the x integral? Do you mean complete squares in the x variable and then inegrate? But then there shouldn't be any q after integration right? But it looks like q is being treated as a constant.

I made a typo when I wrote $e^{-ax^2-b(pz-qy)^2} \,dx$; it should have been $e^{-ax^2-b(pz-qx)^2} \,dx$, because $y = pz - qx$; and, of course, $p=1/d$ and $q=c/d$ really are constants. The $x$-integral will be of the form $\int_R e^{-Ax^2-Bx-C} \, dx$ with constants $A,B,C$ (which may contain the un-integrated variable $z$). Certainly the answer will be in terms of $A,B,C$.

The formulas I wrote later are correct; I did them on the computer algebra system Maple, using correct formulas without any typos in the inputs. You should never, never just copy down such things, but should yourself carry through all the steps to see what you get.

Typos fixed in original.