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## Homework Statement

I have a 2D integral that contains a delta function:

##\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp{-((x_2-x_1)^2)+(a x_2^2+b x_1^2-c x_2+d x_1+e))}\delta(p x_1^2-q x_2^2) dx_1 dx_2##,

where ##x_1## and ##x_2## are variables, and a,b,c,d,e,p and q are some real constants.

## Homework Equations

How can one integrate this without resulting in poles. Because integral over, for example, ##x_1## leads to a pole, given the property of delta function.

## The Attempt at a Solution

If inside the delta function, the argument would be ##x_1^2+x_2^2##, and not ##x_1^2-x_2^2##, then one could go to polar coordinates, and get rid of the poles coming from the delta function integral. Sadly, this is not the case here.

Further one can get rid of pole by going into hyperbolic coordinates ##x_1=RCosh(\theta)## and ##x_2=RSinh(\theta)##. The R integral can be done exactly because the poles cancel out due to Jacobian. However, the integral over ##\theta## then becomes unbounded. So this method also has a problem. Is there a smarter way to perform integral with quadratic argument of delta function without encountering poles?