How Do You Solve This Challenging Integral Involving a Modified Bessel Function?

[loloPF]

Funky integral!

This integral is driving me nuts , anyone got a clue?
Given two real variables X and Y, one defines the function:
f(X,Y)=sqrt(a*(X+cY)^2+b*(X-cY)^2)
where a, b and c are reals and a>0, b>0.
Then the function g is defined as:
g(X,Y)=exp(-f(X,Y))
I am looking for:
1- a primitive for g
2- and/or the value of the following integrals I=integral(-infty,+infty;g(X,Y),dX) and J=integral(-infty,+infty;g(X,Y),dY).

 

[HallsofIvy]

Do you have any reason to believe it has an "elementary" primitive?

If you take a=-1, b= c= 0, you have
$$f(x,y)= e^{-x^2}$$
which certainly does NOT have a primitive that can be written in terms of elementary functions. It can of course be written as [itex]2\pi Erf(x)[/tex] where Erf(x) is the error function- but it's not "elementary", it is <b>defined</b> as <br /> $$\frac{1}{2\pi}\int e^{-x^2}dx$$[/itex]

 

[loloPF]

You are right HallsofIvy and the answer to your question is: "No I do not.", this is why my second point starts with: "and/or [...]".
I have made some (but little) progress on this and I'll let you know in a coming post where I stand now.

 

[loloPF]

HallsofIvy, you might have missed the square root in the definition of f:
$$f(X,Y)=\sqrt{a(cY+X)^2+b(cY-X)^2}$$

Definition:
$$I(X)=\int_{-\infty}^{\infty} e^{-f(X,Y)}dY$$
Changes of variable:
first $$u=cY+X$$
then $$v=u\sqrt{a+b}$$
and $$w=v-\frac{2bX}{\sqrt{a+b}}$$
leading to:
$$I(X)=\frac{1}{c\sqrt{a+b}}\int_{-\infty}^{\infty} e^{-\sqrt{w^2+4X^2\frac{ab}{a+b}}}dw$$

I am now considering a trig transformation:
$$w=2|X|\sqrt{\frac{ab}{a+b}}tan(\theta)$$
to get rid of the square root but I am then stuck again

 

[loloPF]

I just found the answer: this integral is well known as the "modified Bessel function of second kind", period.