# why no change of variable to polar coordinates inside multi-loop integral ??

by zetafunction
Tags: coordinates, inside, integral, multiloop, polar, variable
 P: 399 given a mul,ti-loop integral $$\int d^{4}k_{1} \int d^{4}k_{2}.................\int d^{4}k_{n}f(k_{1} , k_{2},.....,k_{n})$$ which can be considered a 4n integral for integer n , my question is why can just this be evaluated by using a change of variable to 4n- polar coordinates ? one we have made a change of variable and calculated the Jacobian, and integrated over ALL the angular variables we just have to make an integral $$\int_{0}^{\infty}drg(r)r^{4n-1}$$ which is just easier to handle
 Sci Advisor P: 5,774 I don't what specific integral you have in mind, but it depends very much on the form of f as it depends on the k's. You seem to imply that it can be represented as a function g of one variable. This may be true for some particular f, but it certainly is not true in general.
 P: 399 for example $$\iint dx dy \frac{x^{3}}{1+xy}$$ its divergent if taking the limits (0,oo) making a change of variable to polar coordinates one gets $$\int du \int_{0}^{\infty}dr\frac{r^{4}cos^{3}(u)}{1+(1/2)r^{2}sin(2u)}$$ integrating over the angular variable 'u' you have now a simple one dimensional integral