Integral on R^3: Change of Variable and Montecarlo Integration

[zetafunction]

let be an integral on R^3 (imporper integral over all space)

$$\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy \int_{-\infty}^{\infty}dz f(x,y,z)$$

the integral is convergent , my question is if i can make a change of variable to spherical coordinates and then use MONTECARLO INTEGRATION to get rid of the angles so in the end we have an (approximate) sum of one dimensional integrals o the form

$$\int_{0}^{\infty} r^{2}drg(r)$$

for some g(r)

 

[mathman]

let be an integral on R^3 (imporper integral over all space)

$$\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy \int_{-\infty}^{\infty}dz f(x,y,z)$$

the integral is convergent , my question is if i can make a change of variable to spherical coordinates and then use MONTECARLO INTEGRATION to get rid of the angles so in the end we have an (approximate) sum of one dimensional integrals o the form

$$\int_{0}^{\infty} r^{2}drg(r)$$

for some g(r)

You didn't finish the question. Are you asking if this will work? There isn't anything wrong as far as I can see. g(r) must be independent of angle.

 

[zetafunction]

thanks mathmatn , i said that if you can always make a change of variable to polar coordinates (even the integral is divergent) and then use numerical methods to integrate over the angles to get a set (or sum) of 1-dimensional integrals

$$\int_{0}^{\infty}drg(r)r^{n-1}$$

i used or try to discuss it with my physics teacher i order to obtain the renormalization for multi-loop integrals but they said it wouldn't work :( but gave no argument

 

[chiro]

let be an integral on R^3 (imporper integral over all space)

$$\int_{-\infty}^{\infty}dx \int_{-\infty}^{\infty}dy \int_{-\infty}^{\infty}dz f(x,y,z)$$

the integral is convergent , my question is if i can make a change of variable to spherical coordinates and then use MONTECARLO INTEGRATION to get rid of the angles so in the end we have an (approximate) sum of one dimensional integrals o the form

$$\int_{0}^{\infty} r^{2}drg(r)$$

for some g(r)

I don't know how a convergent integral can become divergent under any valid substitution.

 

[zetafunction]

i meant that we began with an initial DIVERGENT integral :S the question is that even for divergent integrals we can use polar coordinates

divergent integrals appear in RENORMALIZATION the idea is to reduce multiple integrals to 1-dimensional integrals by change to polar coordinates.

 

[mathman]

You can always change from Cartesian coordinates to spherical or vice versa. It won't effect convergence. If it appears that it does, there was an error.

 

[Mute]

Let me try to get this straight:

You have a divergent integral in Cartesian coordinates that is difficult to regularize. You want to know if it's possible that you can formally change to spherical coordinates and integrate out the angular variables without immediately getting an infinite result, leaving you with a radial integral that is still divergent but perhaps easier to regularize or otherwise deal with. Is this what you're hoping to do?

If so, I do not think this will work in general. I forget the details, but I remember learning in QFT that when you have a divergent integral, even a change of variables as innocent as $x \rightarrow x + a$ can be disastrous. You need to regularize the integral first, before you can make any changes of variables.

 

[zetafunction]

yes mute , is what you said

however if i introduce a regulator (in polar coordinates) so there will exist any big 's' with the regulator $$(1+q_{a}q_{a})^{-s}$$ so that for big enough 's' the multiple integral will be convergent, is the method right ??